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Magnetovolume effects in transition metal alloys near the critical composition for ferromagnetism
Journal of Magnetism and Magnetic Materials 10 (1979) 259-264 0 North-Holland Publishing Company
MAGNETOVOLUME EFFECTS IN TRANSITION FOR FERROMAGNETISM
METAL ALLOYS NEAR THE CRITICAL COMPOSITION
J.J.M. FRANSE Natuurkundig Laboratorium der Universiteit van Amsterdam, Amsterdam, The Netherlands
Results of magnetovolume studies are presented on intermetallic compounds and binary alloys with weakly ferromagnetic properties of the following compositions: NisAl, (3, Fe)+Sa, Ti(Fe, Co), (Ti, Al)Co, Ni-Pt and Pd-Ni. Data for the pressure dependence of the spontaneous magnetization at 4.2 K and the shift of the Curie temperature with pressure follow roughly the relations that can be derived in the model for weak itinerant ferromagnetism in most of the investigated systems. Invar-type of anomalies in the thermal expansion have already been observed in several cases. Experimental values for the magnetovolume parameter are derived and compared with theoretical predictions in the Stoner model.
1. Introduction Magnetovolume effects in transition metals have a long tradition of theoretical interest (electron correlation) and technical applications (Invar alloys). Our study of transition metal alloys is directed to a systematic investigation of magnetovolume effects in alloys around the critical composition for ferromagnetism. These magnetovolume effects include the pressure dependence of the spontaneous magnetization at 4.2 K, the shift with pressure of the Curie temperature, the magnetic contribution to the linear thermal expansion coefficient and the field induced linear magnetostriction. In some cases (non-stoichiometric Ni3Al compounds in the range of 74 to 76 at% Ni, disordered Ni-Pt alloys around the critical composition of 42.5 at% Ni) the set of existing magnetovolume data can be represented by a single magnetovolume parameter that turns out to be only slightly temperature dependent. An interpretation of this parameter in the band model of magnetism yields information about the volume dependence of microscopic parameters such as the density of states at the Fermi level and the . effective interaction between the itinerant electrons. This latter information may again be used for an estimate of electron correlation effects in transition metals.
Magnetovolume effects in Ni3Al and Ni-Pt are positive and influence strongly the low temperature ther-
ma1 expansion of the ferromagnetic alloys close to the critical composition. Thermal expansion measurements in the Ni-Pt system have now been performed over the whole range of compositions from pure Ni to pure Pt. These experiments show once more the discontinuity in the low temperature thermal expansion coefficient at the critical composition. The Invar-type of anomaly (clearly present in the composition range 42.5-65 at% Ni) is understood very well and can be described with data for the magnetovolume parameter derived from magnetostriction or high pressure experiments. A negative magnetic contribution to the thermal expansion can also be derived from high pressure data for the weakly ferromagnetic &l--Ni alloys. Negative values for the thermal expansion coefficient, however, have not been observed in this system. The present paper will deal with further results of high pressure studies on intermetallic compounds with different types of crystal structures: Ni3Al and (3, Fe)sGa (Cu3Au-structure), Ti(Fe, Co) and (Ti, Al)Co (CsClstructure). In all these intermetallic systems one or two critical compositions for ferromagnetism have been observed. A few pressure data on the magnetic parameters of some prototypes are shown in table 1. These pressure studies were extended in Ni3Al and Ti(Feo. s Coo. 5) to the non-stoichiometric compounds; in the other cases the ratios Ni/Fe, Co/Fe or Ti/Al were changed. In our discussion of the high pressure results we
J.J.M. Franse / Magnetovolume effects in transition metal alloys
260
Table 1 Effects of pressure on the magnetic parameters of some intermetallic compounds and some binary alloys with weakly ferromagnetic properties aln tro/b p ( 1 0 - 4 kbar -1)
Mn Tc/a p ( 1 0 - 4 kbar -1)
T c a Tc/b p (K2/kbar)
Compounds or alloy
o0 (G cm3/g)
Tc (K)
NiT sA12s Ni74.8Feo.2Ga2 S TiFe0.sCo0. 5 Ti0.sA10.5Co
6.62 4.8 8.5 20.6
43 46 59 130
- 87 - 69 -165 - 42
-116 -105 -226 - 38
-21.5 -22 -80 -65
[1,2]" [2] [31 [4]
4.2 6.5
54 69
-190 - 32
-130 - 41
-38 -19
[6]
Ni4s.2Pts4.8 Pd9sNis
will pay special attention to values for the relative pressure dependence of the spontaneous magnetization o 0 at 4.2 K and the relative shift of the Curie temperature T c with pressure. The experimental data for both quantities are of the same order of magnitude, a characteristic property for a weakly itinerant electron ferromagnet. Absolute values for ~ In Tc/~p turn out to be in general larger than those for ~ln Oo/~p, contrary to what one expects from a simple analysis as given in the next section.
Our phenomenological description of magnetovolume effects in alloy systems with a small uniformly distributed magnetization starts with the following expression for the thermodynamic potential per unit of mass G [7]: (1)
where Go is independent of o and tO, a is the magnetization per unit of mass, 60 the relative volume change, p the density, K the compressibility, p the external pressure, H the external magnetic field and A and B ~material constants. Taking, in the first approximation, the parameter B to be independent of temperature and volume and writing for the derivatives of A: A '= aA/~T
and
C =-½~A/atO,
[5]
dependence of the spontaneous magnetization Oo Oo a O o / 3 p = - K C / B ,
(3)
for the shift with pressure of the Curie temperature
Tc 3 T c / a p = - 2 r C / A ' (Te) ,
(4)
and for the magnetic contribution to the volume toM tO M = p K C o 2 .
(5)
This latter expression is used to describe the volume magnetostriction and thermal expansion measurements. The volume coefficient of thermal expansion is defined
2. Phenomenology and microscopic theory
G = G O + I A o 2 + 1Bo4 + tO2/2pg + ptO/p - H a ,
Ref.
(2)
we derive for the experimental quantities that are determined in our experiments a set of expression in terms of the parameters A', B and C: for the pressure
as
L?q 3 = V\OT~o"
(6)
is in general the sum of an electron and a phonon contribution. At sufficiently low temperatures 3 can be written as 3 = aT + bT 3 ,
(7)
where a and b represent the electron and phonon terms, respectively. Following eq. (5) we expect for a ferromagnetic material an additional contribution to 3 equal t o pgC~)o2/~T. With a quadratic temperature dependence of o 2 we can still use eq. (7). In that case we write a = ae +aM, where ae and aM refer to the electronic and magnetic parts, respectively. In the band model of magnetism expressions can be derived for the material constants A and B in eq. (1) in terms of microscopic quantities such as the density
ZJ.M. Franse / Magnetovolume effects in transition metal alloys of states at the Fermi level and the effective interaction between the itinerant electrons [8]. In a simple version of the microscopic theory one can write
(8)
.4' ( T c ) = - 2 , 4 ( r = o ) / T c .
Using this expression the pressure dependence of the spontaneous magnetization at 0 K, Oo0, can be related to the shift of the Curie temperature with pressure: aln O0o
ap
_
C(0) aln Tc
C(Tc) ap
(9)
'
with C(O)[C(Te) < 1 from the microscopic theory, see ref. [9]. Eq. (8) does not always agree with the experimental observations in which case the temperature dependence of the parameter C can not be derived from eq. (9). A thermodynamic description of materials in which the magnetization is inhomogeneously distributed requires an additional term in eq. (1), representing the spatial variation of G. This is certainly the case for giant moment systems such as P__d-Ni.and (Ni, Fe)3Ga. A microscopic description of magnetovolume effects in these systems is outside the scope of the present paper.
3. Experimental data and discussion 3.1. Ni3Al and (Ni, Fe)3Ga The existing data on magnetovolume effects in Ni3A1 can be found in refs. [1,9,10]. New experimen10"2kbar 1) -2
~ Ni3AI ,
'
.~ {Ni,Fel3Go 'x
-1 &
oi
0
50
girl
I00
Fig. 1. The relative change with pressure of Curie temperature and spontaneous magnetization at 4.2 K plotted as a function of Te for non-stoichiometric Ni3AI compounds and stoichiometric (Ni, Fe)3Ga compounds; data from [1] and [2].
261
Table 2 The magnetovolume parameter C(106 g/cm3) as obtained from pressure and volume magnetostriction measurements for different ferromagnetic Ni3A1 compounds Ni content (at%) 74.8 75 75.5 76
Pressure
Magnetostriction
T=4.2 K
T= Te
T=4.2K
T= Tc
0.17 0.19 0.16 0.29
0.15 0.16 0.14 0.12
0.20 0.21 0.22 0.25
0.14 0.12 0.13 0.13
tal results have been obtained for the pressure dependence of the spontaneous magnetization at 4.2 K. Fig. 1 represents the high pressure data on four ferromagnetic NiaA1 compounds. In order to evaluate the parameter C at different temperatures we made use of eqs. (3) and (4), in combination with experimental values of the parameters A'(Tc) and B(4.2 K). The temperature dependence of the parameter C was determined before from magnetostriction measurements and discussed in the band model of magnetism [9]. The results of both experimental methods for the parameter C are compared in table 2 and found to be in close agreement. It shows again that for NiaA1 (just as for Ni-Pt) the volume magnetostriction can be determined directly from linear magnetostriction measurements. This is not always the case as we observed, for instance, in the dilute P d - N i alloys. The NiT s_x FexGa2s compounds with x values of 0.2 and 1.0 show ferromagnetic order below the Curie temperatures of 46 K and 100 K, respectively. High pressure data on the magnetic parameters of these compounds are very much the same as those in NiaA1, see fig. 1. The magnetovolume parameter KC for x = 0.2 amounts to 78 (g/cm 3 kbar) at 4.2 K; nearly the same value is found for NiTsAl2s. Differences in magnetic behaviour between NiaA1 and (Ni, Fe)aGa are present in compounds with lower iron content where the magnetization curves point to a less homogeneous spatial distribution of the magnetization over the samples. A determination of T c and its pressure derivative for the compound Ni74.9sFeo.osGa2s, for instance, can lead to ambiguous results. The magnetovolume parameter, however, can be derived with fair accuracy for this compound from high magnetic field data and takes on nearly the same value as in Ni74.sFe0.2Ga2s.
J.J.M. Franse /Magnetovolume effects in transition metal alloys
262 100
/
Tc (K)
/," o~--o //: ',
5(3
// /
/
0
o
I
/ /
50
'\ I
;
o'I i
o
- 0.01~
/--'~.5
x --..,,.
.,~'/Y
1
Ti FexCo 1 x
Ti 1.y( Fe0.5C°0.5)1-y
Fig. 2. The Curie temperature as a function of Composition for stoichiometric TiFexCo 1- x compounds and non-stoichiometric Til+y(Fe0.sCoo.s)l_y compounds; data from [2] and [31.
3.2. Ti(Fe, Co)and (Ti, AI)Co The electronic and magnetic properties of the stoichiometric compounds Ti(FexCo l_x) have been extensively studied by Asada and Nos6 [ 11 ]. These authors arrived at the conclusion that a description of the magnetic properties of this system in the band model is appropriate. High magnetic field and high pressure experiments confirm this conclusion, at least for the
( 10-2k bcl r 1 )
compounds at the Co-rich side of the composition range [3]. In our further study of these weakly ferromagnetic intermetallic compounds we prepared a set of non-stoichiometric samples with formula Til + y(Feo, sCoo. 5) t - y. X-ray exposures showed the samples in the interval -0.005 < y < 0.03 to be single phase. A few preliminary results on some non-stochiometric compounds will be reported in this paper. Fig. 2 shows of the Curie temperature as a function of composition in Ti(FexCol -x) and Til+y(Fe0.sCo0.5)l_y. The effect of pressure on the Curie temperature is shown in fig. 3 for both systems. The non-stoichiometric compounds resemble the Corich stoichiometric compounds in so far as aln Oo/ap and aln Tc/a p diverge as the critical composition for ferromagnetism is approached. The magnetic properties of the pseudo-binary alloys Til_xAlxCo have been discussed by Endo, An and Shinogi on the basis of a local environment model [12]. The magnetic behaviour of this system, however, shows some resemblance to that of Ti(Fe, Co); an itinerant model must not be excluded for that reason. Magnetization measurements under high pressures on three ferromagnetic compounds with x-values of 0.4, 0.5 and 0.6 have been performed between 4.2 K and their Curie temperatures of 142, 130 and 57 K, respectively. Saturation of the magnetization is not reached in any of these compounds in fields up to 5 T. Results for aln Tc/ap, also presented in fig. 3, stress the similarity in magnetic behaviour between Ti(Fe, Co) and (Ti, AI)Co. More extended data on the electronic and magnetic properties of these intermetallic compounds will be published elsewhere [4].
-10 Ti FexCo 1_ x ~'L'"~-C o - rich
,
\
-5
--
\k
3.3. Ni-Pt and Pd-Ni
x
Til.y( Feo.5C Oo.5}~_y
+
Til_xAIxCO
x k
0
I
I
I
50
100
150
Tc (KI
Fig. 3. The relative shift of the Curie temperature with pressure as a function of Tc for stoichiometric TiFexCo l_x and Til-x AlxCo compounds and non-stoichiometric Til+y(Feo.sCoo.5)l_y compounds; data from [2-4].
Magnetovolume effects in Ni-Pt alloys around the critical composition for ferromagnetism have been reported in refs. [5,9,10]. Preliminary results on thermal expansion and magnetostriction over the whole range of compositions are shown in figs. 4 and 5. The coefficient a in eq. (7) is plotted in fig. 4 as a function of the nickel content. In the composition range 42.5 to 65 at% Ni negative thermal expansion coefficients are found at sufficiently low temperatures. At the critical composition a sharp discontinuity in the parameter a can be observed. In the phonon contribution to the thermal expansion no such sharp irregularity is
J.J.M. Franse /Magnetovolume effects in transition metal alloys
263
(168911 +
Pdl - x Nix
×
x"
/ ~ x
/
: ~
116~ot(3)"1 b
+
X
Q
/
x
/o
x
/ /
0
( l O S K "1 ) 6
~
- o _
Nil_xPtx
4
i
O 0
~
i
i
I
.05
~X F
x
0.5 0o
"-
i
., ",
-2
Fig. 6. The parameter a, defined by eq. (7), as a function of composition in Pdl_xNix; (X) derived from [ 15 ], (o) from
1i
X
o "d:
[61.
Fig. 4. The parameters a and b, defined in eq. (7), as a function of composition in Nil_xPt x alloys; experimental data from [61 and [91.
detectable. The parameter b in eq. (7) changes linearly with nickel content; the enhanced b value for the alloy with 42.9 at% Ni (not shown in fig. 4) could be ascribed to small fluctuations in the alloy composition. The strong enhancement of the parameter a at the paramagnetic side of the composition range is one of the striking features of the thermal expansion measurements that is also observable in the Pd-Ni system. Spontaneous and forced magnetostriction data for Ni-Pt are shown in fig. 5. The forced magnetostriction data refer to length changes in the field direction on increasing the external field from 1 to 25 T. A detailed 116s)
(10-6)
i
-5
I
-/ //
t I
,
i
AI
i ÷
x
II
~s (.,o)
5
,1 i i
'L
/
Nil_x Pt x
i
x i¢ t/
P
/.
(~,+)
I
'
description of the magnetostriction data around the critical composition for ferromagnetism has been given previously [9]. High pressure experiments on dilute Pd-Ni alloys have been performed by Beille [13]. Around the critical composition of about 2 at% Ni a rather complex model has to be invoked in order to explain the experimental data [14]. We performed thermal expansion measurements around this critical composition on one paramagnetic and two ferromagnetic alloys. Experimental data for the parameter a, defined by eq. (7), are shown in fig. 6. For the ferromagnetic alloys somewhat smaller a-values than that of pure Pd are found due to negative magnetic contributions to this parameter. These magnetic contributions could not be evaluated from magnetostriction measurements in which large anisotropic contributions appeared to be present. From high pressure experiments on the 5 at% Ni alloy a value for aM of - 1.8 X 10 - s (K - l ) can be calculated with the expressions given in section 2, the experimental data of table 1 and a value for the parameter B in eq. (3) of 640 (ga/G2 cm9). Evidently this magnetic contribution is not large enough to make the coefficient a negative.
II t II
¶
I
1
~, \
J
' i 0
.-x
0
t
•
t
~ ~x ,L 05
~v X
Fig. 5. The spontaneous magnetostriction hs and the forced magnetostriction A1/1 (see text) at 4.2 K as a function of composition in Nil_xPtx; experimental data from [6] and [9].
4. Concluding remarks In the present paper recent data on magnetovolume effects in intermetallic compounds of 3d transition metals have been taken from experimental work that is still in progress. Our high pressure studies suggest that
264
J.J.M. Franse /Magnetovolume effects in transition metal alloys
Table 3 The magnetovolume parameter KC in intermetaUic compounds and binary alloys with weakly ferromagnetic properties Compound
B (g3/G2 cm 9)
KC (g/cm 3 kbar)
Ni7 sA12s Ni74.8Feo.2Ga25 TiFeo. 5C°O. 5 Ti0.5 AIo. sCo
208 493 185 59
79 78 220 105
Ni4s.2Pt54.8 Pd95Ni5
675 640
226 87
respectively. Shiga and Nakamura [17] quote a value for KCin 3d transition metals derived from a paper by Janak and Williams [18] of 5 - 1 0 X 10 - 8 (cm2/dyne). The temperature dependence of the parameter C i n Ni3A1 and N i - P t is in agreement with a more extended expression for this parameter in the band model [9]. We note that the pressure data in table 2 are even in better agreement with the theoretical predictions on this temperature dependence for three of the four Ni3A1 compounds.
Acknowledgements Invar-type of anomalies can be found in a wider range of materials. These anomalies have been observed indeed in compounds o f the NiaA1 phase and in disordered N i - P t alloys. The parameter aM itself, defined in section 2, can not be derived with much accuracy in NiaA1 and N i - P t from thermal expansion measurements because o f uncertainties in the extrapolation o f the coefficient ae from the paramagnetic to the ferromagnetic region. Values for the coefficient aM, calculated from high pressure data, are almost a factor o f two smaller in P d - N i than in the N i - P t alloys with corresponding weakly ferromagnetic properties. By using the aevalue of pure Pd one would expect nevertheless, a negative thermal expansion coefficient at low temperature for Pd9sNis, contrary to what is observed experimentally. This means that ae for this particular alloy is about a factor of two larger than in pure Pd. In our final table we present data for the magnetovolume parameter ~ C derived from high pressure experiments at 4.2 K, see table 3. These numbers may be compared with theoretical predictions that can be derived from the Stoner model. Following an expression given by Wohlfarth and Bartell [16] that is based on a Kanamori-type o f interaction between the itinerant electrons, Kortekaas and Franse [9] calculated the following upper values for the magnetovolume parameter in Ni3A1 and N i - P t ; for Ni3AI: C ~ 0.4 × 106 (g/cm 3) or KC~-- 170 (g/cm 3 kbar) for N i - P t : C "~ 0.6 × 106 (g/cm 3) or KC "" 258 (g/cm 3 kbar). Expressing KC per unit of volume these numbers can also be denoted as 2.3 X 10 - 8 (cm2/dyne) and 1.5 X 10 - 8 (cm2/dyne),
Finishing this paper I wish to acknowledge the nice cooperation with N. Buis, H. Hblscher, P. Disveld and E. Waterman in the magnetovolume studies presented in this paper.
References [1] N. Buis, J.J.M. Frame, J. van Haarst, J.P.J. Kaandorp and T. Weesing, Phys. Lett. 56A (1976) 115. [2] N. Buis et al., to be published. [3] N. Buis, J.J.M. Frame and C.J. Schinkel, AlP Conf. Proc. 39 (1978) 389. [4] E. Waterman et al., to be published. [5] H.L. Alberts, J. Beille, D. Bloch and E.P. Wohlfarth, Phys. Rev. B9 (1974) 2233. [6] H. H~51scheret al., to be published. [7] J.J.M. Franse, Physica 86-88B (1977) 283. [8] E.P. Wohlfarth, J. Phys. C2 (1968) 68. [9] T.F.M. Kortekaasand J.J.M. Frame, J. Phys. F6 (1976) 1161. [10] T.F.M. Kortekaas and J.J.M. Frame, Phys. Stat. Sol. A40 (1977) 479. [11 ] Y. Asada and H. Nos~, J. Phys. Soc. Japan 35 (1973) 409. [12] K. Endo, I. An and A. Shinogi, J. Phys. F7 (1977) L99. [13] J. BeiUe, Phys. Lett. 49A (1974) 63. [14] J. Beille and G. Chouteau, J. Phys. F5 (1975) 721. [15] E. Fawcett, E. Bucher, W.F. Brinkman and J.P. Maita, Phys. Rev. Lett. 21 (1968) 1183. [16] E.P. Wohlfarth and L.C. Bartell, Phys. Lett. 34A (1971) 303. [17] M. Shiga and Y. Nakamura, AlP Conf. Proc. 39 (1978) 540. [18] J.F. Janak and A.R. Williams, Phys. Rev. B14 (1976) 4199.
MAGNETOVOLUME EFFECTS IN TRANSITION FOR FERROMAGNETISM
METAL ALLOYS NEAR THE CRITICAL COMPOSITION
J.J.M. FRANSE Natuurkundig Laboratorium der Universiteit van Amsterdam, Amsterdam, The Netherlands
Results of magnetovolume studies are presented on intermetallic compounds and binary alloys with weakly ferromagnetic properties of the following compositions: NisAl, (3, Fe)+Sa, Ti(Fe, Co), (Ti, Al)Co, Ni-Pt and Pd-Ni. Data for the pressure dependence of the spontaneous magnetization at 4.2 K and the shift of the Curie temperature with pressure follow roughly the relations that can be derived in the model for weak itinerant ferromagnetism in most of the investigated systems. Invar-type of anomalies in the thermal expansion have already been observed in several cases. Experimental values for the magnetovolume parameter are derived and compared with theoretical predictions in the Stoner model.
1. Introduction Magnetovolume effects in transition metals have a long tradition of theoretical interest (electron correlation) and technical applications (Invar alloys). Our study of transition metal alloys is directed to a systematic investigation of magnetovolume effects in alloys around the critical composition for ferromagnetism. These magnetovolume effects include the pressure dependence of the spontaneous magnetization at 4.2 K, the shift with pressure of the Curie temperature, the magnetic contribution to the linear thermal expansion coefficient and the field induced linear magnetostriction. In some cases (non-stoichiometric Ni3Al compounds in the range of 74 to 76 at% Ni, disordered Ni-Pt alloys around the critical composition of 42.5 at% Ni) the set of existing magnetovolume data can be represented by a single magnetovolume parameter that turns out to be only slightly temperature dependent. An interpretation of this parameter in the band model of magnetism yields information about the volume dependence of microscopic parameters such as the density of states at the Fermi level and the . effective interaction between the itinerant electrons. This latter information may again be used for an estimate of electron correlation effects in transition metals.
Magnetovolume effects in Ni3Al and Ni-Pt are positive and influence strongly the low temperature ther-
ma1 expansion of the ferromagnetic alloys close to the critical composition. Thermal expansion measurements in the Ni-Pt system have now been performed over the whole range of compositions from pure Ni to pure Pt. These experiments show once more the discontinuity in the low temperature thermal expansion coefficient at the critical composition. The Invar-type of anomaly (clearly present in the composition range 42.5-65 at% Ni) is understood very well and can be described with data for the magnetovolume parameter derived from magnetostriction or high pressure experiments. A negative magnetic contribution to the thermal expansion can also be derived from high pressure data for the weakly ferromagnetic &l--Ni alloys. Negative values for the thermal expansion coefficient, however, have not been observed in this system. The present paper will deal with further results of high pressure studies on intermetallic compounds with different types of crystal structures: Ni3Al and (3, Fe)sGa (Cu3Au-structure), Ti(Fe, Co) and (Ti, Al)Co (CsClstructure). In all these intermetallic systems one or two critical compositions for ferromagnetism have been observed. A few pressure data on the magnetic parameters of some prototypes are shown in table 1. These pressure studies were extended in Ni3Al and Ti(Feo. s Coo. 5) to the non-stoichiometric compounds; in the other cases the ratios Ni/Fe, Co/Fe or Ti/Al were changed. In our discussion of the high pressure results we
J.J.M. Franse / Magnetovolume effects in transition metal alloys
260
Table 1 Effects of pressure on the magnetic parameters of some intermetallic compounds and some binary alloys with weakly ferromagnetic properties aln tro/b p ( 1 0 - 4 kbar -1)
Mn Tc/a p ( 1 0 - 4 kbar -1)
T c a Tc/b p (K2/kbar)
Compounds or alloy
o0 (G cm3/g)
Tc (K)
NiT sA12s Ni74.8Feo.2Ga2 S TiFe0.sCo0. 5 Ti0.sA10.5Co
6.62 4.8 8.5 20.6
43 46 59 130
- 87 - 69 -165 - 42
-116 -105 -226 - 38
-21.5 -22 -80 -65
[1,2]" [2] [31 [4]
4.2 6.5
54 69
-190 - 32
-130 - 41
-38 -19
[6]
Ni4s.2Pts4.8 Pd9sNis
will pay special attention to values for the relative pressure dependence of the spontaneous magnetization o 0 at 4.2 K and the relative shift of the Curie temperature T c with pressure. The experimental data for both quantities are of the same order of magnitude, a characteristic property for a weakly itinerant electron ferromagnet. Absolute values for ~ In Tc/~p turn out to be in general larger than those for ~ln Oo/~p, contrary to what one expects from a simple analysis as given in the next section.
Our phenomenological description of magnetovolume effects in alloy systems with a small uniformly distributed magnetization starts with the following expression for the thermodynamic potential per unit of mass G [7]: (1)
where Go is independent of o and tO, a is the magnetization per unit of mass, 60 the relative volume change, p the density, K the compressibility, p the external pressure, H the external magnetic field and A and B ~material constants. Taking, in the first approximation, the parameter B to be independent of temperature and volume and writing for the derivatives of A: A '= aA/~T
and
C =-½~A/atO,
[5]
dependence of the spontaneous magnetization Oo Oo a O o / 3 p = - K C / B ,
(3)
for the shift with pressure of the Curie temperature
Tc 3 T c / a p = - 2 r C / A ' (Te) ,
(4)
and for the magnetic contribution to the volume toM tO M = p K C o 2 .
(5)
This latter expression is used to describe the volume magnetostriction and thermal expansion measurements. The volume coefficient of thermal expansion is defined
2. Phenomenology and microscopic theory
G = G O + I A o 2 + 1Bo4 + tO2/2pg + ptO/p - H a ,
Ref.
(2)
we derive for the experimental quantities that are determined in our experiments a set of expression in terms of the parameters A', B and C: for the pressure
as
L?q 3 = V\OT~o"
(6)
is in general the sum of an electron and a phonon contribution. At sufficiently low temperatures 3 can be written as 3 = aT + bT 3 ,
(7)
where a and b represent the electron and phonon terms, respectively. Following eq. (5) we expect for a ferromagnetic material an additional contribution to 3 equal t o pgC~)o2/~T. With a quadratic temperature dependence of o 2 we can still use eq. (7). In that case we write a = ae +aM, where ae and aM refer to the electronic and magnetic parts, respectively. In the band model of magnetism expressions can be derived for the material constants A and B in eq. (1) in terms of microscopic quantities such as the density
ZJ.M. Franse / Magnetovolume effects in transition metal alloys of states at the Fermi level and the effective interaction between the itinerant electrons [8]. In a simple version of the microscopic theory one can write
(8)
.4' ( T c ) = - 2 , 4 ( r = o ) / T c .
Using this expression the pressure dependence of the spontaneous magnetization at 0 K, Oo0, can be related to the shift of the Curie temperature with pressure: aln O0o
ap
_
C(0) aln Tc
C(Tc) ap
(9)
'
with C(O)[C(Te) < 1 from the microscopic theory, see ref. [9]. Eq. (8) does not always agree with the experimental observations in which case the temperature dependence of the parameter C can not be derived from eq. (9). A thermodynamic description of materials in which the magnetization is inhomogeneously distributed requires an additional term in eq. (1), representing the spatial variation of G. This is certainly the case for giant moment systems such as P__d-Ni.and (Ni, Fe)3Ga. A microscopic description of magnetovolume effects in these systems is outside the scope of the present paper.
3. Experimental data and discussion 3.1. Ni3Al and (Ni, Fe)3Ga The existing data on magnetovolume effects in Ni3A1 can be found in refs. [1,9,10]. New experimen10"2kbar 1) -2
~ Ni3AI ,
'
.~ {Ni,Fel3Go 'x
-1 &
oi
0
50
girl
I00
Fig. 1. The relative change with pressure of Curie temperature and spontaneous magnetization at 4.2 K plotted as a function of Te for non-stoichiometric Ni3AI compounds and stoichiometric (Ni, Fe)3Ga compounds; data from [1] and [2].
261
Table 2 The magnetovolume parameter C(106 g/cm3) as obtained from pressure and volume magnetostriction measurements for different ferromagnetic Ni3A1 compounds Ni content (at%) 74.8 75 75.5 76
Pressure
Magnetostriction
T=4.2 K
T= Te
T=4.2K
T= Tc
0.17 0.19 0.16 0.29
0.15 0.16 0.14 0.12
0.20 0.21 0.22 0.25
0.14 0.12 0.13 0.13
tal results have been obtained for the pressure dependence of the spontaneous magnetization at 4.2 K. Fig. 1 represents the high pressure data on four ferromagnetic NiaA1 compounds. In order to evaluate the parameter C at different temperatures we made use of eqs. (3) and (4), in combination with experimental values of the parameters A'(Tc) and B(4.2 K). The temperature dependence of the parameter C was determined before from magnetostriction measurements and discussed in the band model of magnetism [9]. The results of both experimental methods for the parameter C are compared in table 2 and found to be in close agreement. It shows again that for NiaA1 (just as for Ni-Pt) the volume magnetostriction can be determined directly from linear magnetostriction measurements. This is not always the case as we observed, for instance, in the dilute P d - N i alloys. The NiT s_x FexGa2s compounds with x values of 0.2 and 1.0 show ferromagnetic order below the Curie temperatures of 46 K and 100 K, respectively. High pressure data on the magnetic parameters of these compounds are very much the same as those in NiaA1, see fig. 1. The magnetovolume parameter KC for x = 0.2 amounts to 78 (g/cm 3 kbar) at 4.2 K; nearly the same value is found for NiTsAl2s. Differences in magnetic behaviour between NiaA1 and (Ni, Fe)aGa are present in compounds with lower iron content where the magnetization curves point to a less homogeneous spatial distribution of the magnetization over the samples. A determination of T c and its pressure derivative for the compound Ni74.9sFeo.osGa2s, for instance, can lead to ambiguous results. The magnetovolume parameter, however, can be derived with fair accuracy for this compound from high magnetic field data and takes on nearly the same value as in Ni74.sFe0.2Ga2s.
J.J.M. Franse /Magnetovolume effects in transition metal alloys
262 100
/
Tc (K)
/," o~--o //: ',
5(3
// /
/
0
o
I
/ /
50
'\ I
;
o'I i
o
- 0.01~
/--'~.5
x --..,,.
.,~'/Y
1
Ti FexCo 1 x
Ti 1.y( Fe0.5C°0.5)1-y
Fig. 2. The Curie temperature as a function of Composition for stoichiometric TiFexCo 1- x compounds and non-stoichiometric Til+y(Fe0.sCoo.s)l_y compounds; data from [2] and [31.
3.2. Ti(Fe, Co)and (Ti, AI)Co The electronic and magnetic properties of the stoichiometric compounds Ti(FexCo l_x) have been extensively studied by Asada and Nos6 [ 11 ]. These authors arrived at the conclusion that a description of the magnetic properties of this system in the band model is appropriate. High magnetic field and high pressure experiments confirm this conclusion, at least for the
( 10-2k bcl r 1 )
compounds at the Co-rich side of the composition range [3]. In our further study of these weakly ferromagnetic intermetallic compounds we prepared a set of non-stoichiometric samples with formula Til + y(Feo, sCoo. 5) t - y. X-ray exposures showed the samples in the interval -0.005 < y < 0.03 to be single phase. A few preliminary results on some non-stochiometric compounds will be reported in this paper. Fig. 2 shows of the Curie temperature as a function of composition in Ti(FexCol -x) and Til+y(Fe0.sCo0.5)l_y. The effect of pressure on the Curie temperature is shown in fig. 3 for both systems. The non-stoichiometric compounds resemble the Corich stoichiometric compounds in so far as aln Oo/ap and aln Tc/a p diverge as the critical composition for ferromagnetism is approached. The magnetic properties of the pseudo-binary alloys Til_xAlxCo have been discussed by Endo, An and Shinogi on the basis of a local environment model [12]. The magnetic behaviour of this system, however, shows some resemblance to that of Ti(Fe, Co); an itinerant model must not be excluded for that reason. Magnetization measurements under high pressures on three ferromagnetic compounds with x-values of 0.4, 0.5 and 0.6 have been performed between 4.2 K and their Curie temperatures of 142, 130 and 57 K, respectively. Saturation of the magnetization is not reached in any of these compounds in fields up to 5 T. Results for aln Tc/ap, also presented in fig. 3, stress the similarity in magnetic behaviour between Ti(Fe, Co) and (Ti, AI)Co. More extended data on the electronic and magnetic properties of these intermetallic compounds will be published elsewhere [4].
-10 Ti FexCo 1_ x ~'L'"~-C o - rich
,
\
-5
--
\k
3.3. Ni-Pt and Pd-Ni
x
Til.y( Feo.5C Oo.5}~_y
+
Til_xAIxCO
x k
0
I
I
I
50
100
150
Tc (KI
Fig. 3. The relative shift of the Curie temperature with pressure as a function of Tc for stoichiometric TiFexCo l_x and Til-x AlxCo compounds and non-stoichiometric Til+y(Feo.sCoo.5)l_y compounds; data from [2-4].
Magnetovolume effects in Ni-Pt alloys around the critical composition for ferromagnetism have been reported in refs. [5,9,10]. Preliminary results on thermal expansion and magnetostriction over the whole range of compositions are shown in figs. 4 and 5. The coefficient a in eq. (7) is plotted in fig. 4 as a function of the nickel content. In the composition range 42.5 to 65 at% Ni negative thermal expansion coefficients are found at sufficiently low temperatures. At the critical composition a sharp discontinuity in the parameter a can be observed. In the phonon contribution to the thermal expansion no such sharp irregularity is
J.J.M. Franse /Magnetovolume effects in transition metal alloys
263
(168911 +
Pdl - x Nix
×
x"
/ ~ x
/
: ~
116~ot(3)"1 b
+
X
Q
/
x
/o
x
/ /
0
( l O S K "1 ) 6
~
- o _
Nil_xPtx
4
i
O 0
~
i
i
I
.05
~X F
x
0.5 0o
"-
i
., ",
-2
Fig. 6. The parameter a, defined by eq. (7), as a function of composition in Pdl_xNix; (X) derived from [ 15 ], (o) from
1i
X
o "d:
[61.
Fig. 4. The parameters a and b, defined in eq. (7), as a function of composition in Nil_xPt x alloys; experimental data from [61 and [91.
detectable. The parameter b in eq. (7) changes linearly with nickel content; the enhanced b value for the alloy with 42.9 at% Ni (not shown in fig. 4) could be ascribed to small fluctuations in the alloy composition. The strong enhancement of the parameter a at the paramagnetic side of the composition range is one of the striking features of the thermal expansion measurements that is also observable in the Pd-Ni system. Spontaneous and forced magnetostriction data for Ni-Pt are shown in fig. 5. The forced magnetostriction data refer to length changes in the field direction on increasing the external field from 1 to 25 T. A detailed 116s)
(10-6)
i
-5
I
-/ //
t I
,
i
AI
i ÷
x
II
~s (.,o)
5
,1 i i
'L
/
Nil_x Pt x
i
x i¢ t/
P
/.
(~,+)
I
'
description of the magnetostriction data around the critical composition for ferromagnetism has been given previously [9]. High pressure experiments on dilute Pd-Ni alloys have been performed by Beille [13]. Around the critical composition of about 2 at% Ni a rather complex model has to be invoked in order to explain the experimental data [14]. We performed thermal expansion measurements around this critical composition on one paramagnetic and two ferromagnetic alloys. Experimental data for the parameter a, defined by eq. (7), are shown in fig. 6. For the ferromagnetic alloys somewhat smaller a-values than that of pure Pd are found due to negative magnetic contributions to this parameter. These magnetic contributions could not be evaluated from magnetostriction measurements in which large anisotropic contributions appeared to be present. From high pressure experiments on the 5 at% Ni alloy a value for aM of - 1.8 X 10 - s (K - l ) can be calculated with the expressions given in section 2, the experimental data of table 1 and a value for the parameter B in eq. (3) of 640 (ga/G2 cm9). Evidently this magnetic contribution is not large enough to make the coefficient a negative.
II t II
¶
I
1
~, \
J
' i 0
.-x
0
t
•
t
~ ~x ,L 05
~v X
Fig. 5. The spontaneous magnetostriction hs and the forced magnetostriction A1/1 (see text) at 4.2 K as a function of composition in Nil_xPtx; experimental data from [6] and [9].
4. Concluding remarks In the present paper recent data on magnetovolume effects in intermetallic compounds of 3d transition metals have been taken from experimental work that is still in progress. Our high pressure studies suggest that
264
J.J.M. Franse /Magnetovolume effects in transition metal alloys
Table 3 The magnetovolume parameter KC in intermetaUic compounds and binary alloys with weakly ferromagnetic properties Compound
B (g3/G2 cm 9)
KC (g/cm 3 kbar)
Ni7 sA12s Ni74.8Feo.2Ga25 TiFeo. 5C°O. 5 Ti0.5 AIo. sCo
208 493 185 59
79 78 220 105
Ni4s.2Pt54.8 Pd95Ni5
675 640
226 87
respectively. Shiga and Nakamura [17] quote a value for KCin 3d transition metals derived from a paper by Janak and Williams [18] of 5 - 1 0 X 10 - 8 (cm2/dyne). The temperature dependence of the parameter C i n Ni3A1 and N i - P t is in agreement with a more extended expression for this parameter in the band model [9]. We note that the pressure data in table 2 are even in better agreement with the theoretical predictions on this temperature dependence for three of the four Ni3A1 compounds.
Acknowledgements Invar-type of anomalies can be found in a wider range of materials. These anomalies have been observed indeed in compounds o f the NiaA1 phase and in disordered N i - P t alloys. The parameter aM itself, defined in section 2, can not be derived with much accuracy in NiaA1 and N i - P t from thermal expansion measurements because o f uncertainties in the extrapolation o f the coefficient ae from the paramagnetic to the ferromagnetic region. Values for the coefficient aM, calculated from high pressure data, are almost a factor o f two smaller in P d - N i than in the N i - P t alloys with corresponding weakly ferromagnetic properties. By using the aevalue of pure Pd one would expect nevertheless, a negative thermal expansion coefficient at low temperature for Pd9sNis, contrary to what is observed experimentally. This means that ae for this particular alloy is about a factor of two larger than in pure Pd. In our final table we present data for the magnetovolume parameter ~ C derived from high pressure experiments at 4.2 K, see table 3. These numbers may be compared with theoretical predictions that can be derived from the Stoner model. Following an expression given by Wohlfarth and Bartell [16] that is based on a Kanamori-type o f interaction between the itinerant electrons, Kortekaas and Franse [9] calculated the following upper values for the magnetovolume parameter in Ni3A1 and N i - P t ; for Ni3AI: C ~ 0.4 × 106 (g/cm 3) or KC~-- 170 (g/cm 3 kbar) for N i - P t : C "~ 0.6 × 106 (g/cm 3) or KC "" 258 (g/cm 3 kbar). Expressing KC per unit of volume these numbers can also be denoted as 2.3 X 10 - 8 (cm2/dyne) and 1.5 X 10 - 8 (cm2/dyne),
Finishing this paper I wish to acknowledge the nice cooperation with N. Buis, H. Hblscher, P. Disveld and E. Waterman in the magnetovolume studies presented in this paper.
References [1] N. Buis, J.J.M. Frame, J. van Haarst, J.P.J. Kaandorp and T. Weesing, Phys. Lett. 56A (1976) 115. [2] N. Buis et al., to be published. [3] N. Buis, J.J.M. Frame and C.J. Schinkel, AlP Conf. Proc. 39 (1978) 389. [4] E. Waterman et al., to be published. [5] H.L. Alberts, J. Beille, D. Bloch and E.P. Wohlfarth, Phys. Rev. B9 (1974) 2233. [6] H. H~51scheret al., to be published. [7] J.J.M. Franse, Physica 86-88B (1977) 283. [8] E.P. Wohlfarth, J. Phys. C2 (1968) 68. [9] T.F.M. Kortekaasand J.J.M. Frame, J. Phys. F6 (1976) 1161. [10] T.F.M. Kortekaas and J.J.M. Frame, Phys. Stat. Sol. A40 (1977) 479. [11 ] Y. Asada and H. Nos~, J. Phys. Soc. Japan 35 (1973) 409. [12] K. Endo, I. An and A. Shinogi, J. Phys. F7 (1977) L99. [13] J. BeiUe, Phys. Lett. 49A (1974) 63. [14] J. Beille and G. Chouteau, J. Phys. F5 (1975) 721. [15] E. Fawcett, E. Bucher, W.F. Brinkman and J.P. Maita, Phys. Rev. Lett. 21 (1968) 1183. [16] E.P. Wohlfarth and L.C. Bartell, Phys. Lett. 34A (1971) 303. [17] M. Shiga and Y. Nakamura, AlP Conf. Proc. 39 (1978) 540. [18] J.F. Janak and A.R. Williams, Phys. Rev. B14 (1976) 4199.