- Email: [email protected]
Workshop on thermodynamic models and data for pure elements and other endmembers of solutions
00005-3
Workshop on THERMODYNAMIC MODELS AND DATA FOR PURE E:LEMENTS AND OTHER ENDMEMBERS OF SOLUTIONS S&lo6 Ringberg, Febr. 26, to March 3, 1995
Group 4:
h - lkansitions
Group members:
Didier de Fontnine Dept. Mater. Sci. & Engg., Univ. of California, Berkeley, USA Suzana G. Fries Lehrstuhl f. theor. Htittenkunde, RWTH-Aachen,
Germany
Gerhard Inden (group leader) Max-Planck-Institut
fur Eisenforschung
GmbH, Dusseldorf, Germany
Peter Miodownik ThermoTech.
Ltd., Surrey Technol. Centre, Guildford, U.K. Rainer Schmid-Fetzer
Metall. Zentrum, AG Elektron. Mater., TU-Clausthal,
Germany
Shuang-Lin Chen Dept. Mater. Sci. & Engg. Univ. of Wisconsin, Madison, USA
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D. DE FONTAINE et al.
500
1. Introduction 2. Current magnetic treatments 2.1 Description 2.2 Comparison of existing treatments 2.3 Interim conclusion 3. Methods of extracting cy 3.1 3.2
from experimental total cp
Simple metals Compounds (End-members)
4. Magnetic specific heat capacity obtained from experiments 4.1 Simple metals 4.2 Oxides, fluorides, chlorides 4.3 Conclusions 5. Theoretical constraints 5.1 Magnetic entropy 5.2 Magnetic enthalpy 5.3 Conclusions 6. Multiple magnetic states 6.1 Experimental evidence for two y-states 6.2 Theoretical treatment of multiple states 6.3 Thermodynamic consequences of multiple states 6.4 First principles calculation of multiple states 6.5 Potential advantages of including a multiple state model 6.6 Incorporation of multiple states into computing software 7. Reference states 7.1 Choices of reference states 7.2 Conclusions 8. Treatment of metastable phases 8.1 Procedure in the absence of c, data 8.2
Comparison of values for p in different allotropes
9. Overall conclusions 10.
References
11.
Appendices 11.1 AppendixA 11.2 Appendix B
12.
Tables
13.
Figures
501
A-TRANSITIONS
1.Qoduction
Group 4 was assigned the task of incorporating specific heat capacity, the representation
c,(T),
the contributions
into the overall temperature dependence of from so-called ,,h-transitions”, in particular
of the magnetic specific heat capacity CF.
It is generally not possible to
treat the problem in a rigorous theoretical manner: even if magnetic effects would be described by an Ising model, a detailed mathematical treatment would be out of question as it is known that (a) the three dimensional Ising model cannot be treated exactly, even in principle, and (b) finding approximate solutions is one of the hardest problems in theoretical physics. We are forced, therefore, to adopt a phenomenological approach consisting of optimal fits of experimental data to an empirical analytical expression. The choice of this mathematical expaession should, however, not be completely arbitrary, and indeed its form should reflect as muclh as possible the essential physics of the problem. If that can be achieved, (a) at least some of the fitting parameters will have physical meaning and these derived values can be checked against known experimental information, (b) parameters for metastable phases can be predicted with some chance of success. A priory, the total specific heat capacity can be assumed as the sum of various contributions: cp
=C;b+C;+C;+C:,
The first term represents the vibrational contribution, the second one the electronic, the third one ,that due to the h-transition, and the fourth one an excess contribution. Other groups of this workshop have treated in detail the vibrational and electronic contributions, so that what remains is the effect of the h-transition and of the excess contribution. The h-transition represents
mainly
magnetic
and/or
atomic
order/disorder
contributions,
CT
respectively. If the first three terms can be handled properly the excess term will most cases, except perhaps at very high temperatures, and its explicit form discussed further in this report. In some cases special electronic effects may considered which would then need to be subsumed in this term like those due magnetic and/or electronic states. Such effects will be discussed later.
and
c;~
be small in will not be have to be to multiple
Because order/disorder transitions can also be mapped generally onto the Ising model, it will suffice to consider the magnetic Ising model as the prototype. Hence, order/disorder transitions do not require a separate description. A major problem is that the various contributions appearing in Eq. (1) &e not mutually independent. Nor is it easy to extract from overall cP data that portion due to magnetic effects alone. To acomplish that separation, one would need precise theoretical what we are lacking.
formulation
of the various contributions,
which is precisely
An essential feature of h transitions is the appearance of a singularity in the function c?
as a
function of temperature. This is used to define the critical temperature T,. On either side of T, are formed continuous low- and high-temperature ,,wings“ of the curve associated with long range and short range order respectively. As mentioned earlier, no exact expressions are available for describing c;f”8 in the vicinity of T,, hence approximate empirical formulas have been proposed. One such formulation for the magnetic transition was suggested by Inden [76Ind, 81Ind], and further details are given below. This formulation has the merit of
502
D. DE FONTAINE eta/.
presenting the singularity at T,, but also has draw backs for practical numerical analysis. Hence, present versions of THERMO-CALC [85Sun], ChemSage [9OEri(b)], MTDATA [90Dav], employ a Taylor expansions of the Inden expressions truncated to the third term as originally proposed by Hillert and Jar1 [78Hil]. Another option has been proposed by Chuang, Schmid and Chang [85Chu] featuring exponential functions. In the past, various attempts have been made to isolate the magnetic contribution
c,”
by
subtracting out the other contributions, assuming approximate forms for the thermal components. Not surprisingly, conflicting results were often obtained by different investigators due to the process of fitting noisy data by somewhat arbitrary empirical models. Final results and calculated values of physical properties depend critically on the method of separation used. In section 3 the deconvolution procedure which has been used to derive the mathematical expressions for clf”8will be outlined in detail. There does not appear, at present, to be any compelling reason to abandon the Hillert and Jar1 formulation, even for materials whose magnetic properties deviate significantly from those of the ,,classic“ elements Fe, Co and Ni. However, in order to comply with ferro- or antiferromagnetic systems like oxides, fluorides, chlorides, some of the parameters which have so far been kept constant might have to be varied. It is therefore suggested that the Hillert-Jar1 form be retained in an extended version as the default option. However, it is also suggested that the original Inden formulation be made available as an option where a good fit at the singularity at Tc needs to be included, and that the option proposed by Chuang et al. [85Chu] is also made available, since the exponential expressions are particularly convenient to work with algebraically. Once the expressions for cy
and the parameters subject to variation have been identified, it
is essential to apply global fitting whereby all the terms appearing in Eq. (1) are treated together in such a way that each reflects in an approximate manner the essential physics of the problem. For example, the vibrational part will be approximated by a Debye or Einstein function supplemented by a polynomial in absolute temperature of the type aT + bT* . The magnetic part will likewise be given its own mathematical expression derived in section 4. In the ensuing least-squares fit of the parameters of the postulated mathematical expression to the known data, decisions will need to be made concerning the weight to be given to various data and how the relative weights should be varied in an iteration process so as to obtain the best overall fit and so that the parameters have clear physical meaning. Such a process of optimization necessarily involves subjective decision-making, but this is part of the basic philosophy of the Calphad approach. In such an approach the separation of the magnetic component from cP will not be unique, but the subjectivity should be minimized. Detailed descriptions of these empirical expressions will now be given with the methodology of their implementation in the optimization process. Further sections deal with the process of estimating values for metastable phases where this protocol cannot be used and thoughts on the incorporation of a module that can handle multiple magnetic states.
A-TRANSITIONS
Descriph
2.1
The treatment of the magnetic contribution to Gibbs energy has been based so far on an analytical description of the magnetic specific heat capacity c? as a function of the reduced temperature o = f
. The two branches below and above T, are described separately and will c be denoted here as c; and ci, respectively, and their associated constants as K* (or k*).
Four empirical approaches from the literature have been discussed: l
Logarithmic representation of c?
[76Ind, 81Ind]:
l
Truncated power series expansion of Eq. (2) [78Hil]: c; = 2K’R z-” +++’
c, = 2K-R l
Representation of cy
5
(3)
by exponential functions [85Chu]:
c; = k-Rrexp(-p(ll
+17””
2)) , ci = k’Rrexp(q(l-
7))
(4)
Representation of clf”8 scaled to the experimental values for pure Fe [Y&n, 79Nis]: this fourth empirical approach refers everything to the cy
67Hi1,
function of pure Fe.
This is useful for bee Fe-base systems, but is considered less applicable for other systems and therefore has not been considered further in this report. Expressions (2)-(4) have been written here in a more general form than originally proposed, leaving not only the coefficients K* (or k*) open for adjustment but also the exponents m,n (or p,q) which have been used in current treatments with constant values for ferromagnetic 3delements (Fe, Co, Ni). Inden [76Ind] suggested m=3, n=5 in Eq. (2) and these values were also taken by Hillert and Jar1 [78Hil] in Eq. (3), while in Eq. (4) the values p=4 (for both fee and bee) and q=8 (bee), 16(fcc) have been used by Chuang et al. [85Chu]. According to [76Ind] the coefficients K* (and likewise k*) should character&e the particular element, alloy or compound via the following constraints: l
The fraction of magnetic enthalpy above T, should depend mainly on coordination number, i.e. on crystal structure. This fraction has to be determined empirically, e.g., by using the experimental data for Fe, Co and Ni. Inden obtained the following values: c;dT AH+ f= AH- +AH+
P 0.4 @cc), 0.28 (fee) ‘= lc;:+
jc;dT Tc
(5)
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504
l
er al.
The total magnetic entropy should be consistent with relation (6) which gives the total magnetic entropy in terms of the mean atomic moment l3 (expressed in Bohr magnetons &,) per atom: ASWg =AS-+AS’=-cdT=Rln@+l) I 0 T
(6)
The conditions (5) and (6) give a recipe for treating elements, compounds and alloys for which experimental c,” values are not available. Curie (or Neel) temperatures and I3 values are usually known in these cases. Condition (6) has been relaxed by Chuang et al. [UChu] during their extraction and optimization of Fe, Co, and Ni. 2.2on
. . of exuUUMaWs
The two treatments (2) and (3) are closely related and can directly be compared since the parameters involved are identical. Treatments (3) and (4) have in common that they do not give a singularity at Tc but a finite step which is typical for a second order transition. In order to be able to compare later on the results of (3) with those of (4) it is useful to define the coefficients K* in such a way that the same step is obtained at T, in both treatments. For this purpose we take arbitrarily c; =3R and ci =R. This leads to k- = 3, k+ = 1 for treatment (4) and to K- = 0.8949, K’ = 0.2983 for treatment (3) taking the expansion up to the fourth term. It has been found very useful to consider not only linear representations of cy
but also
double-log representations. Fig. 1 thus shows the general shape of c,“” in both representations as obtained from (2) for different exponents (m=l, 2, 3 and n=3, 4, 5). Fig.2 shows the comparison of these c,“” data with those of the truncated series (3) using four instead of three terms in the expansion. The agreement between both treatments is fully acceptable for most practical cases when the experimental li peak is not very sharp. It is thus worthwhile examining the differences between both treatments relative to the integrated quantities like enthalpy and entropy. These data are given in Table I. It should be kept in mind that this comparison is purely formal since the coefficients in both treatments have been taken to be the same. However, when it comes to a fit of experimental data using each one of these treatments the coefficients would no longer turn out the same, of course, and the integrated quantities are expected to be much closer. A comparison of the treatments (3) and (4) is shown in both linear and double-log. representation in Fig. 3 for one set of parameters. The parameters for treatment (3) are the same as those of Fig.1. The exponents p and q of treatment (4) have been chosen in such a way that the magnetic enthalpies AH* below and above T, take the same values as previously used in treatement (3), i.e., 50.541, then obtained as T
$=
0.334. The corresponding entropies are
= 0.699 [0.:96] and --R“’ =’ 0.233 [0.264] for (3) and [(4)].
A-TRANSITIONS
In Fig.3b distinct topological differences between the two theoretical treatments become visible. It will be shown later (section 4) that the experimental cy curves have an opposite curvature to that following from (4).
2.3
Interim conclusions
The three treatments (2) to (4) are purely empirical since there is no fundamental physical argument in favour of one of them, except that a logarithmic singularity is more in keeping with theory. It may thus be useful to keep the logarithmic treatment (2) as an option. For most purposes the truncated series, the treatment (3), seems to offer sufficient flexibility, in particular if the exponents m and n are allowed to take other values than 3 and 5 used previously. Therefore it is recommended to continue using treatment (3) as the default option. If th.e parameters in the treatment (4) are allowed to vary the difference between treatments (3) and (4) become negligible as far as the integrated quantities are concerned. Therefore, the treatment (4) should also be included as an option in the same way as suggested for the treatment (2). The expressions of the magnetic contributions to the thermodynamic functions are ,given in appendix A and B for the treatments (3) and (4), respectively.
3. Methpds of BcWffOm
total
P
CP
LLSimDlemetals
In most methods reported in the literature the magnetic contribution CT is separated from measured total c, by subtracting the other physically identifiable contributions[ZTau, 64&a, 65Bra, IlMes, 85Chu].
64Lyt,
These are the harmonic vibrational c, taken from the Debye model, a dilatational term C ” = cp - c, , which may be calculated from the Griineisen relation, an electron contribution cc which may be simplified to be proportional to T, and finally the magnetic term CF. One may extend Eq. (1) to include more contributions such as electron-phonon enhancement contributions to the low temperature cc [75Gri, 76Gri], anharmonic effects [83Ros], effect of magnetic ordering on Debye temperature via its effect on Young’s modulus [64Lyt], effects of spin waves [55Tau, 64Lyt, 85Chu], effects from equilibrium vacancies [81Mes, 83Ros]. However, these contributions have to be calculated from independently determined physical quantities which are rarely available. In many cases the various contributions have to be determined by deconvoluting cp . Then there may be interference from the magnetic contribution. E.g. if low temperature c, data are to be used for the determination of 8,, the contribution from spin wave theory [85Chu] has to be taken into account to extract the non-magnetic contribution. On the other hand, the applicability of spin wave theory has been questioned strongly by [64Lyt]. In this respect some help might be available from statistical models. The magnetic contribution is then extracted (sometimes in an iterative procedure) on the basis of cy(extr.)=c,(exp.)-c,
-P
-cc
(7)
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D. DE FONTAINE
It is evident that this extracted c?
et al.
depends strongly on the specific details in the separation
method described above. It also depends on the quality of the experimental data which may vary considerably with date and author (see section 4). Whatever technique of deconvolution has been used and whatever individual contributions have been obtained, a mandatory requirement is to be consistent with the total cP when all the contributions are recombined later. 3.2
Comuounds (End-members)
Up to eleven nonmagnetic compounds (carbides, oxides, nitrides, fluorides) have been analysed in detail by Hofman et al. [56Hofl in order to develop a good representation of the nonmagnetic part of cr. These materials are nonmetallic and hence do not have an electronic contribution. They are more complicated, however, in that the lattice vibrational contribution cannot be represented by a single Debye function. Hofmann et al. were unable to obtain a good fit over an appreciable temperature range using a combination of a single Debye function with a number of Einstein functions. The best representation they could obtain was with a two Debye term of the following form for a binary compound with formula R,X, : c, =m-f,(eE
/T)+n~f,(B~
IT)
(8)
where f,@, / T) is the Debye function and t3: and Q are the Debye temperatures associated with the elements R and X of the compound. Hofman et al. applied the same procedure to ferro- and antiferromagnetic salts.
4.
Magnetic specific heat capacity obtained from experiments
It has already been pointed out that the magnetic specific heat capacity cannot be accessed directly by experiments but has to be derived from total c, data by deconvolution. Therefore the cy data, although directly derived from experimental data, cannot be called experimental in the strict sense. In order to distinguish these data from those obtained from some formula or model, these ,,derived experimental“ cy data will be denoted hereafter as ,,deconvoluted”. The (deconvoluted) c,“” results presented in this section will show that these data exhibit a shape which is similar to what follows from Eq. (2)-(4). This is particularly convincing in the double-log representation, see Figs lb and 3b. Therefore it is expected that these equations can provide a reasonable fit to cy .
4.1
Simule metals
The 3d metals Fe, Co and Ni are the most widely studied ferromagnetic elements. In Figs. 4-6 the cy data for Fe, Co and Ni according to Braun and Kohlhaas [6SBra] are shown in Iinear and double-log representations. The numerical values of the physical parameters used in deconvoluting cP the reader is referred to [64Bra, 65Bra]. In all three cases the double-log representations of the data follow the shape which is obtained from the descriptions (2)-(3)
A-TRANSITIONS
shown in Figs lb and 3b. On the contrary, treatment (4) yields a concave curvature for the low temperature branch and a convex curvature for the high temperature branch of c? (Fig. 3b) which is opposite to what has been observed in the represented cases and in all instances studied so far. On the other hand, the double-log representation overemphasizes small values so Ithat in practice this may not be prohibitive in view of the accuracy limits imposed by the experiments.
Deconvoluted c? calculated c ,”
data for bee Fe [64Bra] are shown in Fig. 4. The lines represent the
according to Eq. (3) using the parameters for Fe according to the current
des’cription in the thermodynamic databases (e.g. SGTE [91Din]). The values of the parameters are given in Table II. In this description the magnetic moment per Fe atom is p E z 2.2 CI,[60Goo] which is the value obtained from magnetic measurements.
Deconvoluted c,” calculated c?
data for fee Co are shown in Fig. 5. The solid lines represent the
obtained from Eq. (3) using the parameter values in Table II. The magnetic
moment attributed to Co is p=1.35 &, a value close to spin 112.From magnetic measurements a magnetic moment of 0=1.7 [56Boz] is obtained. This value was originally used by Inden [76Ind]. The value p=1.35 & gives a better fit to the experimental data.
In Fig. 6 the deconvoluted c ,” data for Ni from two different sources, [64Bra, 65Bra] and [81 Mes] are shown. At temperatures below T< the two sets of data differ considerably. The deconvolution was performed by the two groups, respectively. It is to be checked with the total cP-data whether this difference is due to the difference in the raw data or due to the differences in the deconvolution. The calculated c y is shown in Fig. 6 as a solid line. The parameters are given in Table II. The low value of magnetic moment (0 3 0.52pJ attributed to Ni has been explained in terms of the Van Vleck model [52Vle] as being due to an average electronic 3d9.4configuration with 40 % of the Ni atoms taken as 3d’O(spin 0) and 60 % as 3d9 (spin l/2) [56Hofl. It is supposed that the Ni atoms continuosly redistribute between these states among the lattice sites.
Fig. 7 shows the ,,experimental“ cy -data according to [54Gri]. The magnetic moment has been obtained from low temperature magnetization measurements, p z 7 b (spin 7/2) [80I,eg]. c b of Gd exhibits a pronounced hump. This hump has to be associated with the high spin number of Gd. This has been shown for Gd by a statistical thermodynamic treatment based on the Ising model and using a tetrahedron approximation of the Cluster Variation Method [96Sch]. It is evident from both the linear and the log representations that c 7 cannot be approximated with sufficient accuracy by Eq. (3). An improved treatment is definitely required in such instances.
507
508
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et al.
A second element causing problems with the currently accepted treatment (3) is Cr. Fig, 8 shows total c,-values according to [79Wil]. It is difficult to get a reliable representation of the nonmagnetic terms of c, since Cr (like many more 3d-elements, e.g. Ti, V, Mn) exhibits special electronic effects discussed in [78Ben(a)-(b), 79Ben] and also in section 6 of this report. The conflict with Eq. (3) comes from the facts that (a) the Neel temperature is rather high, (b) the h-curve is extremely narrow and c, tends to very high values in the vicinity of T,, while the average magnetic moment derived from neutron diffraction [62Shi] is rather high (p P 0.6&s). This p-value is conflicting with (3) if Hq. (6) is assumed to hold since it is hard to see how the corresponding total magnetic entropy could be obtained from such a narrow hcurve. Faced with this problem the SGTE database works with an extremely low value B = 0.008 H [91Din] which is sometimes called a ,,thermodynamic p,, to make a distinction with the magnetically determined value. Cr is not the only case with this behaviour. MnO is another candidate which will be discussed in the next paragraph. It should be noticed that there are antiferromagnetic elements and compounds (as well as alloys) which cannot be treated with (3). Obviously the h-shape of c rmgmay become very narrow and tend to very high values in particular cases. At present it is n:t clear which properties controle this sharpening effect. 4.2
Oxides. fluorides. chlorides
In Figs 9-12 cy
data for antiferromagnetic salts are shown. These data have been derived by
Inden [95Ind] using a deconvolution of published total c, values according to the procedure worked out and recommended by Hofman et al. [56Hofl. The Debye temperatures used in this deconvolution are given in Table III together with other relevant data. The double-log representations show that an extension of (3) (and (2), of course) away from constant exponents m and n is mandatory. It is interesting to compare the cy
data of compounds which have the same crystal structure,
e.g., MnO, NiO (Fig. 9), Fe0 (Fig. lo), and NiF,, MnF, (Fig. 11). It turns out from these results that the exponents m and n cannot be taken the same for a given crystal structure. Likewise no direct correlation with the spin number can be found: while MnO and NiO (Fig. 9) seem to exhibit rather similar data at temperatures away from T, their spin numbers are very much different. In Fig. 9 the log-scale of CT similarities of cy
has been limited to two decades in order to show the
of NiO and MnO. One experimental point of MnO close to Tc is beyond
the upper scale and has thus been excluded in this figure. Fig. 13 shows cy
of MnO in a
linear scale including this point. MnO seems to behave similar to Cr in that cy
becomes
narrow and very high at the critical point. 4.3
Conclusipns
The vibrational part of c, of many compounds cannot be described with sufficient accuracy by means of a single Debye term. A two-term description instead gives a good description. It is to be checked whether the two Debye term approach used here can be made equivalent to a
A-TRANSITIONS
single Debye term approach with temperature-dependent Debye temperature. Such an approach has been recommended for other reasons by Group 1 of this workshop. The general trend of the cy
values in the double-log representations indicates that Eq. (3) is
to be preferred against (4). A possible overemphasis in the range of small values should be kepi in mind. Some advantage of (4) is the simplicity of derived integrated thermodynamic functions, see appendix B, although this aspect is marginal in numerical treatments. The.refore, Eq. (3) should be taken as the default representation of the magnetic effect. There rem,tin exceptional cases, like Cr (Fig. 8) and MnO (Fig. 13), which cannot be fitted reasonably well with Eq. (3) due to the strong singularity at T,. It may be valuable to come back to Eq. (2) to take this behaviour into account. These cases need further thorough studies. The presented deconvoluted data indicate that the exponents m and n should not be treated as constants (as it has been done up to now for 3d-elements and alloys) but rather left free to vary in an optimization procedure. The hump in the cy
curve of elements with p-values higher than 1 cannot be neglected and
needs further consideration. Further studies, e.g., using statistical thermodynamic models are needed to get more detailed knowledge of this effect and its dependence on spin number and exchange energy. Before such informations become available it is not possible to work out any empirical mathematical description of this effect. uheoretical
constraint5
In principle the parameters of the treatment of the magnetic effects can be taken as being free for variation in an optimization procedure until a perfect reproduction of the experimental data is obtained. In reality, experimental c, values show a serious scatter (e.g., Fig. 6) and there may also be deficiencies in the deconvolution process (e.g., if reliable data of physical parameters like 0,, Griineisen constant, electronic effects etc. are not available). Therefore, deconvoluted CT data can hardly provide the basis for such an optimization process. It is thus recommended, and it has been practiced to some extent in the past, to fix some of the parameters by means of consistency relations which are based on general principles. 5.1
Magnetic entropy
The total entropy gain in going from the completely ordered to the disordered state is given by Eq. (6) if a system is considered with one mole of atoms, each with magnetic moment p. This expression follows from theory in the case of integral Bohr magneton numbers for p. This expression can be generalized to account for different spins of the atoms (e.g., due to crystallographically nonequivalent positions) [81Ind]: ASms8= RExi I where (R) = xx,&,
ln(l3, +l)n
Rln((R)+l)
(9)
xi and Oi are the mole fractions and magnetic moments (in pB) of
atoms of kind i, respectively. The summation in (9) includes all atoms. Non-magnetic eleml:nts do not contribute since their p-value is zero.
D. DE FONTAINE
510
et al.
The latter expression in (9) can be used as an approximation if the difference between the pi is not too large. This approximate expression is of practical relevance in alloys where the individual moments pi are usually not known while the average moment
is often available from magnetization measurements. It is not possible to make a general statement about the applicability of (9) to real systems with their complex electronic structure covering the range from delocalized band behaviour to fully localized spin systems. Taking Ni as an example, it seems more a question of defining the fraction of atoms with a given state rather than of the validity of (9). Hofman et al. [56 Hofl investigated Fe, Co, Ni, Gd, and a large number of compound systems. They used the constraint (9) successfully in their deconvolution procedure in order to obtain the Debye temperatures from an entropy matching. Since they used (9) only for determining the physical parameters of nonmagnetic contributions the deconvoluted ca as such were not subjected to this constraint. Yet the resulting AS”= values came out very close to the values expected from (9). Consequently, Inden [76Ind, 81Ind] suggested to use (9) as a constraint for determining K* ,as did Hillert and Jar1 [78Hil]. That is the reason why the calculated cy values derived from Eq. (3) do not perfectly fit the experimental data of Fe, Co and Ni (Figs 4-6). Constraint (9) has been relaxed by Chuang et al. [85Chu] in their deconvolution of c, of Fe, Co and Ni. As a result they obtained a closer fit to their deconvoluted cy data which were based on assessed data in Hultgren’s tabulation rather than on the original c,‘s. However, in view of the considerable scatter of experimental c, data from different sources, (e.g., Fig. 6) there seems to be no point to give up constraint (9) for a more close fitting of one particular set of data. The major drawback would be the loss of predictability. In view of the scarcity of experimental c, data for elements, alloys and compounds, predictability cannot be sacritied. 5.2
Magnetic enthalny
Another constraint can be formulated on the basis of theoretical models. Starting from the work of Brown and Luttinger [55Bro], Paskin [57Pas] developed an expression for the fraction of magnetic enthalpy above T, (or T,): f=-
s+l s(z - 1)
( 10)
where s is the spin number and z the coordination number. Applying this relation to the fee structure and taking spin % (applicable to Co and Ni) one obtains f=O.27. For the bee structure the values f=O.43 (spin l/2) and 0.29 (spin 1) are obtained from (10). At present, the magnetic effects are treated with f=O.28 for fee and f=O.4 bee, irrespective of spin number, as suggested by Inden [76Ind, 81IndJ. This can be improved with constraint (10). 5.3
Conclusions
No argument has been found which could impair the validity of (9). Therefore, the description of c? should be made consistent with the two constraints (9) and (10). This makes sure that
511
k-TRANSITIONS
a high level of predictability is maintained to cover the majority of cases (particularly alloys and compounds) where cPdata are not existing.
6.1
Exnerimental evidence for two y-states
It has been assumed in all the previous sections that each element has only one magnetic state pertinent to a particular crystal structure. As the fee allotrope of pure iron, ‘y, is not stable at room temperature, various techniques have been used to extract the relevant magnetic properties. These include extrapolation from alloys [63Wei], the examination of a variety of phycical properties [78(a) and (b)Ben], and measurements on very small particles where the fee phase has been retained by coherency effects. When the results from these various techniques are compared, it is evident that, at low temperatures, fee y-iron behaves predominantly like an anti-ferromagnetic material with a low magnetic moment, state yz, but there is also evidence for a much higher moment at higher temperatures, state *(I. In Re-Ni and Fe-Co, there is compelling evidence for the co-existence of two magnetic states [78h4io], and this concept has also been extended to other 3d elements [79Wei]. 6.2
Theoretical treatment of multiple states
The simplest way to describe the equilibrium between various competing states is to use a Schottky model [63Wei, 63Kau], where two states are considered, the ground state yz and an excited state Tl, with an energy difference AE between them. If (g,, g,) are the degeneracies of the two states, respectively, the fraction v of excited states is given by
JLJL,, 1-v
g,
__AE
( 1 RT
(11)
Clearly if AE is large, there is effectively only one state (and one moment). However as AE becomes smaller there can be some quite marked effects, as evidenced by Invar alloys where AE is not far from zero. The Schottky model does not lead to a h-transition per se, but several variants of the Schottky model have been developed to take into account the situation where one of the two states undergoes magnetic ordering [77Mio, 8OMio], see Fig. 14. In such cases AH -will become temperature dependent, and there is a further change in the temperature dependence of the effective magnetic moment. 6,2
Thermodvnamic consequences of multiple states
Although there is extensive circumstantial evidence for the existence of multiple magnetic states, the necessary formalism has not, as yet, been incorporated into the current software for phase diagram calculations. This is despite the fact that its inclusion has been suggested as essential to the proper description of the phase transformations in iron base alloys [63Kau, 82Ben], and the explanation of a number of thermodynamic anomalies is facilitated by this concept [78Mio]. This can be attributed to the lack of theoretical backing for the concept, particularly in the early stages of its development.
D. DE FONTAINE
512
5.4
et al.
First urincinle calculations of multinle states
This situation has however changed dramatically in recent years. First principle calculations have not only confirmed the existence of multiple states, but have also confirmed the value of the moments, Table IV, and energy gaps (see Fig. 15) inferred previously by experimental techniques [88Mor, 90Mor (a,b,c), 93Asa(a,b)].
6.5 Potential advantac s of Muding amti-state model (a) A better descriptioi of the transition between ferromagnetic and anti-ferromagnetic behaviour in iron-base alloys and more appropriate values for the magnetic properties of metastable allotropes. (b) A better description of the low temperature behaviour of iron, which impinges on the calculation of other properties at low temperatures, such as stacking fault energies and martensite transformation temperatures. (c) A potential resolution of some of the discrepancies that seem to occur between magnetic parameters extracted from c, curves and determined by magnetic measurements.
6.6
Incornoration of multinle states into comnutinrr software
One can consider adding the necessary formalism either within the existing magnetic framework or as a separate module. A separate Schottky module would have the advantage that it could also be used in other areas such as a description of the glass transition. Such a module will not need to be invoked in the default situation where only one magnetic state is involved. While separation of multiple state effects and the h-transition would necessarily be an approximation, it is adequate as an interim solution, and much simpler to implement. The general availability of a Shottky function could also have further applications in cases where there is equilibrium between two states that differ with respect to some other electronic transitions and not just for the magnetic case.
7.
Reference states
7.1
Choices of reference states
The magnetic effect has been treated as a separate effect which has to be added nonmagnetic part of c,. Therefore, the integrated functions like enthalpy, entropy, energy also take up the magnetic effect as additive terms. In principle, there should difference whether the magnetic terms are added to the ground state at OK or paramagnetic high temperature state.
to the Gibbs be no to the
In practice, the thermodynamic functions are derived from high temperature data. ,,High temperatures” is synonymous with the range of temperatures where equilibria can be attained during the experiments. The assessment work that has been performed so far has not included,
A-TRANSITIONS
at high temperatures, a magnetic entropy contribution. If then the magnetic term is added there may appear, at low temperatures, a violation of the third law. So far, this has not been considered to be a problem for the calculation of phase equilibria since these. are usually not made at such low temperatures. However, if the low temperatures are being considered in the future, e.g., to treat the thermodynamic effects of martensitic transformations, then the relialoility of the thermodynamic functions needs to be extended to low temperatures, that means into the ranges where the mentioned conflicts may appear. Similar arguments can be put forward if the magnetic term is added, starting from the ground state. The various consequences have been discussed in [91Ind]. 7.2 Conclusions It is recommended to include, in the future a magnetic entropy contribution into the high temperature description of the thermodynamic functions during the assessment of high temperature thermodynamic data.
8.1
Procedure in the absence of cP&Q
If the necessary cP data are not experimentally available, the protocol suggested for stable phas#escannot be implemented and some alternative proposals are therefore necessary. In the absence of any other information, it has been common practice to set the magnetic properties of metastable phases as either identical to the values for the stable phases, or to zero. However this is clearly liable to significant error. As first principle calculations for stable phases now yield values for the Bohr magneton numlber p that are close to the values obtained by experiment, increasing confidence is given to the values calculated for metastable structures. As Debye temperatures can also be derived by the same route, this would mean that there can be significant input into leading terms in the generalexpression cP =cy +f(B,)+aT+bT’ from first princple calculations, even in the absence of experimental cP information. However, there is as yet no theory for calculating the critical temperature for magnetic disordering, so empirical methods have still to be used to obtain an estimation from the Bohr mag,neton number WTau]. It is therefore unavoidable that there will be more divergence between starting values entered from first principle calculations and the final values that may emerge from the optimization process. Nonetheless it is considered this is a better option than the :previously mentioned tendency to set values either equal to those of the stable allotrope or to zero.
Table IV combines data for both stable and metastable phases as well as data relating to altelmative magnetic states for a given phase. The occurrence of multiple states in practice will depend on the energy gap between the states, which can be estimated from [Fig. 151 for some
513
D. DE FONTAINE eta/.
of the cases (a more exhaustive treatment is considered to be beyond the remit of this report). However, several interesting features emerge from the data compiled in this table. 1. It is crystal crystal should
clear that the calculated J3values for metastable phases are generally different for each structure and vary between ferro- and anti-ferromagnetic configurations in a given structure. The assumption of identical values for stable and metastable allotropes therefore be considered the exception rather than the rule in most circumstances.
2. The values of l3 estimated indirectly by Weiss have been confirmed to a surprising degree by first principle calculations. The combined picture also confirms previously suggested systematic trends which should help reliable interpolation (Fig. 16). 3. There is a significant difference between the 0 values that have been extracted from the deconvolution of cP measurements and the values derived independently from magnetic measurements. There is also a significant spread between the 0 values obtained by using different methods to extract the magnetic specific heat capacity [64Lyt, SlMes, 85Chu] (as already mentioned in section 3). The earlier compilations of 13values to be used within the suggested formalism [87Chi] were in line with experimental I3 values and did not make the value of 13an adjustable parameter in the optimization process. 4. Some of these differences can be rationalised by the effect of temperature on mixed magnetic states, as most of the data in Table IV essentially refers to zero K. This is almost certainly the case for Cr and Mn [72Wei, 79Wei, 79Ben]. In the case of Co, the hexagonal phase is a close competitor near the Curie temperature and high stacking fault densities may therefore also lead the unusual effects [95Mio]. Other differences may arise because of the effects of magnetic forces on the Debye temperature [74Woh, 64Lyt]. Another possible source of error is the assumption that there are no orbital contributions, but the latter appear to be very small [9OEri]. 5. Although global optimization is recommended, the consistency of Table IV confirms that theoretical B values should be given a substantial weighting in any optimization procedure.
9. Overall conclusions The following procedure is suggested for the determination of ca
including the global
optimization of c,: l Equation (12) is used to describe the total cP in accordance with the most recent recommendation (see report of Group 1): cP =f(B,)+aT+bT’+cpmaS l
l
+c”
(12)
In order to comply with the high temperature c,-data of some particular cases Group 1 suggested to allow for a temperature dependence of f&,.The two Debye temperature approach discussed here (see Eq. (8)) should also be considered as an alternative. The first three terms in (12) are nonmagnetic contributions and require three parameters 8,, a, and b to be fitted in an optimization. The preferred equation for the magnetic term is the extended version of Eq. (3) as already suggested in section 4 (at least four terms, see appendix A). T, is fixed from
A-TRANSITIONS
l
experiment. According to the present recommendation not only the parameters K- and K+, but also m and n should be allowed to vary and thus be included in the optimization scheme. This optimization can be made with the contraints (A-9) and (A-10) (see appendix A) (P>+2 ~~ This ensures (P).(z-1) * that independent magnetic properties like (p) are included in the optimization and can imposed by the relations ASW(exp) = Rln((P)+ 1) and f =
l
act as a constraint against arbitrary changes. c” should be set to zero as the fist approximation.
The optimization may then have to be repeated with different weightings to obtain a meaningful overall picture, as is the standard practice in phase diagram optimization. The description of the magnetic effect has to be extended to include the effect of high spin numbers.
515
516
D. DE FONTAINE
et al.
10. References 42sto: SIT& 52Bus: 52Vle: 54Gri: 55Bro: 55Cat: 55Tau: 55Tom:
J.W. Stout and H.E. Adams, J. Amer. Chem. Sot. 64 (1942) 1535 S.S. Todd and K.R. Bonnickson, J. Amer. Chem. Sot. 73 (195 1) 3894 R.H. Busey and W.F. Giauque, J. Amer. Chem. Sot. 74 (1952) 4443 J.H. Van Vleck, Rev. Mod. Phys. 25 (1952) 221 M. Griffel, R.E. Skochdopole and F.H. Spedding, Phys. Rev. 93 (1954) 657 H.A. Brown and J.M. Luttinger, Phys. Rev. 100 (1955) 685 E. Catalano and J.W. Stout, J. Chem. Phys. 23 (1955) 1284 K.J. Tauer and R.J. Weiss, Phys. Rev. 100 (1955) 1223 J.R. Tomlinson, L. Domash, R.G. Hay and C.W. Montgomery, J. Amer. Chem. sot. 77 (1955) 909 55Zen: C. Zener, Trans. AIME 203 (1955) 619 56Boz: R.M. Bozorth in Ferromagnetism, van Nostrand (1956) J.A. Hofman, A. Paskin, K.J. Tauer and R.J. Weiss, J. Phys. Chem. Solids 1 56Hof: (1956) 45 R.J. Weiss and K.J. Tauer in: Theory of Alloys Phases, ASM Symposium, pub. 56Wei: ASM (1956) 290 57Pas: A. Paskin, J. Phys. Chem. Solids 2 (1957) 232 58Wei: R.J. Weiss and K.J. Tauer, J. Phys. Chem. Solids 4 (1958) 135-143 6OGoo: J.B. Goodenough, Phys. Rev. 120 (1960) 67 62Shi: G. Shirane and W.J. Takei, J. Phys. Sot. Japan 17 (Suppl. B-III) (1962) 35 63Kau: L. Kaufman, E. Clougherty and R.J. Weiss, Acta Met. 11 (1963) 323 63Wei: R.J. Weiss, Proc. Phys. Sot. 82 (1963) 281 64Bra: M. Braun, Dissertation, Universitlt Kiiln, 1964 64Lyt: J.L. Lytton, J. Appl. Phys. 35 (1964) 2397-2406 65Bra: M. Braun and R. Kohlhaas, phys. stat. sol. 12 (1965) 429 67Hil: M. Hillert, T. Wada and H. Wada, J. Iron Steel Inst. 205 (1967) 539 72Wei: R.J. Weiss, Phil. Mag. 26 (1972) 261 74Woh: E.P. Wohlfarth, Phys. Stat. Sol. (a) 25 (1974) 285-291 75Gri: G. Grimvall and I. Ebbsjo, Physica Scripta 12 (1975) 168 75Ind(a): G. Inden, Z. Metallk. 66 (1977) 725 75Ind(b): G. Inden and W.O. Mayer, Z. Metallk. 66 (1975) 725 76Gri: G. Grimvall, Physica Scripta 14 (1976) 63 761nd: G. Inden, Proc. Project Meeting CALPHAD V, Dusseldorf 1976, p. IIIA-1 77Ben: W. Bendick, H.H. Ettwig, F. Richter and W. Pepperhoff, Z. Metallk. 68 (1977) 103-107 77Mio: A.P. Miodownik, Calphad 1(1977) 133-158 77Roy: D.M. Roy and D.G. Pettifor, J. Phys. F 7 (1977) 183 78Ben(a): W. Bendick, H.H. Ettwig and W. Pepperhoff, J. Phys. F 8 (1978) 2525 78Ben@): W. Bendick and W. Pepperhoff, J. Phys. F 8 (1978) 2525 78Chi: S. Chikazumi and M. Matsui, J. Phys. Sot. Japan 45 (1978) 458 78Hil: M. Hillert and M. Jarl, Calphad 2 (1978) 227 78Mio: A.P. Miodownik, The concept of 2 gamma states in: Physics and Application of Znvar Alloys, Ed. Saito, Vol. 3 Maruzen (1978) pp. 288-309 79Ben: W. Bendick and W. Pepperhoff, J. Phys. F 9 (1979) 2 185 79Nis: T. Nishizawa, M. Hasebe and M. Ko, Acta Metall. 27 (1979) 817 79Wei: R.J. Weiss, Phil. Mag. B 40 (1979) 425-428
A-TRANSITIONS
79Wil: 80Leg: 80Mio: 811nd: 8 1Mes: 82Ben: 83Ros: 85Chu: 85Sun: 87Chi: 88Mor: 90Dav:
90Eri(a): 90Eri(b): 90Mor(a): 90Mor(b): 90Mor(c): 91Din: 91Ind: 93P,sa(a): 93&a(b): 94Fri: 95hIio: 951nd: 95Sch:
I.S. Williams, E.S.R. Gopal and R. Street, J. Phys. F 9 (1979) 431 S. Legvold in Ferromagnetic Materials, Vol. 1, E.P. Wohlfarth (Ed.), NorthHolland Publ., Amsterdam 1980, p. 184-295 A.P. Miodownik and M. Hillert, Calphad 4 (1980) 143 G. Inden, Physica B 103 (1981) 82-100 P.J. Meschter, J.W. Wright, C.R. Brooks and T.G. Kollie, J. Phys. Chem. Sol. 42 (1981) 861-871 W. Bendick and W. Pepperhoff, Acta Met. 30 (1982) 679 J. Rosen and G. Grimvall, Phys. Rev. B 27 (1983) 7199 Y.Y. Chuang, R. Schmid and Y.A. Chang, Met. Trans. 16A (1985) 153-165 B. Sundman, B. Jansson and J.-O. Andersson, The THERMO-CALC databank system, CALPHAD 9 (1985) 153-190 Chan-peng Chin, S. Hertzman and B. Sundman, TRITA-MAC 0203, revised Aug. 1987 V.L. Moruzzi et al., Phys.Rev. B 38 (1988) 1613-1520 R.H. Davies, A.T. Dinsdale, S.M. Hodson, J.A. Gisby, N.J. Pugh, T.I. Barry and T.G. Chart, MTDATA-The NPL Databank for Metallurgical Chemistry, in Proc. Conf. User Aspects ofPhase Diagrams, F.H. Hayes (Ed.), Institute of Materials, London 1990, p. 140-152 0. Eriksson et al., Phys. Rev. B 42 (1990) 2707-2709 G. Eriksson and K. Hack, Met. Trans. 21B (1990) 1013-1023 V.L. Moruzzi, Phys. Rev. B 41(1990) 6939-6946 V. L. Moruzzi et al., Phys. Rev. B 43 (1990) 8361-66 E.G. Moroni et al., Phys. Rev. B 41(1990) 9600-02 A.T. Dinsdale, SGTE Data for Elements, CALPHAD 15 (1991) 317 G. Inden, Stand. J. Metall. 20 (1991) 112 T.Asada, K. Terakura, in: Computer Aided Innovation of New Materials, M. Doyama et al. (Eds), Elsevier Publ., Vol. I (1993) 169-172 T. Asada, private communication to A.P. Miodownik K. Frisk and C. Qiu, Z. Metallk. 85 (1994) 60-69 A.P. Miodownik, unpublished work (in preparation) G. Inden, unpublished work (in preparation) C.G. Schon and G. Inden, submitted to Scripta Met.
517
D. DE FONTAINE
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et al.
Aonendices
The thermodynamic functions are given with reference to the complete paramagnetic state for which T== stands. Definitions: R=gas constant T=$
(or +) N
C
1 A=L+p+-+-+... l+m 3(1+3m) 1 B=L+-+-+-+... l-n 3(1-3n)
D=i
n(
1 5(1+5m) 1 5(1-5n)
1 7(1+7m) 1 7(1-7n)
l+$+&+-&+...
ai =2.i-1
11.1 Anpendix A Magnetic contributions to thermodynamic functions based on Eq. (3)
c, =2.K-.R.
=-2.K-
1 1 1 z~+--z~“‘+-~~‘“+-~~~+... 3 5 7
.R.‘&.
T1+Sll?
A-<-z_ l+m
+2.K+.R.T,
(A-l)
3(1+3m)
----...
5(1+5m)
T
*+7m
7(1+7m)
(A-2)
.B
Smag(T)-Srrug(=)= C_~_??-~-r?m-.~. m 9m 25m
(A-3) 49m
A-TRANSITIONS
519
Gw(T)-Gw(=)=Hw(T)-T.Sw(T)-(HMg(m)-T.S~(=+ =-2.R.Tc _2.K-
.(K- .A-K+ .R.Tc .i
.D)q(A-4)
a,*m(l+ aim)
1 ~-n+T~-3"+;+'
=:z.K+.R.
.(K- .C-K+
Z’+a’m
id
c;
.B)+2.R.T,
+1-T-7n+
(A-3
...
7
P
=z.K+.R.‘Q.
-+-+-+-+...
l-n
z I-3” 3(1-3n)
Smg(T)-S*(w)=-2.R.K+.
G’“‘~(T)-Gmg(m)=2.K+
z I-5n 5(1-h)
z l-70 7(1-7n)
n
.R.‘&
(A-7)
“-”
.i i=l
q*n(l - cqn)
K+.B
f=
(A-6)
K+.B-K-.A
(A-8)
(A-9)
(A-10)
J-LL&wndix
E
Magnetic contributions to thermodynamic functions based on Eq. (4)
211 -_ C, :=
R.k-.~.exp(-p(l-7))
(B-l)
H”@(T)-Hmg(=)=
S”“(T)-S’(~)=~exp(-p(l_~))-R. P
(B-3)
520
D. DE FONTAINE
eta/.
Gw (T) - Gmg (w) =
ci = R.k* sTmexp(q(l-T))
(B-5)
exp(q(1 - 2)) Gw(T)-Gm(-)=-k”q~‘Tc
SW(T) - SUE
= -y
exp(q(l-7))
e.xp(q(l - 2))
( I f=;(l-::.p*)t4&.l(l+;)
(B-6)
(B-7)
(B-8)
k’ I+’
ln((p)+l)=;+$-$e-.
(B-9)
(B-10)
521
A-TRANSITIONS
12.
Tables
Table I Integrated quantities below (-) and above (+) T, as obtained from treatments (2) and (3) m
Eq.#
AI-I-
AS-
(2) (3) (2) (3) (2) (3)
RTc 1.233 1.133 0.778 0.728 0.571 0.538
R 2.199 2.093 1.097 1.045 0.729 0.696
-
-
1 1 2 2 3 3
Parameters used for calculation of cy
Eq.#
AH’
AS+
(2) (3) (2) (3) (2) (3)
RTc 0.346 0.333 0.235 0.226 0.178 0.170
R 0.245 0.232 0.183 0.174 0.147 0.139
Table II with Eq. (3) in accordance with the SGTE database [9 lDin]
Table III Debye temperatures and spin values of the elements in various compounds [56HofJ %mpound - WL MnF, NiF, NiCl, NiO MnO Fe0 -
0R,/K
e;rK
P&n
T,
Structure
240 306 325 425 350 450
625 700 390 900 700 600
5 2 2 2 5 4
66.5 73 53 523 117.8 191
SnO, SnO, CdCl, NaCl NaCl NaCl
D. DE FONTAINE
522
et al.
Table IV Comparison of magnetic moments for selected elements Data from SGTE (Scientific Group Thermodata Europe), first principle calculations (FPC) anti-ferromagn. 8 in & Element Cr Cr Cr Cr Cr Cr Cr Mn Mn Mn Mn Mn Mn Fe Fe Fe Fe Fe Fe Fe Fe co
co co co co co co
co Ni Ni Ni Ni Ni Ni Ni Ni Ni
Source J87Chi 91Din (SGTE) 93Asa(b) (FPC) 88Mor, 90Mor(a)-(c) 79Wei (EXP magn.) (EXP cp ) 87Chi 91Din (SGTE) 93Asa(a)-(b) (FPC) 88Mor, 90Mor(a)-(c) 79Wei 1(EXP mag) /87&i 91Din (SGTE) 93Asa(a) (FPC) 88Mor, 90Mor(a)-(c) 79Wei (EXP mag) 85Chu I64Lyt I87Chi 91Din (SGTE) 93Asa(b) (FPC) 88Mor, 90Mor(a)-(c) 78Mio (estim.) (EXP mag) 85Chu 164Lyt I87Chi
88Mor, 90Mor(a)-(c)
BCC
FCC
0 0
0
1.0 0.9
Of4
BCC 0.4 0.008 010.5 010.6 010.7 0.4 O.O+
FCC 0.82 0.82 013.0
CPH 0 0 0
-
-
0
0.09
2.0
2.76
0.62 2.25
0 0.15
2.410 2.3 0.70 0.70 1.21 0.510 0.40 0.70 0.57
0
0.20 -
0
1.2
CPH 0 0 0
4.5 (1) 2.22 2.22 2.32 2.2 2.22 2.05 1.03 1.80 1.35 1.80 1.70 1.70 2.00
0.85 0.84 0.50 0.30 0.85
(1) 2.56 2.7 2.6 -
0 0 2.56
1.70 1.35 1.70 1.80 1.80 1.80
1.70 1.35 1.61
0.62 0.52 0.66
0.62 0.25 0.58 0.53
1.55 1.75
-
1.0
0 0 0
0
1.70 0.89 1.21 0.52 0.60
-
0
0.1
0.25
523
h-TRANSITIONS
Magnetic specific heat capacity versus reduced temperature according to Eq. (2) for different values of exponents m and n using K- = 0.89486, and K’ = 0.29830. These numerical values have been selected in order to provide a given step in cy at T, in the treatments (3) and (4) of Figs 2-3. Fig. 1a linear representation IFig. lb double-log representation
Comparison between Eq. (2) (solid lines) and (3) (broken lines) in linear representation for various values of m and n. The coefficients K* are given the same values as in Fig. 1 so that Eq. (3) yields the limiting values c;(Tc) = 3R and ci(Tc) =R. The series expansion (3) has been extended to include a fourth term. !Fig. 2a low temperature branch IFig. 2b high temperature branch
Comparison between Eq.(3) and Eq. (4) for the same situation as in Fig. 2, i.e., with limiting values c;(Tc) =3R and ci(Tc) =R, K* as in Fig. 1 and k-=3, k’=l. The exponents used are m=3, n=3 in Eq. (3) and p=4.25, q=3.8 in Eq. (4). IFig. 3a linear representation :Fig. 3b double-log representation
Magnetic specific heat capacity of bee-Fe. Deconvoluted
cy
values from [64Bra].
Solid line: calculated with Eq. (3) using the parameters in Table II. IFig. 4a. linear representation. IFig. 4b double-log representation.
Magnetic specific heat capacity of feeCo.
Deconvoluted
CT values from [64Bra].
Solid line: calculated with Eq. (3) using the paramaters in Table II. Fig. 5a linear representation. Fig. Sb double-log representation.
Magnetic specific heat capacity of fee-Ni. Deconvoluted cy values: l [64Bra], A [81Mes]. Solid line: calculated with Eq. (3) using the parameters in Table II. .Fig. 6a. linear representation IFig. 6b double-log representation
D. DE FONTAINE et al.
524
Magnetic specific heat capacity of Gd. Deconvoluted cy
values from [54Grifl.
Fig. 7a linear representation Fig. 7b double-log representation
Magnetic specific heat capacity of beeCr. Experimental total c,, -values from [79Wil].
Magnetic specific heat capacity per mole (formula unit) of MnO and NiO. Deconvoluted cy -values [95Ind] derived from [SlTod] (MnO) and [55Tom] (NiO) using the Debye temperatures in Table III. Fiy 1Q Magnetic specific heat capacity per mole (formula unit) of Fe,,O. Deconvoluted cy -values [95Ind] derived from [SlTod] using the Debye temperatures in Table III. Fig. 11 Magnetic specific heat capacity per mole (formula unit) of NiF, and MnF,. Deconvoluted c,” -values [95Ind] derived from [55Cat] (NiF,) and [42Sto] (MnF,) using the Debye temperatures in Table III. Fig. 12 Magnetic specific heat capacity per mole formular unit of NiCl, Deconvoluted c,“-values [95Ind] derived from [52Bus] using the Debye temperatures in Table III. Fig. 13 Magnetic specific heat capacity of MnO plotted in a linear representation. Data as in Fig. 9. Fig. 14 Relative position of energy levels according to the models of Weiss [63Wei], Hillert [8OMio], Chikazumi [78Chi] and Pepperhoff [78Ben(a)]. Fig. 15 Variation of the energy difference between competing states with electron concentration.
Variation of the magnetic moment 8 with crystal structure and position in the periodic table. (FM: ferromagn., AF: antiferromagn., MAG: from magnetic measurements, Cp: from specific heat capacity)
A-TRANSITIONS
Magnetic specific heat capacity versus reduced temperature according to Eq. (2) for different values of exponents m and n using K- = 0.89486, and K’ = 0.29830. These numerical values have been selected in order to provide a given step in c w p at T, in the treatments (3) and (4) o:I Figs 2-3. Fig. la linear representation Fig. lb double-log representation
525
D. DE FONTAINE
526
et al.
4-
a)
p! a VP
r ‘T/T=
T=T/T
C
Comparison between Eq. (2) (solid lines) and (3) (broken lines) in linear representation for various values of m and n. The coefficients K’ are given the same values as in Fig. 1 so that IQ. (3) yields the limiting values c;(Tc) = 3R and ci(T,-) =R. The series expansion (3) has been extended to include a fourth term. Fig. 2a low temperature branch Fig. 2b high temperature branch
527
A-TRANSITIONS
a.)
-~~ 2
0
r=T/Tc
Cotnparison between Eq.(3) and Eq. (4) for the same situation as in Fig. 2, i.e., with limiting values $(T,)=3R and ci(Tc) =R, K* as in Fig. 1 and k-=3, k+=l. The exponents used are m=3, n=3 in Eq. (3) and p=4.25, q=3.8 in Eq. (4). Fig. 3a linear representation Fig. 3b double-log representation
D. DE FONTAINE
528
et al.
b)
r=T/Tc
Fig. Magnetic specific heat capacity of bee-Fe. Deconvoluted c,“” values from [64Bra]. Solid line: calculated with Eq. (3) using the parameters in Table U. Fig. 4a. linear representation. Fig. 4b double-log representation.
A-TRANSITIONS
1
a)
0
b)
loo t
0.1
co
;
1
I
10
T’T/Tc
Magnetic specific heat capacity of fee-Co. Deconvoluted
cy
values from [64Bra].
Solid line: calculated with Eq: (3) using the paramaters in Table II. Fig. 5a linear representation. Fig. 5b double-log representation.
D. DE FONTAINE
530
et al.
0
7=T/T
C
b)
1 7=T/Tc
Magnetic specific heat capacity of fee-Ni. Deconvoluted cPW values: l [64Bra], A [SlMesJ. Solid line: calculated with Eq. (3) using the parameters in Table II. Fig. 6a. linear representation Fig. 6b double-log representation
531
A-TRANSITIONS
a)
8
Gd
i
Tc=BOK
i
r=T/T
C
10
Gd
P 3 E .=1.
i..
1
.
H o=
.
.
. ,
0.1
0.1
I
1
10
r =T/Tc
Magnetic specific heat capacity of Gd. Deconvoluted c,” values from [54Grifl. Fig. 7a linear representation Fig. 7b double-log representation
532
D. DE FONTAINE
et a/.
35
Cr TN=311K
. 30 P 5 E zd
0 0
O-25
20 0.6
1.4
1
r=T/T
N
Magnetic specific heat capacity of bee-Cr. Experimental total cP -values from [79Wil].
0.1
1
10
r=T/TN
Magnetic specific heat capacity per mole (formula unit) of MnO and NiO. Dcconvoluted c? -values [95Ind] derived from [51Tod] (MnO) and [STom] (NiO) using the Debye temperatures in Table III.
A-TRANSITIONS
533
Magnetic specific heat capacity per mole (formula unit) of Fe,,,,O. Deconvoluted c,“‘-values [95Ind] derived from [SlTod] using the Debye temperatures in TabIe III.
loo L
0.1
NiF2 1 : l ......._......_...... -I._._......_.__.._._,.__........,..,..................... TN=73.2K;
0.01 0.1
1 7 =
a
10
T/TN
Fig. 11 Magnetic specific heat capacity per mole (formula unit) of NiF, and MnF,. Deconvoluted c,” -values [951nd) derived from [55Cat] (NiF,) and [42Sto] (MnFJ using the Debye temperatures in Table III.
534
D. DE FONTAINE et al.
10 l-
NiClp a 0
rN=53K
l
B
8
?
0
‘ij
E
8
8
2 P
8
6a 0
l
8
0.1
10
1 r=T/TN.
EiLK Magnetic specific heat capacity per mole formular unit of NiCl, Deconvoluted cy -values [95Ind] derived from [52Bus] using the Debye temperatures in Table III.
0
1
r=T/T
2 N
Magnetic specific heat capacity of MnO plotted in a linear representation. Data as in Fig. 9.
A-TRANSITIONS
535
HILLERT
WEISS
-PEPPERHOFF
Relative position of energy levels according to the models of Weiss [63Wei], Hillert [80Mi0], Chikazumi [78Chi] and Pepperhoff [78Ben(a)].
Energy dlfterence between Two Gamma States 20
- FM Ground State 10 ‘; B E 3
..= 0.
i$ (10) -
lnvar Composition
C Q (20)
AF Ground State
-
I @Ok.5
&
7
,
I 7.5
8
,
,
8.5
9
electron-atom 63wEI
78MlO
WpI$~efl “*BI’”
77SEN
“p.“”
,
,
,
9.5
10
ratio
&EN(b)
7SSEN(J
QOMOR(b)
Elac~M
Sp.$sd
F,’
BOMOR(O)
%?A.%
1
‘.’
Variation of the energy difference between competing states with electron concentration.
536
D. DE FONTAINE
et al.
Variation of Magnetic Moment in the periodic table
0
V
CR
MN
FE
co
NI
CU
Element CPH
Fip. 16 variation of the magnetic moment I3with crystal structure and position in the periodic tat&,