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HOMOGENEOUS AND HETEROGENEOUS WEAKLY FERROMAGNETIC ALLOYS
E. Wohlfarth
To cite this version:
E. Wohlfarth. HOMOGENEOUS AND HETEROGENEOUS WEAKLY FERROMAGNETIC AL- LOYS. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-636-C1-637. �10.1051/jphyscol:19711217�.
�jpa-00214043�
JOURNAL DE PHYSIQUE Colloque C 1, supplkment au no 2-3, Tome 32, Fkvrier-Mars 1971, page C 1 - 636
HOMOGENEOUS AND HETEROGENEOUS WEAKLY FERROMAGNETIC ALLOYS
E. P. WOHLFARTH
Department of Mathematics, Imperial College, London, S. W. 7, England
Rkumi3. - Mathon en utilisant la thkorie de Landau des transitions de phase a donne un ensemble de resultats thboriques, concernant I'aimantation M(H, T, c) et les quantitks associees. Ces resultats ne sont applicables que si M est homogkne dans l'alliage de concentration c et pres d'une concentration critique co pour laquelle M = 0. Le degre dIhet6- rogkntite peut ainsi Stre atteint experimentalement et quelques observations dans les alliages Zr I - ~ X ~ Z ~ Z , NisAl, NlsGa,
%,In, Invar, Ni-Cu, NI-Rh et Ni-Cr sont discutees sur cette base. Nous discutons aussi des resultats de neutrons.
Abstract. - Use by Mathon of the Landau theory of phase transitions has given a set of theoretical results, concer- ned with the magnetization M(H, T, c) and related quantities, which are only applicable if M is homogeneous in alloys with concentration c and close to a critical concentration co where M = 0. The degree of heterogeneity may thus be asses- sed experimentally, and some observations on Zrl-zXzZn~, Ni3A1, Ni3Ga, Scdn, Invar, Ni-Cu, Ni-Rh and Ni-Cr alloys are discussed on this bas& Neutron observations are also discussed.
I. Introduction. - The itinerant electron model is difficult to apply to ferromagnetic alloys since in most cases the magnetization M is not homogeneous throughout the material. It should, however, be possible in principle to gain information about 'the heterogeneities of M by comparing the results of experiments with theoretical predictions based on an assumed state of homogeneity. In this paper this idea is discussed in a simple qualitative way by referring to weakly ferromagnetic alloys whose concentration c is close to some critical value c, where M vanishes.
11. Methods of observation of heterogeneity. - 1. MAGNETIZATION M(H, T, c). - Ferromagnetic metals and alloys whose magnetization is low compared to the saturation value corresponding to complete alignment of the spins are called weak itinerant ferromagnets. For these Mathon [I], following the Landau theory, obtained the following expansion for the free energy
+ C T c V M r V M r ] (I) Here f, is a constant free energy density, M(r) the magnetization density depending on position and A, B and C are independent of M. Minimizing (1)
2. M(0, T, c). - The stable solutions of (3) cor- respond to
The index 9 in (5) is analogous to the critical index /3 for the magnetization as a function of temperature near Tc ; p only equals 3 in the mean field approxi- mation underlying the Landau theory with M homo- geneous ; for heterogeneous alloys P # &.
3. Tc(c). - Since [2] T, - M(0, 0, c) in this case it follows that
T,(c) - [c - co(0)]% ; (6)
an explicit relation in terms of band parameters is also known [3].
4. ~ ( 0 , T, c). - From (3)
the index 1 in (7) is now analogous to the indices and y, respectively, for the corresponding tempera- ture case where they have this value only if M is homogeneous. Plots of M(c)' T,(C)', and x(c)-' against c, which may be called Mathon plots, thus give an indication of the appropriateness of the Lan- dau theories with hon~ogeneous magnetization.
5. NEUTRON DIFFUSE CROSSECTION. - Under the present conditions [4] this is given approximately by gives
[A(T, c) + B(T, c) M2(r) + C(T, C) v2] M(r) = H(r) , daldQ - [l + K'/K;J-~ , (8)
(2) where K is the neutron scattering vector and thus giving a relation for M(H, T, c, r), a heterogeneous K ; ~ = 11 - I / - 1
quantity. Only if M is homogeneous does (2) reduce to (9)
the usual expression for weak itinerant ferromagnets [2] is the usual Stoner enhancement factor. Under the (3) same conditions as before, therefore,
A(T,c) + B(T,c) M 2 (H, T, c) = H / M ( H , T, c),
KO - [C - co(0)]% .
giving a linear relation (Arrott plot) between M' (101
and HIM. Neutron scattering experiments on weakly magnetic
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711217
HOMOGENEOUS AND HETEROGENEOUS WEAKLY FERROMAGNETIC ALLOYS C 1 - 637
alloys allow this relation to be tested. They also give a vast amount of other information. For dilute Pd-Fe alloys [5] it was found that the ferromagnetic polarization clouds around the Fe atoms extend a long way into the Pd matrix and thus the measure- ments reveal a very specific heterogeneity in M. Giant moment behaviour and magnetization clouds of this type were also found for weakly magnetic Ni-Cu alloys [6].
111. Some representative results. - 1. Zr, -,X,Zn,.
- These very weak itinerant ferromagnets [2] with X = Ti, Hf give [7] linear Arrott plots over a wide range of x and temperature. The Mathon plot of M(0, 0, c ) ~ against c - co (0) is straight for X = Hf but deviates for X = Ti and x > 0.08. The static magnetic measurements thus point to a large degree of homogeneity in M.
2. Ni3A1 AND Ni,Ga. - According t o stoichiome- try these materials [8] are weakly ferromagnetic or strongly paramagnetic. The Arrott plots are linear over wide ranges of temperature and composition.
Mathon plots for I - ' , M2 and T: fit the linear rela- tionships for homogeneity with respective values of
c, = 74.60, 74.45 and 74.48 % Ni. Similar plots for Ni,Ga are also linear but with an apparent influence of structural defects [9] leading to a break in the linear Mathon plots at stoichiometry.
3. Sc31n. -This material is also regarded as a very weak itinerant ferromagnet [2] but the observed Arrott plots [lo] are very non-linear on either side of T,. Using the present criterion this points to a state of heterogeneity, but it is not clear if this is metallurgical.
The pressure dependence of T, [lo] is also anomalous
in having a sign opposite to that for other similar materials [I I].
4. INVAR ALLOYS. - For Fe-Wi alloys with Ni concentrations c between about 25 and 50 % the Mathon plot for T: against c was found [12], [3] to be remarkably straight implying a homogeneous state of magnetization in these alloys which may be regar- ded as weak itinerant Ill].
5. Ni-Cu, IVi-Rh, Ni-Cr. -- For Ni-Cu alloys near co there is now much evidence for heterogeneities in M. Firstly, the saturation magnetization and T, do not vary according to relations (5) and (6). Secondly, the high field (ferromagnetic) susceptibility [13] by no means varies according to (7). Thirdly, neutron investigations [6] reveal the presence of polarization clouds near Ni atoms. Fourthly, they also [14] reveal that the expression for K O is not as in (10) but rather
For Ni-Cr alloys the situation is slightly less clear.
Susceptibility measurements [15] give quite good agree- ment with (7) above and below c, = 11.0 % Cr.
On the other hand neutron measurements [14] give KO again not obeying (10) but rather a relation like (lo)*
with a critical index about 0.20. They also [4] imply a heterogeneous state of magnetization around the Cr atoms. The present approach is thus inconclusive.
I am grateful to P. F. de Chatel, B. D. Rainford and W. E. Gardner for submitting unpublished data and helpful discussions.
Note added in proof : The papers given a t this Conference by Aldred et al. (Ni-Cr), Fawcett et a].
(ZrZn, and Sc,In) and Yamada et al. (Invar) support fully the philosophy of the present paper.
References
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Roy. Soc., 1968, 303, 127 ; WOHLFAKTH (E. P.), [Ill WOHLFARTH (E. P.), Coll. Int. C. N. R. S., 1970,
J. Appl. phys., 1968, 39, 1061. 188, 363.
[3] MATHON (J.) and WOHLFARTH (E. P.), Phys. Stat. [12] BOLLING (G. F.) et al., Phys. Stat. Sol., 1968,26, 743 ; Sol., 1968, 30, K 131. CRANGLE (J.) and HALLAM (G. C.), Proc. Roy.
[4] Low (G. G.), Adv. in Phys., 1969, 18, 371. SOC., 1963, 272, 119.
[S] Low (G, G.) and HOLDEN (T. M.), Proc. Phys. Soc., [13] FONEK (S.) and MCNIFF (E. J.), J. Appl. Phys., 1970,
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[6] HICKS (T. J.) et al., Phys. Rev. Letters, 1969, 22, 531. [14] RAINFORD (B. D.), private communication.
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