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Effect of MgMg pairs on the low-temperature lattice specific heat of AlMg alloys
5 August 1994
CHEMICAL PHYSICS
Chemical Physics Letters 225 (1994) 542-546
Effect of Mg-Mg pairs on the low-temperature lattice specific heat of Al-Mg alloys Francis Marinelli, Michel Roche Centre de Thermodynamiqueet deMicrocalorimktrie,CNRS, 26 Rue du 14lPme RIA, 13003 Marseille,France
Received 19 January 1994; in tinal form 8 June 1994
We discuss the order cz correction of the lattice specific heat of All-N& substitutional alloys for different concentrationsc and temperatures. To do so we introduce Mg-Mg pairs in the calculation of the phonon density of states, and estimate their contribution to different relative configurations.
1. Introduction
In two recent papers [ 1,2] we have reported the change in the low-temperature lattice specific heat ACVof Al-based FCC substitutional alloys with Li [ 1 ] and Mg or Be [ 2 ] as solute. The calculations of AC” were carried out for an isolated atom of solute and then simply multiplied by the number of solute atoms per mole; the change in atomic volume was also included for Al-Mg where it is important. This standard procedure has been applied before by many authors and is correct to first order in the atomic concentration c of the solute for a random alloy. In order to introduce the interaction of solute atoms, which results in order c” corrections (n> 1) for n-atoms impurity clusters, several methods based on averaging the Dyson equation for the alloy’s Green functions have been proposed a long time ago. A review of these methods may be found in Ref. [ 3 1, and further theoretical efforts in Refs. [ 4,5 1, for example. The simplest of the above methods is that of Elliot and Taylor [ 61 and is exact in c only, but contains higher-order corrections in an approximate way. A theory exact in c2, plus higher-order apElsevier Science B.V. SSDZOOO9-2614(94)00660-I
proximate corrections, was proposed by Aiyer, Elliott, Krumhansl and Leath [ 7 1. Unfortunately, although formally elegant, it seems that none of these methods have so far been found to be tractable for computation nor been applied to a realistic case including force constant changes and direct interaction of the impurities. Here we adopt a simple approach that should be correct in c2, with no attempt at c” terms (n > 2)) by systematically including pair effects in the Al-based Mg alloy. We chose Mg as solute rather than Be or Li for practical reasons, namely, because it was found [ 2 ] that the local contraction (expansion) of the lattice in the vicinity of an Mg atom could be ignored, which results in less computational work.
2. Interaction of two Mg atoms The method used here is well known [ 8 ] and based on Green function theory whose fundamentals will not be repeated here. When substituting some host
F. Marineiii, M. Roche / Chemical Physics Letters 225 (1994) 542-546
lattice atoms by impurities, we write for the total change in the phonon density of states M(02)
= c M?J(@2) 2
(1)
Y
where Yis an irreducible representation (IR) of the point group of the system. We have AN,(o2)=
g$,
tancs,=-ImU
ReQ,(z)
(2) (3)
’
8, is the phase shift, I, the degeneracy of the IR v and D(z)=detlZ+GO(z)P(oZ) D(z) = n Q(z)
Y
I,
(4)
9
where P( 0’) is the perturbation matrix, z= w2 + ie, and Go(z) is the host lattice Green function matrix. In the above equations o2 goes from 0 to c.&, w, being the maximum frequency of the host lattice. In the case of two light impurities, we may find one or two locahsed modes for w> o, and those modes oI introduce additional 6 functions 6(w: -w2) in Ah7(w2). The technical details and the practical determination of Go(z) and the force constants of P(o’) are
543
to be found in Refs. [ 1,2 1. Here P(02) includes two Mg atoms. Changes in force constants from Al-Al to Al-Mg or Mg-Mg are taken into account for first and second neighbours around each Mg atom, using the interatomic pair potential of Dagens et al. [ 9 ] . The respective positions of the Mg atoms are from first to fourth neighbours (positions 1,2,3 and 4 below). Because of the large variation in the atomic volume D of the alloy with c according to Q= ti( 1 + 0.4~) [ 10 ] at each concentration c we calculate kF which in turn determines the force constants used for Go(z) and P(w2) at S, as was done in Ref. [ 21, with the right number of electrons. The total phase shifts S,6 ls2for a pair ( 1,2 ) : WC
F sy )
(5)
where v is an IR of D2,, or Dab, as well as 26’=2 c s; Y
(6)
(Oh symmetry) for two independent Mg atoms are reported in Fig. 1, with the force constants calculated at kF= 0.9098 au and a = 4.1197 A for the fee lattice parameter, i.e. for c= 0.15. The phase shifts differ perceptibly only in the last tenth of the frequency spectrum (0.9
-7.50
Fig. 1. Total phaseshifts8. Curves( 1) to (4) arc for pairswhenthe respective positions of the two Mg atoms vary from fti neighbours. Curve (5) is for two independent Mg atoms.
to fourth
F. Marinelli, hf. Roche / Chemical Physics Letters 225 (1994) 542-546
544
crease significantly from position 1 to position 2, but then slowly and quasi-linearly from position 2 to position 4. Since we have not considered pairs of Mg atoms beyond position 4, an extrapolative scheme will be considered in the next calculations of ACV.
3. Calculation of AC,/ ACV for the alloy is linear in AN(w’). If we call AC; the variation of lattice specific heat due to an isolated substitutional Mg impurity, and ACb2 the same quantity for two interacting Mg, ( 1, 2) standing for one respective configuration, then we compute AC” for one mole by ACV=N cAC$+ tc2 c (ACti -2AC:) 1.2
nconf
>
,
(7) where nconf is the number of equivalent ( 1,2) configurations (twelve for first neighbours, for example); N is the Avogadro number and c, the atomic concentration. This is correct in c2 provided that we include all the ( 1,2) interacting configurations. At each temperature the order c2 correction 6(ACy)=iNc2C
(ACti2-2AC:) 1.2
‘0 I
nconf
(8)
was estimated by fitting a parabola at positions 2, 3, 4 and then extrapolating the quantity A’L*2’=ACb2-2AC:
(9)
as a function of the distance d of 1 and 2 or of the nth respective position of ( 1,2). A(‘v2) decreases from a positive value when d (or n) increases and we take into account all the ( 1,2) configurations with A(‘p2) positive since the two impurities may be considered independent when A(‘p2) reaches zero: the results thus obtained for 6( AC”) are quite the same whether we use d or n and are reported for n in Figs. 2 and 3 for c= 0.05 and c= 0.15, respectively. For ~~0.05, we found that it was not necessary to go beyond position 4, whereas positions 5 and 6 had to be included for c=O. 15 where 6(ACy) is about twenty times larger. However, the correction brought about by positions 5 and 6.is at most 15% of G(ACv) for T= 100 K and we believe that the approximate character of this correction does not affect the final AC” in a significant way. In Fig. 4 we give ACyfor ~~0.05 and ~~0.15, as well as the corresponding first-order results NcAC:, for temperatures up to 100 K. The ratio second-order/first-order is negative and its absolute value is illustrated in Fig. 5. It is at most about 5% for ~~0.05 but reaches 30% for T= 100 K and c= 0.15. 1
Fig. 2. Second-o&r cOrrectionJ(AC,) at c~~O;05in mJ K-’ mol-’ when one progressively includes more Mg pairs. Curve (1) is 6(ACV) when one only includes all Mg pairs in position 1. Curve (2) when one includes positions 1 and 2, and so forth.
545
F. Marine& M. Roche /Chemical Physics Letters 225 (1994) 542-546
Fig. 3. Second-order correction 6( AC”) at c= 0.15 with the same conventions as in Fig. 2.
-540 L 0
I
I
I
I
I
I
I
I
/
I
10
20
30
40
50
60
70
80
90
100
T (K) Fig. 4. AC” in mJ K-’ mol-’ at c=O.O5 and c=O. 15: (0) only.
or (0) including second-order correction &AC,);
The second-order terms reduce ACy and this reduction which increases with T may be by a factor of almost 1.5: by comparison the correction due to the introduction of changes in force constants with respect to the change of mass only model was found in Ref. [ 2 ] to multiply AC: by a factor of about 3. In conclusion, we have computed at a reasonable degree of accuracy the order c2 correction of the change in lattice specific heat of Al for Al, _&I& al-
(A ) or (A )first-order
loys for c= 0.05 and c= 0.15, and for T ranging from Oto 100K. The correction is negligible for c= 0.05 and for this concentration it is reasonable to use the simple order c theory. For c= 0.15 on the other hand, at T= 100 K, the c2 correction is about half the first-order correction due to the changes in force constants and of the opposite sign, and thus becomes necessary.
546
F. Marineili, M. Roche /Chemical Physics Letters 225 (1994) 542-546
/’
1’ 0
/’
/’
d
/’
I/” //
-0.M
//I’ d/’ ~_---b_---d----
0
10
20
30
40
SO
60
70
80
90
100
-UK) Fig. 5. Absolute value of the ratio second-order/first-order
Unfortunately, it seems that no attempt has yet been made to determine the vibrational change AC” of this alloy [ 111 and verify these considerations.
Acknowledgement We thank the C3NI of the Centre National Universitaire Sud de Calcul (CNUSC) for their financial support.
References [ 1] F. Marinelli and M. Roche, J. Phys. Condens. Matter 4 (1992) 747.
for ACv.
[2] I. Baraille, C. Pouchan, F. Marinelli and M. Roche, Chem. Phys. Letters 207 ( 1993) 203. [ 31 R.J. Elliot, J.A. Krumhansl and P.L. Leath, Rev. Mod. Phys. 46 (1974) 465. [4] R. Mills and P. Ratanavavaraksa, Phys. Rev. B 18 ( 1978) 3291, [S]T.Raplan,P.L.Leath,L.J. GrayandH.W.Diehl,Phys.Rev. B 21 (1980) 4230. [ 61 R.J. Elliott and D.W. Taylor, Proc. Roy. Sot. London A 296 (1967) 161. [7]R.N. Aiyer, R.J. Elliott, J.A. Krumhansl and P.L. Leath, Phys. Rev. 181 (1969) 1006. [8] M.D. Tiwari, R.N. Keshanvani and B.K. Agrawal, Phys. Rev.B7 (1973) 2378. [ 91 L. Dagens, M. Rasolt and R. Taylor, Phys. Rev. B 11 ( 1975) 2726, [ lo] B. Noble, S.J. Harris and R. Dinsdale, J. Mater. Sci. 17 (1982) 461. [ 111 V.G. Vaks andN.E. Zein, J. Phys. Condens. Matter 2 (1990) 5919.
CHEMICAL PHYSICS
Chemical Physics Letters 225 (1994) 542-546
Effect of Mg-Mg pairs on the low-temperature lattice specific heat of Al-Mg alloys Francis Marinelli, Michel Roche Centre de Thermodynamiqueet deMicrocalorimktrie,CNRS, 26 Rue du 14lPme RIA, 13003 Marseille,France
Received 19 January 1994; in tinal form 8 June 1994
We discuss the order cz correction of the lattice specific heat of All-N& substitutional alloys for different concentrationsc and temperatures. To do so we introduce Mg-Mg pairs in the calculation of the phonon density of states, and estimate their contribution to different relative configurations.
1. Introduction
In two recent papers [ 1,2] we have reported the change in the low-temperature lattice specific heat ACVof Al-based FCC substitutional alloys with Li [ 1 ] and Mg or Be [ 2 ] as solute. The calculations of AC” were carried out for an isolated atom of solute and then simply multiplied by the number of solute atoms per mole; the change in atomic volume was also included for Al-Mg where it is important. This standard procedure has been applied before by many authors and is correct to first order in the atomic concentration c of the solute for a random alloy. In order to introduce the interaction of solute atoms, which results in order c” corrections (n> 1) for n-atoms impurity clusters, several methods based on averaging the Dyson equation for the alloy’s Green functions have been proposed a long time ago. A review of these methods may be found in Ref. [ 3 1, and further theoretical efforts in Refs. [ 4,5 1, for example. The simplest of the above methods is that of Elliot and Taylor [ 61 and is exact in c only, but contains higher-order corrections in an approximate way. A theory exact in c2, plus higher-order apElsevier Science B.V. SSDZOOO9-2614(94)00660-I
proximate corrections, was proposed by Aiyer, Elliott, Krumhansl and Leath [ 7 1. Unfortunately, although formally elegant, it seems that none of these methods have so far been found to be tractable for computation nor been applied to a realistic case including force constant changes and direct interaction of the impurities. Here we adopt a simple approach that should be correct in c2, with no attempt at c” terms (n > 2)) by systematically including pair effects in the Al-based Mg alloy. We chose Mg as solute rather than Be or Li for practical reasons, namely, because it was found [ 2 ] that the local contraction (expansion) of the lattice in the vicinity of an Mg atom could be ignored, which results in less computational work.
2. Interaction of two Mg atoms The method used here is well known [ 8 ] and based on Green function theory whose fundamentals will not be repeated here. When substituting some host
F. Marineiii, M. Roche / Chemical Physics Letters 225 (1994) 542-546
lattice atoms by impurities, we write for the total change in the phonon density of states M(02)
= c M?J(@2) 2
(1)
Y
where Yis an irreducible representation (IR) of the point group of the system. We have AN,(o2)=
g$,
tancs,=-ImU
ReQ,(z)
(2) (3)
’
8, is the phase shift, I, the degeneracy of the IR v and D(z)=detlZ+GO(z)P(oZ) D(z) = n Q(z)
Y
I,
(4)
9
where P( 0’) is the perturbation matrix, z= w2 + ie, and Go(z) is the host lattice Green function matrix. In the above equations o2 goes from 0 to c.&, w, being the maximum frequency of the host lattice. In the case of two light impurities, we may find one or two locahsed modes for w> o, and those modes oI introduce additional 6 functions 6(w: -w2) in Ah7(w2). The technical details and the practical determination of Go(z) and the force constants of P(o’) are
543
to be found in Refs. [ 1,2 1. Here P(02) includes two Mg atoms. Changes in force constants from Al-Al to Al-Mg or Mg-Mg are taken into account for first and second neighbours around each Mg atom, using the interatomic pair potential of Dagens et al. [ 9 ] . The respective positions of the Mg atoms are from first to fourth neighbours (positions 1,2,3 and 4 below). Because of the large variation in the atomic volume D of the alloy with c according to Q= ti( 1 + 0.4~) [ 10 ] at each concentration c we calculate kF which in turn determines the force constants used for Go(z) and P(w2) at S, as was done in Ref. [ 21, with the right number of electrons. The total phase shifts S,6 ls2for a pair ( 1,2 ) : WC
F sy )
(5)
where v is an IR of D2,, or Dab, as well as 26’=2 c s; Y
(6)
(Oh symmetry) for two independent Mg atoms are reported in Fig. 1, with the force constants calculated at kF= 0.9098 au and a = 4.1197 A for the fee lattice parameter, i.e. for c= 0.15. The phase shifts differ perceptibly only in the last tenth of the frequency spectrum (0.9
-7.50
Fig. 1. Total phaseshifts8. Curves( 1) to (4) arc for pairswhenthe respective positions of the two Mg atoms vary from fti neighbours. Curve (5) is for two independent Mg atoms.
to fourth
F. Marinelli, hf. Roche / Chemical Physics Letters 225 (1994) 542-546
544
crease significantly from position 1 to position 2, but then slowly and quasi-linearly from position 2 to position 4. Since we have not considered pairs of Mg atoms beyond position 4, an extrapolative scheme will be considered in the next calculations of ACV.
3. Calculation of AC,/ ACV for the alloy is linear in AN(w’). If we call AC; the variation of lattice specific heat due to an isolated substitutional Mg impurity, and ACb2 the same quantity for two interacting Mg, ( 1, 2) standing for one respective configuration, then we compute AC” for one mole by ACV=N cAC$+ tc2 c (ACti -2AC:) 1.2
nconf
>
,
(7) where nconf is the number of equivalent ( 1,2) configurations (twelve for first neighbours, for example); N is the Avogadro number and c, the atomic concentration. This is correct in c2 provided that we include all the ( 1,2) interacting configurations. At each temperature the order c2 correction 6(ACy)=iNc2C
(ACti2-2AC:) 1.2
‘0 I
nconf
(8)
was estimated by fitting a parabola at positions 2, 3, 4 and then extrapolating the quantity A’L*2’=ACb2-2AC:
(9)
as a function of the distance d of 1 and 2 or of the nth respective position of ( 1,2). A(‘v2) decreases from a positive value when d (or n) increases and we take into account all the ( 1,2) configurations with A(‘p2) positive since the two impurities may be considered independent when A(‘p2) reaches zero: the results thus obtained for 6( AC”) are quite the same whether we use d or n and are reported for n in Figs. 2 and 3 for c= 0.05 and c= 0.15, respectively. For ~~0.05, we found that it was not necessary to go beyond position 4, whereas positions 5 and 6 had to be included for c=O. 15 where 6(ACy) is about twenty times larger. However, the correction brought about by positions 5 and 6.is at most 15% of G(ACv) for T= 100 K and we believe that the approximate character of this correction does not affect the final AC” in a significant way. In Fig. 4 we give ACyfor ~~0.05 and ~~0.15, as well as the corresponding first-order results NcAC:, for temperatures up to 100 K. The ratio second-order/first-order is negative and its absolute value is illustrated in Fig. 5. It is at most about 5% for ~~0.05 but reaches 30% for T= 100 K and c= 0.15. 1
Fig. 2. Second-o&r cOrrectionJ(AC,) at c~~O;05in mJ K-’ mol-’ when one progressively includes more Mg pairs. Curve (1) is 6(ACV) when one only includes all Mg pairs in position 1. Curve (2) when one includes positions 1 and 2, and so forth.
545
F. Marine& M. Roche /Chemical Physics Letters 225 (1994) 542-546
Fig. 3. Second-order correction 6( AC”) at c= 0.15 with the same conventions as in Fig. 2.
-540 L 0
I
I
I
I
I
I
I
I
/
I
10
20
30
40
50
60
70
80
90
100
T (K) Fig. 4. AC” in mJ K-’ mol-’ at c=O.O5 and c=O. 15: (0) only.
or (0) including second-order correction &AC,);
The second-order terms reduce ACy and this reduction which increases with T may be by a factor of almost 1.5: by comparison the correction due to the introduction of changes in force constants with respect to the change of mass only model was found in Ref. [ 2 ] to multiply AC: by a factor of about 3. In conclusion, we have computed at a reasonable degree of accuracy the order c2 correction of the change in lattice specific heat of Al for Al, _&I& al-
(A ) or (A )first-order
loys for c= 0.05 and c= 0.15, and for T ranging from Oto 100K. The correction is negligible for c= 0.05 and for this concentration it is reasonable to use the simple order c theory. For c= 0.15 on the other hand, at T= 100 K, the c2 correction is about half the first-order correction due to the changes in force constants and of the opposite sign, and thus becomes necessary.
546
F. Marineili, M. Roche /Chemical Physics Letters 225 (1994) 542-546
/’
1’ 0
/’
/’
d
/’
I/” //
-0.M
//I’ d/’ ~_---b_---d----
0
10
20
30
40
SO
60
70
80
90
100
-UK) Fig. 5. Absolute value of the ratio second-order/first-order
Unfortunately, it seems that no attempt has yet been made to determine the vibrational change AC” of this alloy [ 111 and verify these considerations.
Acknowledgement We thank the C3NI of the Centre National Universitaire Sud de Calcul (CNUSC) for their financial support.
References [ 1] F. Marinelli and M. Roche, J. Phys. Condens. Matter 4 (1992) 747.
for ACv.
[2] I. Baraille, C. Pouchan, F. Marinelli and M. Roche, Chem. Phys. Letters 207 ( 1993) 203. [ 31 R.J. Elliot, J.A. Krumhansl and P.L. Leath, Rev. Mod. Phys. 46 (1974) 465. [4] R. Mills and P. Ratanavavaraksa, Phys. Rev. B 18 ( 1978) 3291, [S]T.Raplan,P.L.Leath,L.J. GrayandH.W.Diehl,Phys.Rev. B 21 (1980) 4230. [ 61 R.J. Elliott and D.W. Taylor, Proc. Roy. Sot. London A 296 (1967) 161. [7]R.N. Aiyer, R.J. Elliott, J.A. Krumhansl and P.L. Leath, Phys. Rev. 181 (1969) 1006. [8] M.D. Tiwari, R.N. Keshanvani and B.K. Agrawal, Phys. Rev.B7 (1973) 2378. [ 91 L. Dagens, M. Rasolt and R. Taylor, Phys. Rev. B 11 ( 1975) 2726, [ lo] B. Noble, S.J. Harris and R. Dinsdale, J. Mater. Sci. 17 (1982) 461. [ 111 V.G. Vaks andN.E. Zein, J. Phys. Condens. Matter 2 (1990) 5919.