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Unpolarized neutron diffuse scattering from magnetic clusters in nearly and weakly ferromagnetic alloys
Solid State Communications,Vol.
25, pp. 493-498,
Pergamon Press.
1978.
UNPOLARIZED NEUTRON DIFFUSE SCATTERING FROM MAGNETIC CLUSTERS IN NEARLY AND WEAKLY FERROMAGNETIC
Printed in Great Britain
ALLOYS
J .C. Ododo The Blackett Laboratory,
Imperial College, London, SW7 2BZ
(Received 2 March 1977; in revised form 21 September 1977 by R.A. Cow@) It is suggested that the observed unpolarized neutron elastic diffuse mag netic scattering from alloys in the critical concentration region for the onset of ferromagnetism may be usefully considered as the paramagnetic scattering from an assembly of nearly independent magnetic clusters. hence by plotting the experimental data at small K values as (do/dfl)-“2 vs K2 both KOand (do/da),, can be determined. Furthermore the polarization clouds are assumed to contribute additively to the spontaneous magnetization, psp, of the system so that
ELASTIC DIFFUSE magnetic scattering cross-sections of unpolarized neutrons from CuNi [I], PdNi [2] and CrNi [3] alloys in the critical concentration region for the onset of ferromagnetism exhibit a marked forward peak similar to that observed for PdFe and PdCo alloys [4,5]. It was found necessary to assume that the forward scattering was due to the presence of identical but nearly independent polarization clouds. The concentration, c*, of these clouds is taken to be zero at the critical concentration but increases steadily as the impurity concentration, c, increases. The diffuse differential magnetic cross-section, in mb/sr. atom, is given by do dn
= 73 sin’& c*(l - c*)~(K)}~
ccSP
(1)
where K is the scattering vector, (Yis the angle between K and the direction of magnetization, and M(K) is the Fourier transform of the average moment density, p(r), within a polarization cloud, i.e. M(K) = j dr p(r) eiK *I.
(2)
Thus M(O) = s drp(r) gives the average total integrated the forward direction
(3) moment
per cloud. In
= 73 sin*& c*(l - c*){M(O)}* and this is obtained from the experimental assuming that p(r) has the Yukawa form dr)
-
;
1
e
(4) data by
-K,r (5)
where Kc is a range parameter characterizing the extent or spread of the polarization cloud. It follows that
(6) 493
=
c*M(o).
(7)
The values of (da/da), and pgp are then used to obtain c* and M(0). While we may agree that the sharp forward peak that is observed in the unpolarized neutron diffuse crosssection arises from the presence of polarization clouds and also that such neutron data provide strong evidence for the inhomogeneous nature of the onset of ferromag netism in transition metal alloys, we reject one aspect of the above model which is physically unsatisfactory. This refers to the fact that the model requires that while the total moment within a cloud remains constant (or approximately so) the cloud concentration goes to zero, with the spontaneous magnetization, at the critical concentration. However, a number of experimental observations notably low temperature resistance minima [6], heat capacity [7-91 and susceptibility measurements [IO] all indicate that the magnetic clusters persist well into the non-ferromagnetic regime below the critical concentration. In order to explain this discrepancy, it has been argued [ 1 l] that the cloud concentration as determined from neutron diffraction data is merely the effective concentration of “magnetically coupled” clouds while the other experiments mentioned above probe the “uncoupled” polarization clouds. For example, the uncoupled or “free” clouds may be regarded as being in nearly zero molecular fields and consequently are thermally excitable. The apparently constant term observed in the low temperature specific heat of such systems has been attributed to these free polarization clouds [ 12, 131. More importantly Muellner and Kouvel [ 141 have demonstrated quite clearly, in the case of the RhNi system, that even in the ferromagnetic region above the critical concentration, cf, uncoupled
494
UNPOLARIZED
NEUTRON DIFFUSE SCATTERING
magnetic clusters still exist because there is a significant difference between the spontaneous magnetization, psp, which (by definition) vanishes at cf and the saturation magnetization, &,t, which is the total aligned moment of all (i.e. coupled and uncoupled) clusters in a given alloy. Thus for ferromagnetic Rh 64% Ni (all % are atomic) y,, = 0.122 /+Jatom while psp = 0.053 pn/ atom which shows that only 43% of the magnetic clusters is effectively fully aligned. The proportion of the coupled clusters increases rapidly as the alloys become more strongly ferromagnetic (for Rh 65% Ni 63% of the clusters is aligned [ 141) so that at some concentration psat and lisp become identical. It is thus obvious that acruss the critical concentration uncoupled magnetic clusters exist and one must therefore consider their contribution to the unpolarized neutron diffuse scattering from nearly and weakly ferromagnetic alloys. It is this problem that is the primary object of this communication. The fact that the uncoupled magnetic clusters contribute to the observed neutron diffuse scattering of alloys in the critical concentration region for the onset of ferromagnetism may be easily seen in the case of CuNi [ 1 ] and CrNi [3] alloys. For CuNi a recent semiquantitative analysis [ 151 of existing data has shown that cf = 47.6 + 0.1% Ni which is significantly higher than the value of = 44% Ni which has been hitherto generally accepted [ 11. Thus we can attribute the scattering observed for Cu 44 and 46% Ni alloys [ 11, which are non-ferromagnetic, to the presence of uncoupled magnetic clusters. Similarly for CrNi cf= 89.4 f 0.1% Ni [ 161 but again, there was appreciable amount of neutron diffuse scattering from a Cr 87.5% Ni alloy [3]. As an illustration of the necessity to consider the scattering from uncoupled magnetic clusters for weakly ferromagnetic alloys, we present below unpolarized neutron diffuse scattering data for a Rh 64% Ni alloy. A preliminary investigation of this alloy had been briefly mentioned by Hicks et al. [l] who observed a peak in the forward direction. The elastic diffuse magnetic crosssections from this and several other Ni-rich RhNi alloys have been measured at the Pluto reactor, AERE, Harwell. The experiments were carried out at 4.2 K using neutrons of wavelength 5 a and following a now standard procedure which has been previously described [ 1, 18, 193 except that the scattered neutrons are counted by a fixed bank of twenty-two detectors mounted in an arc at the end of the neutron flight path. The detectors cover a range of scattering angles, 0 (K = 4n sin S/X, where h is the neutron wavelength) varying from - 11.5” to + 49.5’ with respect to the straight-through position. Also in order to increase the observable range of K-values the measurements have been carried out at two values (0 and 45’) of the angle,
Vol. 25, No. 7
loo< 80 I-l
0
0.4
0.8 -
1.2
1.6
KW’)
Fig. 1. Unpolarized neutron diffuse scattering crosssection for Rh 64% Ni. (do/dSl) is the Taximum switchabk cross-section. A: $I = 0”; n : 4 = 45 . The solid line is a fit of the data to equation (6) allowing for a “background” cross-section due to the second moment of the fluctuations of the local Ni moments (equation (16)). @,between the normal to the plane of the specimen and the direction of the incident neutron beam. At each specimen angle 4, counting was carried on continuously for 72 hr alternately with the horizontal field off and on. The results are shown in Fig. 1 in which do/da is the maximum switchable cross-section, i.e. the results have been normalised to sin CY= 1. The data have been fitted to equation (6) allowing for the scattering due to variations in the local moment at the lattice sites (see below) to obtain = 177 + 13 mb/sr. atom and K. = 0.19 + 0.01 A-‘. If we adopt the regard c* as the clusters so that 0.053 &/atom = 0.116% C*
procedure used by Hicks et al. [ 1 ] and concentration of magnetically coupled equations (4) and (7) apply with /J,~ = [ 141 we obtain and M(0) = 45.8~~
which are in a marked disagreement
with the values
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Vol. 25, No. 7
NEUTRON DIFFUSE
c* = 0.595% and M(0) = 20.4 &J obtained by Muellner and Kouvel [ 141 from magnetization data. On the other hand, if we regard c* as the total concentration of magnetic clusters (coupled and uncoupled) then with psat = O.l22&atom [14], we get that c* = 0.61% andM(O)= 201.(B in better agreement with the magnetization values. We may justify the proposition that c* should be the total concentration of all magnetic clusters in a given alloy as follows: Suppose ci and c2 are the concentrations of the coupled and uncoupled clusters respectively. In keeping with previous authors [l-3, 10, 141 it will be assumed that the clusters are identical and bear the same moment, Mel. This assumption is only approximately valid as in reality a distribution of cluster sizes is expected - and observed. For example, in CuNi large clusters with moments of N 40-220~~ [20] and - 756~~ [21] have been reported. However, the concentration of such large clusters, which ideally should arise only through statistical concentration fluctuations, should be small and has been estimated as about 10% of the total number of clusters in CuNi [20]. We further assume that the scattering from the coupled clusters is given by equation (4) with c* replaced by cl, sin’ar by its average value (3). and M(0) by MC2, i.e. / 2 _ \ coupled
(8) We then suggest that the scattering from the uncoupled clusters is similar to that from a paramagnet. Thus in the absence of an applied magnetic induction, Bo, the “paramagnetic” scattering cross-section for the uncoupled clusters in the forward direction is [22], in mb/sr. atom,
When a magnetic induction is applied, say in the zdirection parallel to K, this forward cross-section now becomes = 73c*{M,r(M,r + 1) - ((M:*)2)}. Tne difference cross-section
=
(10)
is therefore
k2{((M:~)“)- ~Mc@‘fc~ + 1)).
, A _\ uncoupled
n 73c&l4,21- SMc,}.
From equations section is -do = 73 i da 1o
(12)
(8) and (12) the total forward cross-
X &(i
-
cl)Mzl +
73c2{$&
= 73ci(l
-
$Mcl} (13)
but since the observed cross-sections sin (Y= 1, we should have
are normalized
- c,)M,: + 7k2(M,:
- fMc~>.
to
(14)
To this equation should be added the scattering due to fluctuations in the magnitude of the local moments at lattice sites resulting from concentration fluctuations. If pi and ,.& are the local solute and host moments respectively and pi, ,!?hare their corresponding average values then this scattering cross-section is given by = 73 sin2a! [c(l - c){Fipi -Fh/&}2 +
Ffd(6/.ii)2) + F;(l -c)((6,&)2)]
(15)
where Fi and Fh are the solute and host atom form factors. Since /Ji and ,.,&when non-zero are usually d-like moments, the K-dependence of (da/dfi)fl is slow in the normal experimental range (< 2 A-‘) and equation (15) can be regarded as merely contributing a nearly constant “background” to the observed cross-section. In fitting the neutron data for Rh 64% Ni to equation (6) we corrected for this background by putting fi 73 sin2a cFii((6pNi)2)
(16)
where FNi = e-0.05K ‘, with ((8/JNi)‘) determined by the cross-section at large K. The other terms in equation (15) have been neglected because Cable [23] has found that for Rh 65% Ni &h = 0.053 nB and fiNi = 0.15 /-& showing that both iinh and fiNi are very small in this concentration limit and that PNi 3 ,&h. For Rh 64% Ni the scattering at large K (see Fig. 1) gives that ((6pNi)2) r 0.05 yi - (pNi)2. In further discussion, it will be assumed that when necessary the observed forward cross-section has been corrected for equation (15). It will be noticed that the two terms in equation (14) are similar because cr < 1 and Mel % 1. Therefore, we can write
(11)
Usually M,r s 1 so that the cluster spins are fully aligned in a relatively low magnetic field (- 0.4 T) especially at low temperatures. Consequently, we may write M$-M,r and hence
495
SCATTERING
1 73(cr + c2)M,:
= 73c*M,21
(17)
with I-lsat = c*M,r.
(18)
We thus see that the forward cross-section should represent the scattering from all clusters, coupled and uncoupled. Equation (17) may be directly deduced from
496
UNPOLARIZED
NEUTRON DIFFUSE
equation (9) if c2 is replaced by c*. Therefore the elastic diffuse magnetic scattering from alloys in the critical concentration region may be considered as the “paramagnetic” scattering from an array of nearly independent magnetic clusters. Such an interpretation is useful because (i) it gives estimates of c* and I& which agree with the values obtained from magnetization measurements only; (ii) it obviates the need to differentiate between coupled and uncoupled magnetic clusters. Such a distinction is physically incorrect because with the exception of a relatively small number of very large clusters, there is no a priori reason why the molecular field acting on each cluster should not be the same especially as the clusters are supposed to be randomly dispersed throughout the matrix. Levin and Mills [ 171 have already argued that it is wrong to assume that the constant term in the heat capacity of such systems measures the number of uncoupled clusters. It will be shown elsewhere that the magnetic heat capacity of alloys in the critical concentration region is proportional to temperature at the lowest temperatures but changes over to a Schottky function at higher temperatures. The use of an apparently constant term in the heat capacity [ 12, 131 is an approximation that is only valid under certain conditions which are not always satisfied. The linear temperature dependence of the magnetic heat capacity in the critical concentration region has been observed for PtCo alloys [24] and in the more recent measurements for CuNi [25] ; (iii) it accounts for the observed diffuse scattering from non-ferromagnetic CuNi (1) and CrNi (3) alloys which hitherto had been attributed to either “quasi-elastic scattering of neutrons from slow critical fluctuations” [l-3] or to “intra-cluster spin fluctuations” [ 171. For the weakly ferromagnetic alloys the validity of the “paramagnetic” approximation is less certain but partial justification is provided by the fact that these alloys have very low Curie temperatures implying that the magnetic clusters are only very weakly coupled (i.e. they are in low molecular fields) and also by the need otherwise to consider the separate contributions from “coupled” and “uncoupled” magnetic clusters. It must also be borne in mind that the use of equation (1) was strictly ad hoc [I]. For non-zero K equation (17) should be replaced by do
-
da
fi 73c*F,,M,:
(19)
where FCl is the form factor for a magnetic cluster which may be taken to be a Lorentzian, i.e. F,r = [ 1 + (K’/Kz)] -’ as in equation (6). Hence the extrapolation procedure for obtaining (do/dQ), is unchanged. Two observations about the Rh 64% Ni alloy are worth noting. Firstly, the polarization range deduced above (Ki’ = 5.3 A) is in good agreement with the
SCATTERING
Vol. 2.5, No. 7
average distance between the magnetic clusters. For a cloud concentration c* the average separation between the cluster is [26] r Cl”= 0.554(c*)-“3a0
(20)
where a0 is the lattice parameter. The lattice parameter for this alloy measured by X-ray diffraction is 3.64 A which agrees with published data for other RhNi alloys [27]. Thus r,, = 11 A, and since each cluster has a diameter of - 10.6 A it follows that neighbouring clusters just overlap. Secondly, in his discussion of the neutron data for RhNi, particularly Rh 65% Ni, Cable [23 ] did not consider the explicit role of magnetic clusters in the neutron scattering except to mention that the presence of these clusters, so clearly established by magnetization measurements [ 141, only indicated the importance of local magnetic environment effects. In the critical concentration region the magnetic clusters consist of those Rh and Ni atoms with twelve Ni nearest neighbours. It may be easily checked that the statistical concentration of these clusters (= cti) is in good agreement with the experimental values [ 141 particularly for concentrations just below cf (= 63.4% Ni). Above cf we should expect the experimental values of c* to be greater than the statistical values because of the favourable effect of a finite molecular field on the nearly magnetic clusters. It must be mentioned that the “paramagnetic” scattering model (or simply the cluster model) is one of two models that we are currently considering for the analysis of unpolarized neutron scattering data in the critical concentration region. The other model derives from a suggestion [28] that the onset of ferromagnetism as a function of concentration, in transition metal alloys should be treated as a proper cooperative phase transition similar to the well-known phase transition, which occurs as a function of temperature, at a ferromagnetic Curie point. In the latter case neutrons are scattered by the critical fluctuations of magnetization which are thermally induced. It has been shown [29] that the quasi-elastic neutron diffuse scattering cross-section is given by da = 73 sin2a KBTx(K) dR
(21)
where KB is Boltzmann’s constant and x(K) is the static wavevector-dependent susceptibility. For the phase transition that occurs at cf (and T - 0 K) the critical fluctuations of magnetization are caused by concentration jluctuations so that in Equation (21), we may replace the factor kBT representing the thermal fluctuation energy by the factor c( 1 - c) for the concentration fluctuations to obtain do = 73 sin2a c(1 - c)x(K) dR
(22)
UNPOLARIZED
Vol. 25, No. 7
NEUTRON DIFFUSE
where K,{- (c - cf)1’2} is an inverse correlation range. We have shown that this critical scattering model gives the best fit to our neutron data for several weakly ferromagnetic PtCo and PtFe alloys [33]. Equation (26) also fits the neutron data shown in Fig. 1 for Rh 64% Ni (giving (da/dQ)o = 246 mb/sr. atom and K, fi 0.08 A-l) but the extrapolated forward cross-section appears to be too small for an alloy that is so close to criticality. Also, no measurements were made for other alloys in the critical region so that we are unable to confirm that
(23) where x(O) is, of course, the initial (or zero-field) susceptibility. It may be easily shown [28,30] that x(O)_ l a dp a (c-q)
(24)
so that
0 :i
0
a 73 sin2a c(1 -c)
Thus we recover an expression that is similar to that derived by Marshall [3 1] for dilute alloys (ferromugnetic hosts with dilute concentrations of magnetic or non-magnetic solutes) except that for these alloys psp = psat. If we assume that for small wavevectors x(K) is of the Ornstein-Zernicke form [32] then -
do
da
=
do -’
(25)
12 .
i2
ZO
-
K,” - (c - Cf)
(27)
as in the case of PtCo alloys [33 1. However, we cannot at the moment rule out the possibility that the cluster and critical scattering models may be complementary. Acknowledgements -We would like to thank Vie Rainey for experimental assistance and the U.K. Science Research Council for financial support of this programme.
(26)
73 sin2a c(1 -c)x(O)
497
SCATTERING
REFERENCES B.D., KOUVEL J.S., LOW G.G. & COMLY J.B., Phys. Rev. Lett. 22,53 1 (1969).
1.
HICKS T.J., RAINFORD
2.
ALDRED A.T., RAINFORD
3.
RAINFORD
4.
LOW G.G. & HOLDEN T.M.,Proc.
5.
HICKS T.J., HOLDEN T.M. & LOW G.G.,J. Phys. Cl, 528 (1968).
6.
HOUGHTON R.W., SARACHIK M.P. & KOUVEL J.S., Phys. Rev. Lett. 25,238
7.
WOLCOTT N.M. & FALGE R.L.,J. Low Temp. Phys. 2, 329 (1970).
8.
FALGE R.L. &WOLCOTT
9.
BECK P.A. & CLAUS H.,NBSJ.
B.D. & STRINGFELLOW
M.W., Phys. Rev. Lett. 24,897
(1970).
B.D., ALDRED A.T. & LOW G.G.,J. Phys. 32, Cl-575 (1971). Phys. Sot. 89,119
(1966). (1970).
N.M.,J. Low Temp. Phys. 5,617 (1971). Rex 74A, 449 (1970).
10.
KOUVEL J.S. 8~ COMLY J.B., Phys. Rev. Lett. 24,598
11.
BENNETT L.H., SWARTZENDRUBER
12.
SCHRODER K.,J. Appl. Phys. 32,880
13.
HAHN A. & WOHLFARTH
14.
MUELLNER W.C. & KOUVEL J.S., Phys. Rev. Bll,
15.
ODODO J.C. & COLES B.R.,J. Phys. F. 11,2393
16.
ODODO J.C. (unpublished
17.
LEVIN K. & MILLS D.L., Phys. Rev. B9,2354
18.
LOW G.G., & COLLINS M.F.,J. Appl. Phys. 34,1195
19.
ALDRED A.T., RAINFORD
(1970).
L.J. & WATSON R.E.,Phys.
Rev. Lett. 23,1171
(1969).
(1961).
E.P.,HeZv. Phys. Acta 41,857
(1968).
4552 (1975).
(1977).
analysis). (1974). (1963).
B.D., HICKS T.J. & KOUVEL J.S., Phys. Rev. B7,218
(1973).
20.
ACKER F. & HUGUENIN R., Phys. Lett. 38A, 343 (1972).
21.
AMAMOU A., GAUTIER
22.
MARSHALL W. & LOVESEY S.W., Theory of 7’hermal Neutron Scattering, p. 111. Oxford University Press (1970).
F. & LOEGEL, B., J. Phys. F5,1342
(1975).
498
UNPOLARIZED
NEUTRON DIFFUSE SCATTERING
Vol. 25, No. 7
23.
CABLE J.W., Phys. Rev. B15,3477
24.
RIBEIRO P.C., PhD Thesis, University of Grenoble (1973).
25.
LANCHESTER P.C., WHITEHEAD N.F, &WELLS P., Physica 86-88B,
26.
KOPP J.,J. Phys. F3,1994
27.
LUO H. & DUWEZ P., J. Less-Common Metals 6,248
28.
ODODO J.C. (to be published).
29.
MARSHALL W. & LOWDE R.D., Rep. Prog. Phys. 21,705
30.
MATHON J.,Proc. Roy. Sot. 306,355
31.
MARSHALL W.,J. Phys. Cl, 88 (1968).
32.
LANDAU L.D. & LIFSHITZ E.M., Statistical Physics, Chapter 12. Pergamon Press, London (1958).
33.
ODODO J.C. & RAINFORD
(1977).
82.5 (1977).
(11973). (1964). (1968); Also [22], Chapter 13.
(1968).
B.D. (to be published).
25, pp. 493-498,
Pergamon Press.
1978.
UNPOLARIZED NEUTRON DIFFUSE SCATTERING FROM MAGNETIC CLUSTERS IN NEARLY AND WEAKLY FERROMAGNETIC
Printed in Great Britain
ALLOYS
J .C. Ododo The Blackett Laboratory,
Imperial College, London, SW7 2BZ
(Received 2 March 1977; in revised form 21 September 1977 by R.A. Cow@) It is suggested that the observed unpolarized neutron elastic diffuse mag netic scattering from alloys in the critical concentration region for the onset of ferromagnetism may be usefully considered as the paramagnetic scattering from an assembly of nearly independent magnetic clusters. hence by plotting the experimental data at small K values as (do/dfl)-“2 vs K2 both KOand (do/da),, can be determined. Furthermore the polarization clouds are assumed to contribute additively to the spontaneous magnetization, psp, of the system so that
ELASTIC DIFFUSE magnetic scattering cross-sections of unpolarized neutrons from CuNi [I], PdNi [2] and CrNi [3] alloys in the critical concentration region for the onset of ferromagnetism exhibit a marked forward peak similar to that observed for PdFe and PdCo alloys [4,5]. It was found necessary to assume that the forward scattering was due to the presence of identical but nearly independent polarization clouds. The concentration, c*, of these clouds is taken to be zero at the critical concentration but increases steadily as the impurity concentration, c, increases. The diffuse differential magnetic cross-section, in mb/sr. atom, is given by do dn
= 73 sin’& c*(l - c*)~(K)}~
ccSP
(1)
where K is the scattering vector, (Yis the angle between K and the direction of magnetization, and M(K) is the Fourier transform of the average moment density, p(r), within a polarization cloud, i.e. M(K) = j dr p(r) eiK *I.
(2)
Thus M(O) = s drp(r) gives the average total integrated the forward direction
(3) moment
per cloud. In
= 73 sin*& c*(l - c*){M(O)}* and this is obtained from the experimental assuming that p(r) has the Yukawa form dr)
-
;
1
e
(4) data by
-K,r (5)
where Kc is a range parameter characterizing the extent or spread of the polarization cloud. It follows that
(6) 493
=
c*M(o).
(7)
The values of (da/da), and pgp are then used to obtain c* and M(0). While we may agree that the sharp forward peak that is observed in the unpolarized neutron diffuse crosssection arises from the presence of polarization clouds and also that such neutron data provide strong evidence for the inhomogeneous nature of the onset of ferromag netism in transition metal alloys, we reject one aspect of the above model which is physically unsatisfactory. This refers to the fact that the model requires that while the total moment within a cloud remains constant (or approximately so) the cloud concentration goes to zero, with the spontaneous magnetization, at the critical concentration. However, a number of experimental observations notably low temperature resistance minima [6], heat capacity [7-91 and susceptibility measurements [IO] all indicate that the magnetic clusters persist well into the non-ferromagnetic regime below the critical concentration. In order to explain this discrepancy, it has been argued [ 1 l] that the cloud concentration as determined from neutron diffraction data is merely the effective concentration of “magnetically coupled” clouds while the other experiments mentioned above probe the “uncoupled” polarization clouds. For example, the uncoupled or “free” clouds may be regarded as being in nearly zero molecular fields and consequently are thermally excitable. The apparently constant term observed in the low temperature specific heat of such systems has been attributed to these free polarization clouds [ 12, 131. More importantly Muellner and Kouvel [ 141 have demonstrated quite clearly, in the case of the RhNi system, that even in the ferromagnetic region above the critical concentration, cf, uncoupled
494
UNPOLARIZED
NEUTRON DIFFUSE SCATTERING
magnetic clusters still exist because there is a significant difference between the spontaneous magnetization, psp, which (by definition) vanishes at cf and the saturation magnetization, &,t, which is the total aligned moment of all (i.e. coupled and uncoupled) clusters in a given alloy. Thus for ferromagnetic Rh 64% Ni (all % are atomic) y,, = 0.122 /+Jatom while psp = 0.053 pn/ atom which shows that only 43% of the magnetic clusters is effectively fully aligned. The proportion of the coupled clusters increases rapidly as the alloys become more strongly ferromagnetic (for Rh 65% Ni 63% of the clusters is aligned [ 141) so that at some concentration psat and lisp become identical. It is thus obvious that acruss the critical concentration uncoupled magnetic clusters exist and one must therefore consider their contribution to the unpolarized neutron diffuse scattering from nearly and weakly ferromagnetic alloys. It is this problem that is the primary object of this communication. The fact that the uncoupled magnetic clusters contribute to the observed neutron diffuse scattering of alloys in the critical concentration region for the onset of ferromagnetism may be easily seen in the case of CuNi [ 1 ] and CrNi [3] alloys. For CuNi a recent semiquantitative analysis [ 151 of existing data has shown that cf = 47.6 + 0.1% Ni which is significantly higher than the value of = 44% Ni which has been hitherto generally accepted [ 11. Thus we can attribute the scattering observed for Cu 44 and 46% Ni alloys [ 11, which are non-ferromagnetic, to the presence of uncoupled magnetic clusters. Similarly for CrNi cf= 89.4 f 0.1% Ni [ 161 but again, there was appreciable amount of neutron diffuse scattering from a Cr 87.5% Ni alloy [3]. As an illustration of the necessity to consider the scattering from uncoupled magnetic clusters for weakly ferromagnetic alloys, we present below unpolarized neutron diffuse scattering data for a Rh 64% Ni alloy. A preliminary investigation of this alloy had been briefly mentioned by Hicks et al. [l] who observed a peak in the forward direction. The elastic diffuse magnetic crosssections from this and several other Ni-rich RhNi alloys have been measured at the Pluto reactor, AERE, Harwell. The experiments were carried out at 4.2 K using neutrons of wavelength 5 a and following a now standard procedure which has been previously described [ 1, 18, 193 except that the scattered neutrons are counted by a fixed bank of twenty-two detectors mounted in an arc at the end of the neutron flight path. The detectors cover a range of scattering angles, 0 (K = 4n sin S/X, where h is the neutron wavelength) varying from - 11.5” to + 49.5’ with respect to the straight-through position. Also in order to increase the observable range of K-values the measurements have been carried out at two values (0 and 45’) of the angle,
Vol. 25, No. 7
loo< 80 I-l
0
0.4
0.8 -
1.2
1.6
KW’)
Fig. 1. Unpolarized neutron diffuse scattering crosssection for Rh 64% Ni. (do/dSl) is the Taximum switchabk cross-section. A: $I = 0”; n : 4 = 45 . The solid line is a fit of the data to equation (6) allowing for a “background” cross-section due to the second moment of the fluctuations of the local Ni moments (equation (16)). @,between the normal to the plane of the specimen and the direction of the incident neutron beam. At each specimen angle 4, counting was carried on continuously for 72 hr alternately with the horizontal field off and on. The results are shown in Fig. 1 in which do/da is the maximum switchable cross-section, i.e. the results have been normalised to sin CY= 1. The data have been fitted to equation (6) allowing for the scattering due to variations in the local moment at the lattice sites (see below) to obtain = 177 + 13 mb/sr. atom and K. = 0.19 + 0.01 A-‘. If we adopt the regard c* as the clusters so that 0.053 &/atom = 0.116% C*
procedure used by Hicks et al. [ 1 ] and concentration of magnetically coupled equations (4) and (7) apply with /J,~ = [ 141 we obtain and M(0) = 45.8~~
which are in a marked disagreement
with the values
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Vol. 25, No. 7
NEUTRON DIFFUSE
c* = 0.595% and M(0) = 20.4 &J obtained by Muellner and Kouvel [ 141 from magnetization data. On the other hand, if we regard c* as the total concentration of magnetic clusters (coupled and uncoupled) then with psat = O.l22&atom [14], we get that c* = 0.61% andM(O)= 201.(B in better agreement with the magnetization values. We may justify the proposition that c* should be the total concentration of all magnetic clusters in a given alloy as follows: Suppose ci and c2 are the concentrations of the coupled and uncoupled clusters respectively. In keeping with previous authors [l-3, 10, 141 it will be assumed that the clusters are identical and bear the same moment, Mel. This assumption is only approximately valid as in reality a distribution of cluster sizes is expected - and observed. For example, in CuNi large clusters with moments of N 40-220~~ [20] and - 756~~ [21] have been reported. However, the concentration of such large clusters, which ideally should arise only through statistical concentration fluctuations, should be small and has been estimated as about 10% of the total number of clusters in CuNi [20]. We further assume that the scattering from the coupled clusters is given by equation (4) with c* replaced by cl, sin’ar by its average value (3). and M(0) by MC2, i.e. / 2 _ \ coupled
(8) We then suggest that the scattering from the uncoupled clusters is similar to that from a paramagnet. Thus in the absence of an applied magnetic induction, Bo, the “paramagnetic” scattering cross-section for the uncoupled clusters in the forward direction is [22], in mb/sr. atom,
When a magnetic induction is applied, say in the zdirection parallel to K, this forward cross-section now becomes = 73c*{M,r(M,r + 1) - ((M:*)2)}. Tne difference cross-section
=
(10)
is therefore
k2{((M:~)“)- ~Mc@‘fc~ + 1)).
, A _\ uncoupled
n 73c&l4,21- SMc,}.
From equations section is -do = 73 i da 1o
(12)
(8) and (12) the total forward cross-
X &(i
-
cl)Mzl +
73c2{$&
= 73ci(l
-
$Mcl} (13)
but since the observed cross-sections sin (Y= 1, we should have
are normalized
- c,)M,: + 7k2(M,:
- fMc~>.
to
(14)
To this equation should be added the scattering due to fluctuations in the magnitude of the local moments at lattice sites resulting from concentration fluctuations. If pi and ,.& are the local solute and host moments respectively and pi, ,!?hare their corresponding average values then this scattering cross-section is given by = 73 sin2a! [c(l - c){Fipi -Fh/&}2 +
Ffd(6/.ii)2) + F;(l -c)((6,&)2)]
(15)
where Fi and Fh are the solute and host atom form factors. Since /Ji and ,.,&when non-zero are usually d-like moments, the K-dependence of (da/dfi)fl is slow in the normal experimental range (< 2 A-‘) and equation (15) can be regarded as merely contributing a nearly constant “background” to the observed cross-section. In fitting the neutron data for Rh 64% Ni to equation (6) we corrected for this background by putting fi 73 sin2a cFii((6pNi)2)
(16)
where FNi = e-0.05K ‘, with ((8/JNi)‘) determined by the cross-section at large K. The other terms in equation (15) have been neglected because Cable [23] has found that for Rh 65% Ni &h = 0.053 nB and fiNi = 0.15 /-& showing that both iinh and fiNi are very small in this concentration limit and that PNi 3 ,&h. For Rh 64% Ni the scattering at large K (see Fig. 1) gives that ((6pNi)2) r 0.05 yi - (pNi)2. In further discussion, it will be assumed that when necessary the observed forward cross-section has been corrected for equation (15). It will be noticed that the two terms in equation (14) are similar because cr < 1 and Mel % 1. Therefore, we can write
(11)
Usually M,r s 1 so that the cluster spins are fully aligned in a relatively low magnetic field (- 0.4 T) especially at low temperatures. Consequently, we may write M$-M,r and hence
495
SCATTERING
1 73(cr + c2)M,:
= 73c*M,21
(17)
with I-lsat = c*M,r.
(18)
We thus see that the forward cross-section should represent the scattering from all clusters, coupled and uncoupled. Equation (17) may be directly deduced from
496
UNPOLARIZED
NEUTRON DIFFUSE
equation (9) if c2 is replaced by c*. Therefore the elastic diffuse magnetic scattering from alloys in the critical concentration region may be considered as the “paramagnetic” scattering from an array of nearly independent magnetic clusters. Such an interpretation is useful because (i) it gives estimates of c* and I& which agree with the values obtained from magnetization measurements only; (ii) it obviates the need to differentiate between coupled and uncoupled magnetic clusters. Such a distinction is physically incorrect because with the exception of a relatively small number of very large clusters, there is no a priori reason why the molecular field acting on each cluster should not be the same especially as the clusters are supposed to be randomly dispersed throughout the matrix. Levin and Mills [ 171 have already argued that it is wrong to assume that the constant term in the heat capacity of such systems measures the number of uncoupled clusters. It will be shown elsewhere that the magnetic heat capacity of alloys in the critical concentration region is proportional to temperature at the lowest temperatures but changes over to a Schottky function at higher temperatures. The use of an apparently constant term in the heat capacity [ 12, 131 is an approximation that is only valid under certain conditions which are not always satisfied. The linear temperature dependence of the magnetic heat capacity in the critical concentration region has been observed for PtCo alloys [24] and in the more recent measurements for CuNi [25] ; (iii) it accounts for the observed diffuse scattering from non-ferromagnetic CuNi (1) and CrNi (3) alloys which hitherto had been attributed to either “quasi-elastic scattering of neutrons from slow critical fluctuations” [l-3] or to “intra-cluster spin fluctuations” [ 171. For the weakly ferromagnetic alloys the validity of the “paramagnetic” approximation is less certain but partial justification is provided by the fact that these alloys have very low Curie temperatures implying that the magnetic clusters are only very weakly coupled (i.e. they are in low molecular fields) and also by the need otherwise to consider the separate contributions from “coupled” and “uncoupled” magnetic clusters. It must also be borne in mind that the use of equation (1) was strictly ad hoc [I]. For non-zero K equation (17) should be replaced by do
-
da
fi 73c*F,,M,:
(19)
where FCl is the form factor for a magnetic cluster which may be taken to be a Lorentzian, i.e. F,r = [ 1 + (K’/Kz)] -’ as in equation (6). Hence the extrapolation procedure for obtaining (do/dQ), is unchanged. Two observations about the Rh 64% Ni alloy are worth noting. Firstly, the polarization range deduced above (Ki’ = 5.3 A) is in good agreement with the
SCATTERING
Vol. 2.5, No. 7
average distance between the magnetic clusters. For a cloud concentration c* the average separation between the cluster is [26] r Cl”= 0.554(c*)-“3a0
(20)
where a0 is the lattice parameter. The lattice parameter for this alloy measured by X-ray diffraction is 3.64 A which agrees with published data for other RhNi alloys [27]. Thus r,, = 11 A, and since each cluster has a diameter of - 10.6 A it follows that neighbouring clusters just overlap. Secondly, in his discussion of the neutron data for RhNi, particularly Rh 65% Ni, Cable [23 ] did not consider the explicit role of magnetic clusters in the neutron scattering except to mention that the presence of these clusters, so clearly established by magnetization measurements [ 141, only indicated the importance of local magnetic environment effects. In the critical concentration region the magnetic clusters consist of those Rh and Ni atoms with twelve Ni nearest neighbours. It may be easily checked that the statistical concentration of these clusters (= cti) is in good agreement with the experimental values [ 141 particularly for concentrations just below cf (= 63.4% Ni). Above cf we should expect the experimental values of c* to be greater than the statistical values because of the favourable effect of a finite molecular field on the nearly magnetic clusters. It must be mentioned that the “paramagnetic” scattering model (or simply the cluster model) is one of two models that we are currently considering for the analysis of unpolarized neutron scattering data in the critical concentration region. The other model derives from a suggestion [28] that the onset of ferromagnetism as a function of concentration, in transition metal alloys should be treated as a proper cooperative phase transition similar to the well-known phase transition, which occurs as a function of temperature, at a ferromagnetic Curie point. In the latter case neutrons are scattered by the critical fluctuations of magnetization which are thermally induced. It has been shown [29] that the quasi-elastic neutron diffuse scattering cross-section is given by da = 73 sin2a KBTx(K) dR
(21)
where KB is Boltzmann’s constant and x(K) is the static wavevector-dependent susceptibility. For the phase transition that occurs at cf (and T - 0 K) the critical fluctuations of magnetization are caused by concentration jluctuations so that in Equation (21), we may replace the factor kBT representing the thermal fluctuation energy by the factor c( 1 - c) for the concentration fluctuations to obtain do = 73 sin2a c(1 - c)x(K) dR
(22)
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Vol. 25, No. 7
NEUTRON DIFFUSE
where K,{- (c - cf)1’2} is an inverse correlation range. We have shown that this critical scattering model gives the best fit to our neutron data for several weakly ferromagnetic PtCo and PtFe alloys [33]. Equation (26) also fits the neutron data shown in Fig. 1 for Rh 64% Ni (giving (da/dQ)o = 246 mb/sr. atom and K, fi 0.08 A-l) but the extrapolated forward cross-section appears to be too small for an alloy that is so close to criticality. Also, no measurements were made for other alloys in the critical region so that we are unable to confirm that
(23) where x(O) is, of course, the initial (or zero-field) susceptibility. It may be easily shown [28,30] that x(O)_ l a dp a (c-q)
(24)
so that
0 :i
0
a 73 sin2a c(1 -c)
Thus we recover an expression that is similar to that derived by Marshall [3 1] for dilute alloys (ferromugnetic hosts with dilute concentrations of magnetic or non-magnetic solutes) except that for these alloys psp = psat. If we assume that for small wavevectors x(K) is of the Ornstein-Zernicke form [32] then -
do
da
=
do -’
(25)
12 .
i2
ZO
-
K,” - (c - Cf)
(27)
as in the case of PtCo alloys [33 1. However, we cannot at the moment rule out the possibility that the cluster and critical scattering models may be complementary. Acknowledgements -We would like to thank Vie Rainey for experimental assistance and the U.K. Science Research Council for financial support of this programme.
(26)
73 sin2a c(1 -c)x(O)
497
SCATTERING
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