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CELLULAR MODEL OF ALLOYS AND ORDER OF THE MAGNETIC TRANSITIONS IN INVAR
L. Billard, M. Natta
To cite this version:
L. Billard, M. Natta. CELLULAR MODEL OF ALLOYS AND ORDER OF THE MAGNETIC TRANSITIONS IN INVAR. Journal de Physique Colloques, 1974, 35 (C4), pp.C4-157-C4-161.
�10.1051/jphyscol:1974428�. �jpa-00215619�
JOURNAL DE PHYSIQUE CoIloque C4, supplbment au no 5, tome 35, Mai 1974, page C4-157
CELLULAR MODEL OF ALLOYS AND ORDER OF THE MAGNETIC TRANSITIONS IN INVAR
L. BILLARD and M. NATTA C. E. A.-C. E. N. Grenoble, DRF/G/Physique du Solide,
BP 85 Centre de Tri, 38041 Grenoble Cedex, France
RbsumC. - Nous discutons la question de l'ordre des transitions magnktiques dans I'INVAR dans le cadre de la mkthode de Smoluchowski que nous avons interpretbe [4,12] comme une theorie de Landau dans le but d'expliquer les spectres Mossbauer.
La comparaison des rksultats avec l'expbience confirme l'hypothese de R. J. Weiss d'une transi- tion du premier ordre au voisinage de 30 % de Nickel ; cependant les mesures de chaleur specifique indiquent qu'il est possible que le dckouplage des deux bandes ne soit pas complet pour les concen- trations leg6rement supkrieures a 30 %.
Abstract. - We discuss the order of the magnetic transitions in INVAR in the frame-work of the Smoluchowski [6] method which we interpreted [4, 121 as a Landau theory in order to explain the Mossbauer spectra [3, 41. The comparison of the results with experiments confirms the R. J. Weiss [7] hypothesis of a first order transition around 30 % of Nickel ; however specific heat data indicate the possibility of occurrence at the Fermi level of a high density of states for the majo- rity spin band [16].
Introduction. - Most of the models for disordered magnetic alloys insist now on the importance of local environment effects [I, 2, 31. This fact is confirmed by the Mossbauer spectroscopy [3, 41. On the other hand most of the calculations are made in the rigid band or coherent potential approximations [5] which suppose that the alloy is homogeneous and so give the free energy fE(x, m) of what we shall call the hypothetical homogeneous alloy (abbreviation : HHA) at concen- tration x[m is the mean spin polarisation per atom ; M = gp, ml2 being the corresponding magnetization ;
I m I varies between 0 and its maximum value m,(x) on the Slater-Pauling curve]. When one attemps to have a global description, as usual, the fluctuations (here of concentration) are the hard things to intro- duce.
1. The cellular model of alloys. - In order to understand the macroscopic magnetization
of Fe(,-,,Cot Smoluchowski [6] proposed a rough but tractable approximation which contains two ideas :
1) The disordered alloy A,-,Bc is sampled into cells of n atoms ; the distribution p(x,c) of the B
concentration xi in each cell i is then roughly a gaussian :
p(x, c) = l j J 2 c ~ exp[(x - c ) '12 a'] (1) with : c2 = c(1 - c)/n.
2) One attributes to the cell i the behaviaur of the HHA at the local concentration xi. This second approximation is equivalent to write the total free energy FT as :
By supposing that for b. c. c. Fe,-,Cot system the HHA is on the Slater-Pauling curve, Smoluchowski obtained the best fit with the experimental data for n = 15 atoms per cell, which corresponds to an atom with its first and second neighbours. In the case of INVAR (i. e. Fe,-,Nit in the f. c. c. phase), with the R. J. Weiss hypothesis [7] (A figure 1) that the HHA undergoes a first order transition between a strong ferromagnetic state F and a paramagnetic P (or possibly antiferromagnetic AF) state at a critical concentration c,
-
0.29, Kachi and Asano obtained the best fit for n = 60 atoms per cell ; this model predicts that the distribution p ( H ) of hyperfine fieldsArticle published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974428
C4-158 L. BILLARD AND M. N,ATTA
20 30 40 50 60
% of Nickel
FIG. 1. - Possible variations of the ground state magnetization of the hypothetical homogeneous alloy (NHA in the tex ).
on the iron nucleus has a strong maximum at H = 0.
The analysis by Rechenberg [3, 4) of the Mossbauer spectra obtained on the alloy at 32 % of Nickel shows this peak but with a shift around 50 k~ which suggests that the paramagnetic regions are polarized by the strong ferromagnetic regions. This leads to introduce in the free energy a coupling term which gives a very natural expression of Landau type :
The average coupling term C i s related to the interato- mic exchange which can be evalutated from the spin wave dispersion constant D ; for a sampling into cubes of lattice side L one finds :
where m(c, 0,O) is the macroscopic magnetization of the ground state (I). In the FeCo system the coup- ling term is not important since at T = 0 K all the mi values which stay on the Slater Pauling curve are comparable. In INVAR that we shall discuss now, since there are large fluctuations the situation is different, so that we must estimate explicity the form of f;(x, m).
2. The INVAR problem. - The magnetic INVAR problem stays, at the microscopic scale, in the behaviour
of f:(x, m) ; it is to understand if the transition from strong ferromagnetism to paramagnetism shown on figure 1 is either a first order (A) or second order (B) transition. On the macroscopic scale, there is no problem since, because the variation of m(c, 0, 0) is not infinitly steep, the transition is of second order.
In fact the question is whether there are Schottky anomalies on a more or less extended scale or not.
2.1 The proposition 2.1 is due to R. J. Weiss [7]
and has been explained in the rigid band approxima- tion by Shimizu and Hirooka [ l l ] ; it is comparable with the Jaccarino-Walker model [l] for impurities of Fe in Nb,-,Mo,. In case 2.1 as the Stoner condi- tion is not satisfied, the analysis of the local para- magnetic susceptibility is useless [3]. In ref. [4] and [12]
we have developped the theory of INVAR along this hypothesis ; in the discussion (9 3) we shall quote some further results.
2.2 The proposition 2.2 due to Mathon and Wohlfarth [I31 and based on the data of m(c, H, T) is that INVAR is a weak ferromagnet with a critical concentration c,
-
0.25. The point is that the theory of weak ferromagnetism is more than a thermodyna- mic theory [lo].2.3 There is a third possibility 2.3 comparable with 2.1, where the first order transition is from an intermediate weak ferromagnetic state f.
3. Discussion. - Generally all the macroscopic data are compatible with the hypothesis 2.2 of weak ferromagnetism i2) ; so our purpose is mainly to look if they can also be fitted in hypothesis 2.1, which we shall work by calculating the free energy f: in the
(1) Our description fails in the case of antiferrornagnetism. In FIG. 2. - Model density of states N(E)and Curie temperature Tc the INVAR case D has been measured on the Slater Pauling calculated in the Hartree Fock and rigid band approximations.
curve by Hatherly et al. [9] down to c = 0.4 and can be approxi- The tricritical points T separate second order from first order
mated by : displacive transitions (Q is a quadricritical point).
D(ms) = - 0.05
+
0.55 c (eV b z )(2) Which is not so clear since the spin waves are generally In (4) with that value of D, one must replace m(c, 0,O) by m ~ ( c ) . absent in these treatments.
CELLULAR MODEL OF ALLOYS AND ORDER OF THE MAGNETIC TRANSITIONS IN INVAR C4-159
rigid band and Hartree-Fock approximations with the simple density of states N(E) shown on figure 2.
This choice greatly simplifies our calculations [4]
which as a function of the intra-atomic Coulomb repulsion U gives the situations 2.1 and 2.2 but not 2.3 ; however its defect which is also present in the choices of Hasegawa and Kanamori [5] is that the real density of states has probably another peak at low energy which turns out to be important in the discussion of the specific heat. We shall compare the three different possibilities and quote the results in the table I in order to choose between them.
Hypothesis C"
CPA
m(c, 0, T ) and am/aP X
cell dimension n
transports P ( H )
Hysteresis
3.1 SPECIFIC HEAT. - Our result for the specific heat per atom C, is :
where
x,,
is the susceptibility in the paramagnetic state [12]. The last term is a Schottky term which is analogous to those occurring in different other disor- dered systems [14, 151 ; because of the n2 factor in the denominator (n 12), it is in fact negligeable so that the specific heath yT, as usual, is a direct estimation of the density of states at the Fermi-level. In our result the y coefficient starts to increase below c < 0.4 which is 10 % below the data.This is in part due to our choice of N ( E ) , but it is difficult to give a conclusion. On the contrary hypo- thesis 2.2 and 2.3 are compatible with the data [16].
3.2 CPA AND RIGID BAND CALCULATION. - The parameters U , and eB of the CPA calculation of Hasegawa and Kanamori [5] were fitted in order to fit the specific heat and the bulk magnetization down to c = 0.4, but without taking into account the local fluctuations. So the result is not a distinct proof from the specific heat for appearance of weak ferromagne- tism. However their result for the displacive Curie temperature which was obtained with the use of the paramagnetic susceptibility xP(T) is very similar with the result of the rigid band calculation (Fig. 2) if one adds to the branch of second order transition obtained
with xP(T) a branch of 1st order transition which cannot be obtained from x,(T) (see 2.1). This indicates that the complete diagram of their CPA calculation is probably of type 2.3.
3 . 3 MAGNETIZATIONS, CELL DIMENSION AND SUS- CEPTIBILITIES. - The data for the spontaneous macroscopic magnetization m(c, 0, 0) can be fitted in the three cases by adjusting co and n. However the num- ber of atoms per cell is only comparable with that of Smoluchowski [6] in case 2.1 or 2.3. One must notice that this reduction of the result of Kachi and Asano [8]
(n = 60) to n N 14 is due to our introduction of the coupling term. At finite T , in order to fit the data on bulk samples of Crangle and Hallam [17], one must introduce in case 2.1 the spin waves [12] ; the fit is then quite good.
Figure 3 gives our results for the field ( x ) and pres- sure (- d log mldp) magnetic susceptibilities. They are in good agreement with the data, although
x
is somewhat too low for c > 0.4 ; this can be discussed on the same line than C,. The data of Rode and Krynetskaja [24] show that the maximum of
x
is around 30 % of nickel ; this disparages the model 2.2 where X , is infinite for x-
25 %.FIG. 3. -Pressure andfield magnetic susceptibilities n = 18 and co = 0.29. The curve - d log mldp is fitted with d log W/da = - 2 A-1 ( W : bandwidth and a : lattice parame- ter) ; thedata are. [18], [19], A [20],
+
[21] ; unit : 10-5 atm-1 the magnetic field susceptibility data are V [22] and 0 x [23]unit : 10-3 u.e.m./mole. The maximum value of x is 9 x 10-3 at co.
3.4 TRANSPORT PROPERTIES. - The transport properties discussed recently by M. C. Cadeville and Loegel [16] indicate that the two s, + d, channels
(o = f or) 1 are open below c = 0.4. Although nobody knows what are the cross-sections, this can be explain- ed in the three situations.
3.5 DISTRIBUTIONS OF MAGNETIZATIONS AND HYPER- FINE FIELDS. - AS Our model [4, 121 was developed
C4-160 L. BILLARD A N D M. NATTA
in order to explain the results of Rechenberg [3, 41 it is not surprising that the distribution p(M) of the local magnetizations (Fig. 4) is in qualitative agree- ment with that p ( H ) of the hyperfine fields. In the case of weak ferromagnetism 2.2, one should not observe a peak around 50 kG but rather a flat distribution.
However such a deconvolution p(H) of the Mossbauer spectra can always be questionned. A careful exami- nation of p(M) (Fig. 4) shows at finite temperature
FIG. 4. - Distribution of local magnetizations. P corresponds to polarized paramagnetic regions and F to strong ferromagnetic
regions. T is due to tunneling states.
the presence of an intermediate third peak T which is due to cells which are tunneling between the states P and F and correspond to the Schottky anoma- lies [14, 151.
3 . 6 HYSTERESIS IN THE MAGNETIC SUSCEPTIBILITY. - In order to have a distinct check for this existence of tunneling states, we made pulsed field experiments on two samples (c = 0.323 and c = 0.35) made of wires of 0.2 mm of diameter immersed in a dielectric.
With this diameter the penetration time of the field is of the order of lo-' s. The rising time up to 270 kG was n/2 o = 50 ps. No relaxation was observed down to 150 K. Figure 5 shows a preliminary result observed at 77 K on the 35 % sample which shows no marten- sitic transformation down to helium temperature.
281
0 1CO 203 250
PULSED FIELD tn kg-
FIG. 5. - Hysteresis of the magnetization in high magnetic pulsed field.
The quantity 6M/AM = 0.1 5
--
oz gives a relaxation time -r = 5 x s, which seems rather high at that temperature but can be understood if one remarks that because of percolation and triggering mechanisms the tunneling regions should contain much more than n = 19 atoms. However this relaxation time did not vary much down to helium temperature and further we did not perform the same experiment on an esta- blished weak ferromagnet like Zn,Zr. So we are not sure of having really observed tunneling states.Conclusion. - In the scope of the cellular model the table I concludes in favor of situation 2.3 ; in agreement with Smoluchowski result [6] this defines a volume 6Q of definition of the order parameter of the Landau theory which contains roughly an atom and its first and second neighbours. However the main result, which is independent of the model, is the confir- mation of the R. J. Weiss hypothesis [7] of two elec- tronic states of iron corresponding to Schottky anoma- lies which do not exist in the theory of weak ferroma- gnetism.
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