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Bonding and Debye temperatures in alkali-earth-metal halides
~
Pergamon
0022-3697(94)E0042-E
Z Phys. Chem. Solids Vol, 55, No. 8. pp. 707-710, 1994 Copyright ~ 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0022-3697/94 $7.00+0.00
B O N D I N G A N D DEBYE TEMPERATURES IN ALKALI-EARTH-METAL HALIDES SHIAN PENG and GO'RAN GRIMVALL Department of Physics, Royal Institute of Technology, S-100 44 Stockholm, Sweden (Receh~ed 20 September 1993; accepted in revised form 24 March 1994)
Abstract--Weconsider 20 compounds of the composition AB 2, where A = Be, Mg, Ca, Sr or Ba, and
B = F, CI, Br or I. From experimental data on their entropy we derive the logarithmic average co (0) of the phonon frequencies in each solid. In this particular average, the atomic masses separate from the interatomic forces. We define a quantity E s, that has the dimension of energy and is directly related to the force--constant part of co 2(0). E s allows us to describe ~o(0), or the equivalent Debye temperature Os, to better than +4% (r.m.s. deviation). Our semi-empirical method is useful to accurately estimate unknown 0s in a class of chemically similar compounds, where 0s of some compounds are known. The connection to the Lindemann melting rule is briefly discussed. Keywords: A. inorganic compounds, D. lattice dynamics, D. thermodynamic properties.
I. INTRODUCTION This paper considers regularities in the bonding of alkali-earth-metal halides. In particular, we shall establish striking semi-empirical results for the logarithmic average of the p h o n o n frequencies in these solids. The compounds considered are of the AB2 stoichiometry, where A is one of the elements Be, Mg, Ca, Sr and Ba, and B is one of the elements F, C1, Br and I. Five of these have the CaF2 (fluorite) lattice structure. The lattice dynamics of the C a F 2 structure is particularly interesting since many compounds with this lattice exhibit a diffuse high-temperature transition to a "fast ion-transport" phase. Here we shall focus on the vibrational properties well below such a transition. A key idea in our analysis is a theorem [1] saying that the logarithmic average of p h o n o n frequencies can be written as the product of two independent factors, referring to the atomic masses and the interatomic forces, respectively. Further, the vibrational entropy at high temperatures depends on only one phonon-related parameter, viz., the logarithmically averaged p h o n o n frequency. These facts allow us to extract from the measured vibrational entropy (i.e. indirectly from the heat capacity) and from crystallographic data a quantity Es, with the dimension of energy. Es only depends on the bonding in the solid, i.e., on the electronic structure. The atomic masses, which affect the temperature dependence of the vibrational heat capacity and hence also the entropy, appear in the analysis as a trivial factor than can be separated out. Intuitively, one may expect that E s varies in a regular way within a group of chemically similar compounds having the
same crystal structure. We have previously observed such regularities, e.g., in alkali halides having the NaCl-type structure [1], in transition-metal carbides and nitrides [2] and also in transition-metal diborides having the A1B2-type structure [3]. It is the purpose of this paper to exploit further the properties of Es, now with the emphasis on how Es may vary in a group of chemically similar ionic compounds that do not all have the same crystal structure, and where also the ratio between the anion and cation radii varies appreciably.
2. FORMALISM Many high-temperature thermodynamic properties of harmonic lattice vibrations only depend on a single frequency moment co (n) of the phonon density of states F(co). We define co (n) by:
[co (n)] n =
co "F(co) do)
r(co) do). (1)
The logarithmic average, 09 (0), is defined with co n replaced by lnco in eqn (1). The high-temperature expansion of the vibrational entropy is: S ( T ) = 3 N k B f l + In[k BT/hco (0)]
,
}
+ ~-~ [hco (2)/kB TI + . . . .
(2)
N is the total n u m b e r of atoms in the sample. It is convenient to introduce an entropy Debye temperature Os(T) such that Os inserted in a Debye-model 707
708
SHIAN PENG and G. GRIMVALL
expression for the entropy reproduces the entropy derived from the full F(co). Of course, Os(T ) then varies with T, unless F(co) happens to be of the Debye form. In a real system, Os(T) varies also at high temperatures, because of anharmonic effects (see, e.g., Refs [4, 5]). The high-temperature entropy, expressed in Os, takes the form:
S(T) =
quantity Es, with the dimension of energy. E s is defined as: Es = ks~'~2/3.
(6)
~a is the average volume per atom in the solid, i.e. the volume of a crystallographic unit cell divided by the number of atoms in that cell.
3NkB {4/3 + ln(T/Os)
+(1/40)(0s/T) 2+...}.
Comparison of (2) and (3) gives, for harmonic vibrations: hco (0) = kBOs(T ~ oo)exp(- 1/3).
(4)
In a real solid, 0 s often decreases appreciably at high T due to anharmonic effects. However, there is usually a temperature region from T ~ 0 . 3 0 s to T ~ 1.50s where Os(T) varies little with T. Then anharmonic effects are not yet important, but the temperature is high enough that 0 s gives a good representation of the logarithmic average co (0). Hence, thermodynamic data on the entropy (after correction for non-vibrational contributions, if any), can be used to get co (0). This forms the first step in the present analysis. Let the p h o n o n frequencies co (q, ~) be labelled by a wave vector q and a mode index ~,, with ~, denoting longitudinal or transverse branches and acoustic or optic branches. ~o 2(q, ~,) are eigenvalues of a dynamical matrix that is a product M - m 17M-1/2 of matrices (M) containing the masses and a matrix (F) containing only interatomic forces. The logarithmic average co (0) results from the sum of all In[co (q, ~,)], i.e., from (1/2) In [HCO2(q, ~)] where 1-ko 2(q, L) is the product of all eigenvalues. This product is also equal to the determinant of the dynamical matrix. Using the fact that the atomic masses appear only through diagonal matrices M - 1/:, it follows that co (0) is the product of one part containing the masses and one part containing the interatomic forces [1]. Like in previous work of our group, we define a quantity ks, with the dimension of a force constant, by: kB0s = h (ks/ Mog) I/2.
3. RESULTS AND DISCUSSION
(3)
(5)
Mc~ is the logarithmic average of the atomic masses, e.g., M e ~ = ( M A M ~ ) 1/3 for a stoichiometric compound AB2. It would be reasonable to assume that k s varies regularly in a group of chemically related compounds, having the same crystal structure. However, an even stronger regularity is found if we consider a
Following the scheme outlined above, we rely on available thermodynamic data for the entropy [6], and calculate 0 s at temperatures T ~ Os. Crystallographic [7] or density [8] data give f~a. Then E s is derived from eqns (5) and (6). The results for Os, f~,, ks and Es are given in Table 1. The quantity E s is remarkably constant, and varies only by + 9% (r.m.s. deviation) for the 20 compounds considered (Fig. 1). Since our f~a is just a plain average of volumes per atom, one might expect a superimposed influence on E s from the relative size of the anion and cation in each compound. Figure 1 shows Es plotted versus r+/r , i.e., the cation (r÷) to anion (r_) radius ratio [8]. It is obvious from this figure that there is no systematic variation in Es with r+/r_. We next look for any systematic variation of Es with the crystal structure. It is then instructive to compare with the plot in Fig. 2, which follows ideas by Villars et al. [9]. It shows a so called structure map where the coordinates are AR (the absolute value of the difference in Zunger pseudopotential radii sums Table 1. The entropy Debye temperature Os, the average volume per atom t~, the effective force constant ks, and the quantity E s Compound
Os (K)
fl (A 3/atom) ks (N/m) E s (Ry)
BeF2 MgF 2 CaF 2 SrF 2 BaF2
634 569 470 390 320
12.2 10.9 13.6 16.3 19.9
170 190 153 137 107
4.13 4.28 4.00 4.03 3.60
BeC12 MgC12 CaCI 2 SrCI2 BaCI2
393 350 286 249 221
23.2 21.9 28.1 28.3 28.6
98 109 86 85 77
3.68 3.91 3.65 3.60 3.33
BeBr2 MgBr2 CaBr2 SrB~ BaBr2
303 244 204 170 156
27.0 26.2 32.6 32.5 33.5
101 91 75 67 70
4.16 3.69 3.52 3.16 3.32
229 202 165 138 128
33.6 34.0 40.3 41.2 42.0
78 85 68 61 61
3.75 4.09 3.64 3.33 3.37
BeI2 MgI2 CaI2 SrI2 Bal2
Alkali-earth-metal halides 1.2
1.0
I
I
I
I
2*
110 I ~
017 71) 8 120 0 1 3 0
16006 0.8
I 3 ~ 4=g
19 014015 010
~a 0.6
0.4 0.2
~
MF2compounds
•
MGI2
0 MBr2 MI2 (M = Be, Mg, Sr, Ba, Ca)
0.0
0.0
I 0.2
I 0.4
0.6
I o.a
I 1.0
1.2
r+/t
Fig. 1. The quantity Es, defined by eqn (6) vs the ionic radius ratio r + / r for the 20 alkali-earth-metal halides BeF2 (1), MgF 2 (2), CaF 2 (3), SrF~ (4), BaF 2 (5), BeC12(6), MgC12 (7), CaC12 (8), SrCI2 (9), BaCI2 (10), BeBr2 (11), MgBr2 (12), CaBr2 (13), SrBr2 (14), BaBr 2 (15), BeI2 (16), MgI 2 (17), CaI 2 (18), SrI 2 (19), BaI 2 (20).
for atoms A and B ) and AX (electronegativity difference), and compounds having the same structure are enclosed by rhombs. Figures 1 and 2 both display a relation between one quantity related to the relative ionic size (r÷/r_ or A R ) and one related to the bonding (Es or AX). The crystal structure of course is correlated to the relative ionic size. However, contrary to AX, there is not a pronounced systematic variation in E s with the crystal structure for the compounds considered here. We see from Table 1 that the entropy Debye temperatures 0 s vary by a factor of five, in the group of 20 compounds. On the other hand, Es is constant
I 3
--r.2
-I
l
a
l
~
~
2
Fig. 2. A Villars-type plot of the 20 alkali-earth-metal halides, where coordinate AR denotes the absolute value in Zunger pseudopotential radii sums and A,rt"is the electronegativity difference. Labels as in Fig. 1. The rfiombic boxes enclose compounds having the same crystal structure. They are labelled by the conventionally chosen compound that has the same crystal structure.
709
to within + 9 % (r.m.s. deviation). The effective mass Me~ is known. If we also assume that the volume per atom, t2a, is known, a variation of + 9 % in Es is equivalent with a variation in Os by only + 4 % . This fact suggests an accurate method to estimate an u n k n o w n Debye temperature 0 s as follows. Consider a class of chemically similar compounds, where experimental information on the entropy and the lattice parameter allow us to get Es for several or most of the compounds. If these E s values are almost constant, or show a regular behavior, we may estimate E s for other compounds in the same class, where entropy information is lacking. The atomic volumes f~a usually show a very regular variation. Then we can obtain Debye temperatures 0 s quite accurately, from eqns (5) and (6). Our Os gives weight to both longitudinal and transverse waves, of all wavelengths. Therefore our method gives a more reliable representation of the p h o n o n spectrum than, e.g. empirical methods using the measured bulk modulus. The Lindemann melting rule is sometimes used to estimate Debye temperatures 0 from known melting temperatures Tin, or vice versa. The rule assumes that the average thermal displacement of an atom around its equilibrium position is a certain fixed fraction of the nearest-neighbour distance when T = Tm. For a monatomic solid, it may be written [10]: T m = (Mk~O 2~2/3/9h 2)x2.
(7)
Here M is the atomic mass and x is assumed to be a constant for a given crystal structure. For elements, all p h o n o n frequencies scale as M -1/2 and the atomic mass cancels in MO 2 in eqn (7). We recall that in a Debye model the average squared vibrational displacement of the atoms at high temperatures is proportional to T/O 2 (see, e.g. Ref. [11]). Hence, for elements, Es being constant is tantamount to a constant x, i.e., the Lindemann melting rule. Berebbi et al. [10] recently applied the Lindemann rule to alkali halides in the NaCl-type crystal structure, with an accuracy comparable to that obtained previously for the same solids by our group using essentially the method of this paper [1], and also comparable to the accuracy obtained in the present paper. Some remarks should therefore be made. We note that in classical statistical mechanics the vibrational energy has a potential part that increases linearly with T and is independent of the atomic masses in the system. However, the sum of all average squared atomic displacements of different atoms need not be equal, and depends on the interatomic forces and the masses. On the other hand, in a crystal with the NaCl-type structure, and for the special case of only point-ion Coulomb forces and nearest-neigh-
710
SHIAN PENG and G. GRIMVALL
bour forces, the average displacements of the anions and the cations are equal [12]. Further, we note that in polyatomic solids the atomic masses enter 0 in different ways, depending on which property and which temperature 0 refers to. There is no a priori choice of the mass M in eqn (7). Berrebbi et al. [10] used the harmonic mean, 2 / M = I/M~ + 1/M2. The experimental Debye temperatures used by them in eqn (7) were obtained in different ways, and therefore should have different mass dependences. Hence there may still be an approximate cancellation of atomic masses in eqn (7). It follows from what has now been said that no precise conclusion can be drawn concerning the connection between their use of eqn (7) and our use of E s to estimate Debye temperatures. It is intriguing that variations in ionic radius ratios and in the crystal structure apparently have no significant influence on E s. We have not been able to find a convincing quantitative theoretical argument for the constancy of E s in the group of 20 compounds considered here. However, the following argument may point at an essential aspect. Consider a class of monatomic solids where the interatomic interaction is represented by potential V ( r ) = V0~P(r/r*). Let the strength V0 and the function ~p be the same for all these compounds, while r* is a length parameter that is allowed to vary. The elements in the force-constant part F of the dynamical matrix are all related to second derivatives of V(r), evaluated at the various atomic positions r = Rt in the lattice. These derivatives scale with r* as (r*) -2. Hence ~o 2(0) and k s also scale as (r*) -2. The atomic volume ~a scales as ( r * ) 3. Therefore Es = ks~2a/3 is proportional to V0 but independent of r*. We conclude by stressing two essential features of our approach. Firstly the quantity Es only depends on the bonding in the solid, with no influence from atomic masses although Es is derived from thermodynamic information on the lattice vibrations. Secondly, the essence of our method lies in the regular
variation of E s within a class of chemically related compounds. Although we have found that E s is approximately constant for alkali halides [1] and for the compounds considered in this paper, we have also found a pronounced but regular variation in E s for NaCl-type structure transition metal carbides and nitrides [2] and for AlB2-structure type transition metal diborides [3]. The present paper shows that in spite of varying crystal structure and strongly varying ionic radius ratios, Es may retain its regularity when the type of bonding (here ionic) is preserved.
Acknowledgements--This work was supported by the
Swedish National Board for Industrial and Technical Development and by the Swedish Natural Science Research Council.
REFERENCES
1. Grimvall G. and Ros~n J., Int. J. Thermophys. 4, 139 (1983). 2. Fernfindez Guillermet A. and Grimvall G., J. Phys. Chem. Solids 53, 105 (1992). 3. Fernfindez Guillermet A. and Grimvall G., J. LessCommon Metals 169, 257 (1991). 4. Grimvall G. and Fernfindez Guillermet A., in Thermodynamic Data (Edited by S. K. Saxena), p. 272. Springer Verlag, New York (1992). 5. Ros6n J. and Grimvall G., Phys. Rev. B 27, 7199 (1983). 6. JANAF Thermochemical Tables, 3rd ed., edited by M. W. Chase, Jr., C. A. Davies, J. R. Downey Jr., D. J., Frurip, R. A. McDonald and A. N. Syverud. [J. Phys. Chem. R e f Data 14, suppl. 1 (1985)]. 7. Wyckoff R. W. G., Crystal Structures, Vol, 1 (2nd edn). Wiley, New York (1963). 8. CRC Handbook of Chemistry and Physics, 73rd edn, edited by D. R. Lide. CRC Press, Boca Raton (1993). 9. Hafner J., Hulliger F., Jensen W. B., Majewski J. A., Mathis K., Villars P. and Vogl P., The Structures of Binary Compounds. North Holland, Amsterdam (1989). 10. Berrebi D., Grigoryan L., Rabinovich S., Shasha E., and Voronel A., J. Phys. Condens. Matter 4, 10139 (1992). 11. Grimvall G., Thermophysical properties of materials. North Holland, Amsterdam (1986). 12. Huiszoon C., and Groenewegen P. P. M., Acta Crystallogr. A28, 170 (1972).
Pergamon
0022-3697(94)E0042-E
Z Phys. Chem. Solids Vol, 55, No. 8. pp. 707-710, 1994 Copyright ~ 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0022-3697/94 $7.00+0.00
B O N D I N G A N D DEBYE TEMPERATURES IN ALKALI-EARTH-METAL HALIDES SHIAN PENG and GO'RAN GRIMVALL Department of Physics, Royal Institute of Technology, S-100 44 Stockholm, Sweden (Receh~ed 20 September 1993; accepted in revised form 24 March 1994)
Abstract--Weconsider 20 compounds of the composition AB 2, where A = Be, Mg, Ca, Sr or Ba, and
B = F, CI, Br or I. From experimental data on their entropy we derive the logarithmic average co (0) of the phonon frequencies in each solid. In this particular average, the atomic masses separate from the interatomic forces. We define a quantity E s, that has the dimension of energy and is directly related to the force--constant part of co 2(0). E s allows us to describe ~o(0), or the equivalent Debye temperature Os, to better than +4% (r.m.s. deviation). Our semi-empirical method is useful to accurately estimate unknown 0s in a class of chemically similar compounds, where 0s of some compounds are known. The connection to the Lindemann melting rule is briefly discussed. Keywords: A. inorganic compounds, D. lattice dynamics, D. thermodynamic properties.
I. INTRODUCTION This paper considers regularities in the bonding of alkali-earth-metal halides. In particular, we shall establish striking semi-empirical results for the logarithmic average of the p h o n o n frequencies in these solids. The compounds considered are of the AB2 stoichiometry, where A is one of the elements Be, Mg, Ca, Sr and Ba, and B is one of the elements F, C1, Br and I. Five of these have the CaF2 (fluorite) lattice structure. The lattice dynamics of the C a F 2 structure is particularly interesting since many compounds with this lattice exhibit a diffuse high-temperature transition to a "fast ion-transport" phase. Here we shall focus on the vibrational properties well below such a transition. A key idea in our analysis is a theorem [1] saying that the logarithmic average of p h o n o n frequencies can be written as the product of two independent factors, referring to the atomic masses and the interatomic forces, respectively. Further, the vibrational entropy at high temperatures depends on only one phonon-related parameter, viz., the logarithmically averaged p h o n o n frequency. These facts allow us to extract from the measured vibrational entropy (i.e. indirectly from the heat capacity) and from crystallographic data a quantity Es, with the dimension of energy. Es only depends on the bonding in the solid, i.e., on the electronic structure. The atomic masses, which affect the temperature dependence of the vibrational heat capacity and hence also the entropy, appear in the analysis as a trivial factor than can be separated out. Intuitively, one may expect that E s varies in a regular way within a group of chemically similar compounds having the
same crystal structure. We have previously observed such regularities, e.g., in alkali halides having the NaCl-type structure [1], in transition-metal carbides and nitrides [2] and also in transition-metal diborides having the A1B2-type structure [3]. It is the purpose of this paper to exploit further the properties of Es, now with the emphasis on how Es may vary in a group of chemically similar ionic compounds that do not all have the same crystal structure, and where also the ratio between the anion and cation radii varies appreciably.
2. FORMALISM Many high-temperature thermodynamic properties of harmonic lattice vibrations only depend on a single frequency moment co (n) of the phonon density of states F(co). We define co (n) by:
[co (n)] n =
co "F(co) do)
r(co) do). (1)
The logarithmic average, 09 (0), is defined with co n replaced by lnco in eqn (1). The high-temperature expansion of the vibrational entropy is: S ( T ) = 3 N k B f l + In[k BT/hco (0)]
,
}
+ ~-~ [hco (2)/kB TI + . . . .
(2)
N is the total n u m b e r of atoms in the sample. It is convenient to introduce an entropy Debye temperature Os(T) such that Os inserted in a Debye-model 707
708
SHIAN PENG and G. GRIMVALL
expression for the entropy reproduces the entropy derived from the full F(co). Of course, Os(T ) then varies with T, unless F(co) happens to be of the Debye form. In a real system, Os(T) varies also at high temperatures, because of anharmonic effects (see, e.g., Refs [4, 5]). The high-temperature entropy, expressed in Os, takes the form:
S(T) =
quantity Es, with the dimension of energy. E s is defined as: Es = ks~'~2/3.
(6)
~a is the average volume per atom in the solid, i.e. the volume of a crystallographic unit cell divided by the number of atoms in that cell.
3NkB {4/3 + ln(T/Os)
+(1/40)(0s/T) 2+...}.
Comparison of (2) and (3) gives, for harmonic vibrations: hco (0) = kBOs(T ~ oo)exp(- 1/3).
(4)
In a real solid, 0 s often decreases appreciably at high T due to anharmonic effects. However, there is usually a temperature region from T ~ 0 . 3 0 s to T ~ 1.50s where Os(T) varies little with T. Then anharmonic effects are not yet important, but the temperature is high enough that 0 s gives a good representation of the logarithmic average co (0). Hence, thermodynamic data on the entropy (after correction for non-vibrational contributions, if any), can be used to get co (0). This forms the first step in the present analysis. Let the p h o n o n frequencies co (q, ~) be labelled by a wave vector q and a mode index ~,, with ~, denoting longitudinal or transverse branches and acoustic or optic branches. ~o 2(q, ~,) are eigenvalues of a dynamical matrix that is a product M - m 17M-1/2 of matrices (M) containing the masses and a matrix (F) containing only interatomic forces. The logarithmic average co (0) results from the sum of all In[co (q, ~,)], i.e., from (1/2) In [HCO2(q, ~)] where 1-ko 2(q, L) is the product of all eigenvalues. This product is also equal to the determinant of the dynamical matrix. Using the fact that the atomic masses appear only through diagonal matrices M - 1/:, it follows that co (0) is the product of one part containing the masses and one part containing the interatomic forces [1]. Like in previous work of our group, we define a quantity ks, with the dimension of a force constant, by: kB0s = h (ks/ Mog) I/2.
3. RESULTS AND DISCUSSION
(3)
(5)
Mc~ is the logarithmic average of the atomic masses, e.g., M e ~ = ( M A M ~ ) 1/3 for a stoichiometric compound AB2. It would be reasonable to assume that k s varies regularly in a group of chemically related compounds, having the same crystal structure. However, an even stronger regularity is found if we consider a
Following the scheme outlined above, we rely on available thermodynamic data for the entropy [6], and calculate 0 s at temperatures T ~ Os. Crystallographic [7] or density [8] data give f~a. Then E s is derived from eqns (5) and (6). The results for Os, f~,, ks and Es are given in Table 1. The quantity E s is remarkably constant, and varies only by + 9% (r.m.s. deviation) for the 20 compounds considered (Fig. 1). Since our f~a is just a plain average of volumes per atom, one might expect a superimposed influence on E s from the relative size of the anion and cation in each compound. Figure 1 shows Es plotted versus r+/r , i.e., the cation (r÷) to anion (r_) radius ratio [8]. It is obvious from this figure that there is no systematic variation in Es with r+/r_. We next look for any systematic variation of Es with the crystal structure. It is then instructive to compare with the plot in Fig. 2, which follows ideas by Villars et al. [9]. It shows a so called structure map where the coordinates are AR (the absolute value of the difference in Zunger pseudopotential radii sums Table 1. The entropy Debye temperature Os, the average volume per atom t~, the effective force constant ks, and the quantity E s Compound
Os (K)
fl (A 3/atom) ks (N/m) E s (Ry)
BeF2 MgF 2 CaF 2 SrF 2 BaF2
634 569 470 390 320
12.2 10.9 13.6 16.3 19.9
170 190 153 137 107
4.13 4.28 4.00 4.03 3.60
BeC12 MgC12 CaCI 2 SrCI2 BaCI2
393 350 286 249 221
23.2 21.9 28.1 28.3 28.6
98 109 86 85 77
3.68 3.91 3.65 3.60 3.33
BeBr2 MgBr2 CaBr2 SrB~ BaBr2
303 244 204 170 156
27.0 26.2 32.6 32.5 33.5
101 91 75 67 70
4.16 3.69 3.52 3.16 3.32
229 202 165 138 128
33.6 34.0 40.3 41.2 42.0
78 85 68 61 61
3.75 4.09 3.64 3.33 3.37
BeI2 MgI2 CaI2 SrI2 Bal2
Alkali-earth-metal halides 1.2
1.0
I
I
I
I
2*
110 I ~
017 71) 8 120 0 1 3 0
16006 0.8
I 3 ~ 4=g
19 014015 010
~a 0.6
0.4 0.2
~
MF2compounds
•
MGI2
0 MBr2 MI2 (M = Be, Mg, Sr, Ba, Ca)
0.0
0.0
I 0.2
I 0.4
0.6
I o.a
I 1.0
1.2
r+/t
Fig. 1. The quantity Es, defined by eqn (6) vs the ionic radius ratio r + / r for the 20 alkali-earth-metal halides BeF2 (1), MgF 2 (2), CaF 2 (3), SrF~ (4), BaF 2 (5), BeC12(6), MgC12 (7), CaC12 (8), SrCI2 (9), BaCI2 (10), BeBr2 (11), MgBr2 (12), CaBr2 (13), SrBr2 (14), BaBr 2 (15), BeI2 (16), MgI 2 (17), CaI 2 (18), SrI 2 (19), BaI 2 (20).
for atoms A and B ) and AX (electronegativity difference), and compounds having the same structure are enclosed by rhombs. Figures 1 and 2 both display a relation between one quantity related to the relative ionic size (r÷/r_ or A R ) and one related to the bonding (Es or AX). The crystal structure of course is correlated to the relative ionic size. However, contrary to AX, there is not a pronounced systematic variation in E s with the crystal structure for the compounds considered here. We see from Table 1 that the entropy Debye temperatures 0 s vary by a factor of five, in the group of 20 compounds. On the other hand, Es is constant
I 3
--r.2
-I
l
a
l
~
~
2
Fig. 2. A Villars-type plot of the 20 alkali-earth-metal halides, where coordinate AR denotes the absolute value in Zunger pseudopotential radii sums and A,rt"is the electronegativity difference. Labels as in Fig. 1. The rfiombic boxes enclose compounds having the same crystal structure. They are labelled by the conventionally chosen compound that has the same crystal structure.
709
to within + 9 % (r.m.s. deviation). The effective mass Me~ is known. If we also assume that the volume per atom, t2a, is known, a variation of + 9 % in Es is equivalent with a variation in Os by only + 4 % . This fact suggests an accurate method to estimate an u n k n o w n Debye temperature 0 s as follows. Consider a class of chemically similar compounds, where experimental information on the entropy and the lattice parameter allow us to get Es for several or most of the compounds. If these E s values are almost constant, or show a regular behavior, we may estimate E s for other compounds in the same class, where entropy information is lacking. The atomic volumes f~a usually show a very regular variation. Then we can obtain Debye temperatures 0 s quite accurately, from eqns (5) and (6). Our Os gives weight to both longitudinal and transverse waves, of all wavelengths. Therefore our method gives a more reliable representation of the p h o n o n spectrum than, e.g. empirical methods using the measured bulk modulus. The Lindemann melting rule is sometimes used to estimate Debye temperatures 0 from known melting temperatures Tin, or vice versa. The rule assumes that the average thermal displacement of an atom around its equilibrium position is a certain fixed fraction of the nearest-neighbour distance when T = Tm. For a monatomic solid, it may be written [10]: T m = (Mk~O 2~2/3/9h 2)x2.
(7)
Here M is the atomic mass and x is assumed to be a constant for a given crystal structure. For elements, all p h o n o n frequencies scale as M -1/2 and the atomic mass cancels in MO 2 in eqn (7). We recall that in a Debye model the average squared vibrational displacement of the atoms at high temperatures is proportional to T/O 2 (see, e.g. Ref. [11]). Hence, for elements, Es being constant is tantamount to a constant x, i.e., the Lindemann melting rule. Berebbi et al. [10] recently applied the Lindemann rule to alkali halides in the NaCl-type crystal structure, with an accuracy comparable to that obtained previously for the same solids by our group using essentially the method of this paper [1], and also comparable to the accuracy obtained in the present paper. Some remarks should therefore be made. We note that in classical statistical mechanics the vibrational energy has a potential part that increases linearly with T and is independent of the atomic masses in the system. However, the sum of all average squared atomic displacements of different atoms need not be equal, and depends on the interatomic forces and the masses. On the other hand, in a crystal with the NaCl-type structure, and for the special case of only point-ion Coulomb forces and nearest-neigh-
710
SHIAN PENG and G. GRIMVALL
bour forces, the average displacements of the anions and the cations are equal [12]. Further, we note that in polyatomic solids the atomic masses enter 0 in different ways, depending on which property and which temperature 0 refers to. There is no a priori choice of the mass M in eqn (7). Berrebbi et al. [10] used the harmonic mean, 2 / M = I/M~ + 1/M2. The experimental Debye temperatures used by them in eqn (7) were obtained in different ways, and therefore should have different mass dependences. Hence there may still be an approximate cancellation of atomic masses in eqn (7). It follows from what has now been said that no precise conclusion can be drawn concerning the connection between their use of eqn (7) and our use of E s to estimate Debye temperatures. It is intriguing that variations in ionic radius ratios and in the crystal structure apparently have no significant influence on E s. We have not been able to find a convincing quantitative theoretical argument for the constancy of E s in the group of 20 compounds considered here. However, the following argument may point at an essential aspect. Consider a class of monatomic solids where the interatomic interaction is represented by potential V ( r ) = V0~P(r/r*). Let the strength V0 and the function ~p be the same for all these compounds, while r* is a length parameter that is allowed to vary. The elements in the force-constant part F of the dynamical matrix are all related to second derivatives of V(r), evaluated at the various atomic positions r = Rt in the lattice. These derivatives scale with r* as (r*) -2. Hence ~o 2(0) and k s also scale as (r*) -2. The atomic volume ~a scales as ( r * ) 3. Therefore Es = ks~2a/3 is proportional to V0 but independent of r*. We conclude by stressing two essential features of our approach. Firstly the quantity Es only depends on the bonding in the solid, with no influence from atomic masses although Es is derived from thermodynamic information on the lattice vibrations. Secondly, the essence of our method lies in the regular
variation of E s within a class of chemically related compounds. Although we have found that E s is approximately constant for alkali halides [1] and for the compounds considered in this paper, we have also found a pronounced but regular variation in E s for NaCl-type structure transition metal carbides and nitrides [2] and for AlB2-structure type transition metal diborides [3]. The present paper shows that in spite of varying crystal structure and strongly varying ionic radius ratios, Es may retain its regularity when the type of bonding (here ionic) is preserved.
Acknowledgements--This work was supported by the
Swedish National Board for Industrial and Technical Development and by the Swedish Natural Science Research Council.
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