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CONTRIBUTION OF GIANT SPIN CLUSTERS TO THE RESISTIVITY, NEUTRON SCATTERING
CROSS SECTION AND SPECIFIC HEAT IN ALLOYS : APPLICATION TO Ni-Cu
K. Levin, D. Mills
To cite this version:
K. Levin, D. Mills. CONTRIBUTION OF GIANT SPIN CLUSTERS TO THE RESISTIVITY, NEU- TRON SCATTERING CROSS SECTION AND SPECIFIC HEAT IN ALLOYS : APPLICATION TO Ni-Cu. Journal de Physique Colloques, 1974, 35 (C4), pp.C4-213-C4-215. �10.1051/jphyscol:1974438�.
�jpa-00215629�
JOURNAL DE PHYSIQUE Colloque C4, supplkment au no 5, tome 35, Mai 1974, page C4-213
CONTRIBUTION OF GIANT SPIN CLUSTERS TO
THE RESISTIVITY, NEUTRON SCATTERING CRO S S SECTION AND SPECIFIC HEAT IN ALLOYS
:APPLICATION TO Ni-Cu (*)
K. LEVIN and D. L. MILLS
Physics Department, University of California, Irvine, California 92664, USA
RCsume. - Un modble simple d'alliage non diluk contenant des grandes assemblees de spins est utilisk pour calculer la contribution des spins
a
la resistivite, a la section efficace de diffusion des neutrons, et la chaleur specifique. Nous croyons que ce modble s'applique aux alliages de Ni-Cu dont la concentration est voisine de celle qui est critique pour le ferromagnetisme. On trouve un accord semi-quantitatif avec les mesures de resistivitk, de diffusion Blastique et de chaleur specifique a basses temperatures dans les alliages Ni-Cu, pour des concentrations et des temperatures voisines des conditions critiques.Abstract. - A simple model for a non-dilute alloy containing giant spin clusters is used to calculate the spin contribution to the resistivity, the neutron cross-section, and the specific heat Cv.
The model is expected to be applicable to Ni-Cu alloys near the critical concentration for ferroma- gnetism. Reasonable, semi-quantitative agreement with resistivity, elastic neutron scattering and low temperature specific heat measurements on Ni-Cu is obtained for a range of concentrations and temperatures.
It is the purpose of the present paper to calculate the resistivity, neutron scattering cross-section and specific heat, in Ni,Cu,-, alloys near the critical concentration for ferromagnetism : x = 0.44. A simple model for a non-dilute alloy which is composed of giant spin clusters is used. The motivation for these calculations is fourfold :
(i) To explain the anomalous temperature depen- dence of the resistivity in Ni-Cu alloys [I] and the behavior of the elastic neutron scattering cross-section in paramagnetic alloys [2]. These experimental obser- vations have not yet been explained. In paramagnetic alloys the resistivity decreases with increasing tempe- rature T [I]. This behavior is common to a variety of other alloy systems [3]. The elastic neutron scatter- ing cross-section in paramagnets has a peak in the forward direction similar to that observed in ferro- magnetic alloys. However, the behavior of the cross- section in paramagnets cannot be explained by the usual Marshall [4] formalism which is applicable only to ferromagnets at temperatures low compared to the Curie temperature, where the spins are pinned tightly along one direction.
(ii) To correlate the results of specific heat, neutron scattering and resistivity measurements. It has pre- viously been assumed [2, 51 (we believe erroneously) that neutron scattering and specific heat measurements
(*) Research sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, USAF, under Grant
NO AFOSR 70-1936.
probe different spin clusters and are thus independent of one another.
(iii) To put the spin cluster model formulated by Beck [6, 71, Kouvel 121, and their co-workers on a firmer theoretical basis.
(iv) To predict the results of new experiments such as inelastic neutron cross-section measurements and measurements of the temperature dependence of the specific heat and of the elastic neutron scatter- ing cross-section over a wider temperature range than that explored until this time.
The model used to calculate the various experimen- tal quantities is based on that proposed by Beck [6]
and Kouvel [2]. While in previous descriptions, the internal dynamics of the spin clusters have not been considered, the present paper focuses on a detailed treatment of intra-cluster interactions. Intra-cluster interactions are described here by a Heisenburg Hamil- tonian. Inter-cluster interactions are treated within the molecular field approximation in ferromagnets and are neglected in paramagnets. However, as in previous papers [6,7] it is assumed that in paramagnets there is a local magnetic field which derives from the anisotropy energy.
In comparing the calculated values for the low temperature specific heat, resistivity and neutron scattering cross-section with experiments on Ni-Cu several other assumptions are made for simplicity only. (i) The Ni atoms are assumed to have a spin of 1/2. This corresponds to exactly 1 d hole per magne-
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974438
C4-214 K. LEVIN AND D. L. MILLS
tic Ni atom which is not too unreasonable if hybri- dization effects are neglected. (ii) The Ni spins within a cluster are assumed to be in the most closely packed configuration on a face centered cubic (fcc) lattice.
Thus no cluster configuration averaging is performed.
As a result of the tendency to cluster magnetically it is clear that the spins will be distributed rather com- pactly. (iii) All clusters are assumed to contain exactly 50 atoms. This number is consistent with neutron scattering measurements [ 2 ] . While it is possible to consider fluctuations in the size of the clusters, this is not necessary if only semi-quantitative agreement with experiment is expected. In view of the number of simplifications necessary in order to make the problem tractable it does not seem reasonable to introduce unnecessary complications. (iv) All clusters are assumed to be in the same local magnetic field. Thus for the same reason as in (iii) fluctuations in the local field are neglected.
The number of Ni near neighbors required for a given Ni atom to have a moment is taken to be 8.
This number is consistent with previous estimates and completely determines the concentration of clusters in the alloy. The cluster concentration is of the order of tenths of a percent near x = 0.5 which is in agree- ment [2] with previous estimates ; however, in contrast to previous treatments [2, 61 it is not a free parameter of the problem.
It is shown that in paramagnetic alloys the spin disorder resistivity p decreases with increasing tem- perature providing the Fermi wavevector times the lattice constant (k, a) is less than z 2.0 in fcc lattices and the spin configuration within a cluster is reaso- nably compact. In comparing the theory with expe- riments it is assumed that in the resistivity form factor the quantity (kF a) is a free parameter. This is reaso- nable in view of the approximate nature of the model.
Good semi-quantitative agreement with experiment is obtained for k, a = 1.0 (this should be compared with the value 2.28 obtained in the free electron approxi- mation with one conduction electron per atom) and for a conduction electron localized spin exchange constant J" of roughly 10 ev-A3.
In figure 1, the theoretical (Fig. la) and the expe- rimental (Fig. lb) resistivity curves are plotted as a function of temperature for four different alloy concentrations. The origin of the resistivity curve for each concentration in both figures 1 is indicated by a horizontal line which is solid, dashed or dot-dashed in the same way as the resistivity curves. It is also shown that if terms of third order in J" are included in calculating the transition rate for the scattering of conduction electrons from the spin clusters, a Kondo-like minimum in the resistivity can appear at low temperatures. As can be seen from figure l b a minimum in the resistivity has been experimentally observed.
In analyzing the low temperature specific heat Cv experiments on Ni-Cu it is found that the cluster
FIG. 1. - Theoretical (Fig. la) and experimental [after ref. [I]]
(Fig. Ib) curves for resistivity in NizCu~-z as a function of temperature for four Ni concentrations. Tc is the Curie tempe- rature of an alloy and J d d is the intracluster exchange constant.
The large potential scattering contribution to the residual resistivity is omitted in the theoretical calculations.
contribution to the specific heat is probably not temperature independent as has been assumed pre- viously [6] but that in ferromagnetic alloys C, increases slowly with increasing temperature while in para- magnets Cv decreases somewhat more rapidly with temperature. However, the anomaly observed by Schroeder [7] which consists of an upturn in Cv/T at low temperatures is reproduced in the present theory.
In figure 2 the specific heat Cv divided by the tem- perature T is plotted as a function of T2 for the same four alloy concentrations as in figure. 1. The dotted line represents the experimental results of Robbins et al. [6] and the solid line represents the calculated spin cluster contribution to C,. If a constant
FIG. 2. - Specific heat divided by temperature T i n Ni,Cul-, as a function of T2 for same four alloy concentrations as in figure la. Solid lines are theoretical results (which include spin contribution only) and dashed lines are experimental results
[after ref. [ 6 ] ] which also include electronic contribution.
CONTRIBUTION OF GIANT SPIN CLUSTERS TO THE RESISTIVITY C4-215 y
-
10.0 x cal .mole-' . K - 2 corresponding tothe electronic contribution to C, is added to the theoretical curves, experiment and theory can be brought into reasonable agreement for all four alloy concentrations.
In figure 3 is shown the elastic neutron scattering cross-section at T = 4 K as a function of ~a for the same four alloy concentrations as in figures 1 and 2.
The solid line represents the theoretical and the dotted line the experimental results [2]. In agreement with experiment, the calculated elastic neutron scattering cross-section do/dQ, in paramagnetic Ni-Cu exhibits a peak in the forward direction. The neutrons are shown to scatter from spin fluctuations in the para- magnetic state and, as in the ferromagnetic alloys, the width of the peak reflects the size of the spin clusters.
It should thus be noted that, contrary to previous suggestions [2], clusters which are not ferromagneti- cally pinned will contribute to the elastic neutron scattering cross-section. It is found, however, that the contribution of spin fluctuations is relatively unimportant at T = 4 K in ferromagnetic alloys slightly above the critical concentration and that the
main contribution in these alloys to the cross-section comes from the usual Marshall term [4] as has been assumed in the analysis of Hicks et al. [2].
FIG. 3. - Elastic neutron cross-section in NizCul-z as function of momentum transfer x times the lattice spacing a for the same four Ni concentrations as in figures l a and 2. Solid lines are theo- retical and dashed lines are experimental results [after ref. [2]].
References
[l] HOUGHTON, R. W., SARACHIK, M. P. and KOUVEL, 3. S., [5] GARLAND, J. W. and GONIS, A., << Magnetism in Alloys >>
Phys. Rev. Lett. 25 (1970) 238. (Metallurgical Society of AIME) 1971.
[2] HICKS, T. J., RAINFORD, B., KOUVEL, 3. S. and Low, G. G.,
Phys. Rev. Lett. 22 (1969) 531. [6] ROBBINS, C. G., CLAUS, H. and BECK, P. A,, J. Appl. Phys.
131 BECK. P. A.. Bull. Am. Phvs. Soc. 18 (1973) 458. 40 (1969) 2269.
[4j MARSHALL, 'w., J. P ~ Y S . c 1 (1968) '88. ' [71 SCHROEDER, K., J. ADPI. P ~ Y S . 32 (1961) 880.
