The magnetic phase transition in ordered and disordered ferromagnets

Journal of Magnetism and Magnetic Materials 78 (1989) 393-402 North-Holland, A m s t e r d a m

393

T H E M A G N E T I C P H A S E T R A N S I T I O N IN O R D E R E D AND DISORDERED FERROMAGNETS M. S E E G E R and H. K R O N M ( ] L L E R lnstitut ]`dr Physik, Max-Planck Institut ]'dr Metallforschung, Heisenbergstr. 1, Pf 800665, D-7000 Stuttgart 80, Fed. Rep. Germany Received 4 October 1988; in revised form 28 November 1988

The magnetic phase transition is investigated for the structurally disordered alloys FeNi 3, Fe3Pt, FePd 3 and for the ordered alloys Fe3Pt and FePd 3. The asymptotic critical exponents B, Y, "r' and 8 are determined by several methods: modified Arrott plot, method of Kouvel and Fisher, scaling plot and In M vs. In H plot. N o influence of structural disorder on the critical exponents is found. Furthermore the temperature dependence of the K o u v e l - F i s h e r exponent y ( T ) is discussed, for which a decisive difference between ordered and disordered materials is expected. The values of the exponents for the alloy FeNi 3 are consistent with the predictions for the 3d Heisenberg model. In contrast, for FeaPt and FePd 3 the exponent values are modified towards the mean field values, presumably because of long-range exchange interactions.

1. Introduction

The critical exponents describing the temperature dependence of the spontaneous magnetization M s and the susceptibilities X and X', or the dependence of the magnetization M on the magnetic field H near the Curie temperature, Tc, are defined as Ms-(T c-T)

#

X -(T-Tc)

-Y

( T - + T c,

T < T c, H = 0 ) ,

( T ~ T c,

T > T c, H = 0 ) ,

( T - - + T c,

T
p

X' - ( T c - T )

-~

M - H 1/~

( T = To).

H=O),

(1)

Only two of these exponents are independent from each other because they are related by the scaling law y = y ' = f l ( 8 - 1).

(2)

Earlier investigations of the magnetic phase transition (cf. refs. [1-5]) have shown, that the asymptotic critical behaviour of ordered as well as disordered Heisenberg ferromagnets can be described by the exponents derived for the 3-dimen-

sional Heisenberg model. In agreement with the Harris criterion [6] no influence of structural disorder on the values of the critical exponents has been found. On the other hand, the temperature dependent Kouvel-Fisher exponent [7], which characterizes the temperature range between the critical regime ( T = T¢) and the mean field regime ( T >> Tc), dx -t y(T)=(T-Tc)X(T) dr (3) shows strong differences between ordered and disordered systems: for all homogeneous isotropic ferromagnets with short-range interactions the quantity y ( T ) decreases monotonically with increasing temperature. The exponent y ( T ) varies from the critical value 7 = 1.387 to the mean field value y = 1. In contrast, for disordered systems the Kouvel-Fisher exponent y ( T ) exhibits a characteristic nonmonotonic temperature dependence with a maximum of y ( T ) at reduced temperatures t = ( T - T c ) / T c of t = 0.1-0.5 with 7(T)-values up to y ( T ) = 1.5-1.8. This behaviour agrees with the predictions of the correlated molecular field theory [3,8,9] and Monte Carlo computer simulations [3,10,11].

0304-8853/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

394

M. Seeger, 1-1. Kronmiiller / Phase transition in ordered and disordered ferromagnets

In this paper we present investigations of the magnetic phase transition in the binary ferromagnetic alloys Fe3Pt, FePd 3 and FeN± 3. For the alloys Fe 3Pt and FePd 3 we investigate both ordered and chemically disordered samples. Whereas m a n y properties of the FeNi-alloy system may be explained by a model of localized magnetic moments with short-range exchange interactions (cf. ref. [12]), it is well known (cf. ref. [13]) that neither the localized nor the itinerant model gives complete agreement with experimental observations for Fe3Pt or FePd 3. For these materials the so-called s - d model has been applied [14] in which it is assumed that there are fairly well-defined 3d local moments on Fe atoms that couple with itinerant electrons of Pd or Pt. As a result, there m a y be long-range effective exchange interactions between the Fe atoms. Indeed, when adopting a localized Heisenberg model, up to six neighbours for F e - F e pairs had to be taken into account for a discussion of the spin wave spectrum [15]. For FeN± 3 the order-disorder transition temperature Tcrit = 773 K is below the Curie temperature Tc = 872 K. It therefore does not exhibit atomic long-range order at the magnetic phase transition, but a quite strong short-range order [16]. Our aim is to clarify the following questions: 1. Are the critical exponents consistent with the values for the 3d Heisenberg model, both for the ordered and the disordered systems? 2. Is there an influence of the disorder on the temperature dependence of 7(T), i.e. do we get the typical monotonic behaviour for the ordered and the nonmonotonic behaviour for the disordered systems? 3. What is the influence of the range of the effective exchange interactions upon the critical behaviour?

analysis of the samples yielded the following compositions: FeN± 3

:

Fe25.0±0.3Ni75.0±0.3,

Fe3Pt

:

Fe73.2±l.3Pt26.8±l.2,

FePd 3

:

Fe24.9±o.3Pd75.l±0.3.

The samples were at first homogenized by an annealing treatment (FEN±3:3 d at 1270 K, Fe3Pt and F e P d 3 : 6 d at 1270 K). The ordered samples were subsequently annealed at temperatures just below the order-disorder transition temperature (Fe3Pt:6 d at 870 K, F e P d 3 : 2 8 d at 770 K). It should be noted that the terms 'ordered' and 'disordered' have only a qualitative meaning in this paper. It may well be that our 'ordered' materials do not exhibit a perfect long-range order, but a certain amount of structural disorder. Accordingly, the term 'disordered' does not necessarily mean perfect atomic disorder, but there may be an even large amount of structural short-range order, as in the case of disordered FeN± 3 (see above). To illustrate this point, we note that our disordered Fe3Pt sample exhibits a rather large Curie temperature of Tc = 432.1 K, whereas the critical temperature for completely disordered Fe3Pt is T c = 287 K [17]. To determine the critical exponents, isothermal magnetization curves M = M ( H , T ) were measured with a vibration magnetometer at applied fields 0 < H a < 9200 Oe in temperature steps of 1 K within the range Tc + 20 K and 5 K in a wider range. The internal field H was calculated by subtracting the demagnetizing field from H a . The temperature was measured by a thermocouple element NiCr vs. NiA1 which was close to the sample. The temperature stability was better than 0.1 K for the isotherms.

3. Analysis of the data 2. Experimental techniques 3.1. Determination of fl and y

The samples were melted inductively using at least 99.9% pure metals. Spheres of about 3 - 5 m m in diameter were obtained from the ingots. An

To determine the exponents /3 and y, we consider the A r r o t t - N o a k e s equation of state [18],

M. Seeger,H. Kronmiiller/ Phasetransitionin orderedanddisorderedferromagnets 0

86818 K 87 3fi K

+

872.t7 k

"14

FeNi3

~

X 872 97 k O 873.75 g

/

c~ t2

x~

"~" 876.24K

Lnolo ~

8

6

4

2 /

• ~.4-~, "2......... ~'~ ....... 6' ........ ; ......... I'; ........ 1'2........ I)4 ........ 1'5....

(HIMIl/li Fig. 1. Modified Arrott plot for isotherms in the critical regime for FeNi 3.

tems we therefore must get rid of the low-field anomalies and we do this by extrapolating from high fields (fig. 1). The intersections of t h e straight lines with the ordinate and the abscissa mark the values M~/# for T < To respectively X -1/~' for T > To The critical isotherm for T = Tc passes through the origin. The advantage of this method is that the exponents can be optimized for a very small temperature range, in principle for one isothermal curve, above and below To respectively. A further possibility to determine the exponents fl and y is the method of Kouvel and Fisher. Starting from the definition of ~,(T), we get (~T(ln

X - 1 ) ) - I = x - 1 / d-~(X -1)

which was first postulated for the magnetic phase transition of pure Nickel single crystals (n)

1/Y

395

1

- - -~(T) (r-

rc).

(5)

T - T c +( M ) 1/# Mll

(4)

with the material parameters M 1 and T 1. Starting from this equation, the experimental data are plotted in the form of the modified Arrott plot, M 1/~ vs. ( H / M ) l/Y, where the exponents B and 3' are chosen in such a way that the isothermal curves in the critical regime represent as well as possible straight lines (see fig. 1). In general, it is impossible to get all data for one isothermal curve on a straight line, because the low-field data ( H a < 500-1000 Oe) deviate from linearity. The reason for this deviation is still unknown. Because these low-field anomalies occur both for the ordered and for the disordered systems (as in the paper of Shen et al. [19] but in contrast to ref. [20]) and even for the case of monocrystalline Ni [1], an interpretation in terms of sample inhomogeneities is not possible. On the other hand there are hints [1,21] that these anomalies have nothing to do with the critical behaviour of an infinitely extended system, i.e. with an equation of state different from eq. (4). We believe that domain effects or stray fields become evident in this low-field regime. To compare with the theoretical results for infinite sys-

When plotting the quantity X - l / d x - a / d T vs. T we get a straight line in the critical regime with the slope ,/-1 and intersection Tc (fig. 2). The onset of deviations of the measured data from the straight line marks the end of the critical regime in the paramagnetic region. In the same way we can determine the values fl and Tc by plotting

2O

1"

F e N i3 18

,~ t6 14

10 8 6

I. 301 T~ = 872.907

4 2 0

875

8~fl

885

890

895

T [Kt

Fig. 2. Kouvel-Fisher plot for y for FeNi 3.

900

-=)

396

M. Seeger, H. Kronmiiller / Phase transition in ordered and disorderedferromagnets 0

FeN

t

FeN

i 3

1' 9 o

10 C13___ 8 5 ./. ..<>'. "~

-20 {Z

./.'.[<"

80

.t'..¢"

T
..,~"

-

1.310

-

0.390

#. D

75 ......

..-

/ ./

T t ~ 872.997

T>T C

Z"

/ .,./

H

Ranse ~, ~ I

~

H 656

~

'

~ 858

'

~

' 860

~

~

' 862

'

~

'

'

864

'

~

'

'

~

066

H

L~I

....

I],,,I

....

P. . . .

I ....

:

872.980

K

!,~,,I,,,,I,,,,I

....

± I ....

1 l

~

868

870 T

t3

872

14

15

16

17

18

19

20

21

22

ln(H/Itl(0+~

[K]

23

24

) }

25

.-~

Fig. 4. Scaling plot for FeNi 3.

Fig. 3. Kouvel-Fisher plot for/3 for FeNi 3.

Ms/(dMs/dT ) vs. T in the ferromagnetic region (fig. 3), starting from the relation

tions

of

the

exponents.

For

large

values

of

ln(H/Itl#+V), i.e. for T = To, the data may be fitted by a straight line (fig. 4) with slope 1 / 6 .

(d-d~(lnMs))-'=Ms/d~ Ms /3(T) - -

(T-

Tc).

3.2. Determination of 8

(6) In contrast to the A r r o t t - N o a k e s plot the K o u v e l - F i s h e r method requires data for a larger temperature range for a useful least mean square fit to a straight line. Another way to evaluate the exponents/3 and y is the scaling plot. If the scaling hypothesis is valid, there exists a reduced equation of state of the form

The exponent 6, which describes the field dependence of the magnetization for T = Tc can be obtained from a In M vs. In H plot. The isotherms for T < Tc are convex and for T > Tc concave (fig. 5). Because we do not measure exactly for T = To we obtain the value of 1/6 by

55

j

I"

Itl # with the two reduced variables M' = M/I t I ¢ and H ' = I-Ill t l ¢÷v. The plus or minus sign denotes the ferromagnetic and the paramagnetic region, respectively. With the correct values for # and ~, we obtain a data collapse into two branches by plotting l n ( M / I t I #) vs. l n ( H / [ t [P+~) (fig. 4). In the case of this plot no data from outside the critical regime should be taken [22]. In this plot also the aforementioned low-field data are not used, because they have nothing to do with the critical behaviour [1,21]. If m a n y data are available, the scaling plot is very sensitive to modifica-

FeNi~

o

5 O @

0

0

@OZx

A

A

A

+

a 45

X

X

A

0

+

A

+

~

x xx xX

X

40

@

X

3 5

+

A

871 36 K 872.17 K

X

873 75 K

X X

3 O ~ I ' H ' " ' " I

2

''~

3

......

I .........

I .........

4

5

I

6 InlHI

......

'",

.........

J ' " " " " l

8

7

9

-')'

Fig. 5. In M vs. In H plot in the critical regime to determine 8 for FeNi 3-

M. Seeger, H. Kronm£dler

1'

/ Phase transition

FeNi3

858.18 K 87t.~ K 872 17 K 872 97 K 873.75 K

0 A + X

14

~

12

0

~o

*0

~

8

% 6 4

in ordered and disordered ferromagnets

397

described in the critical regime by the exponent 7 ' (see eq. (1)). Because of the above discussed lowfield anomalies we could not apply the previously published methods [23,24] to evaluate 7'. We therefore have tested the relation 7 = 7 ' by optimizing the modified Arrott plots separately for T < Tc and T > Tc, using the same value of fl but allowing different values for 7 and 7 ' (fig. 6). In the same way we can optimize separately the two branches in the scaling plot (fig. 7).

2 /

4. Results and discussion Fig. 6. Modified Arrott plot with independent values y and y' for FeNi 3. interpolating the slopes of the approximately straight parts of the near-critical isotherms for large In H (fig. 5). This value for ~ can be compared with the one we get from fl and Y with the scaling law (eq. (2)), where fl and 7 are determined by one of the above discussed methods.

3.3. Determination of Y' The magnetization below Tc consists of two parts: spontaneous magnetization and induced magnetization

M(H, T) = Ms(T ) + x'(H, T)H.

(8)

The temperature dependence of X' (for H --* 0) is

•l~ 9 o

FeNi3

B5

< C

.<:;.

.'<" ~ 0

.'"/'" /~ /"

-T(TC

~J"

y ~

7 5

/

/

/'

T
."

/"

1 • 300

I

T>T C ~ - 1 3 1 0

/J

8 - o. 390

T)Tc

/

~o

/ Ranee

13

14

15

IB

17

18

19

20

872

21

in[H/itl[13+~))

22

900 K i 23

24

1 ~g

25

,,~

Fig. 7. Scaling plot with independent values y and 7' for FeNi 3.

The results for the critical exponents are shown in table 1, together with the values for the 3dimensional Heisenberg model. We estimate an error of + 10% for all exponent values. For the modified Arrott plot there are no significant disapprovements of the quality of the plots when modifying the exponent values up to 10%. For the K o u v e l - F i s h e r method the statistical error for the least mean square fit of the straight line is much smaller than 10%. In this case the error limit results from the uncertainties in the determination of X -1 from the modified Arrott plot, related to the low-field anomalies in the isotherms, as described in full detail in ref. [21]. For the scaling plot the error estimate characterizes the range of exponent values which yields a data collapse of comparable quality, while Tc is modified between its error limits. We note the following points: 1. Th.e different methods applied for a determination of the critical exponents within the error limits lead to consistent results. 2. The scaling law (eq. (2)) is found to be satisfied for all samples. The value 7 ' within the error limits is identical with 7, and the value ~ calculated from fl and 7 using eq. (2) is consistent with that one obtained from the In M vs. In H plot. 3. The exponents for the alloy FeNi 3 within the error bars are consistent with those for the 3d Heisenberg model and with those obtained for Ni (cf. refs. [1,25]) and for disordered crystalline FeToNi30 and Fe66Ni34 [1]. 4. For the alloys FeaPt and FePd3 there is no difference in the exponent values between ordered

M. Seeger, H. Kronmfiller / Phase transition in ordered and disordered ferromagnets

398

Table 1 Values for the exponents r, 3', 3" and 8 as derived from different methods, as well as the values for T¢ and the width of the asymptotic critical regime as determined from the Kouvel-Fisher plots

Tc(K ) Kouvel-Fisher, fl modif. Arrott fl Kouvel-Fisher fl scaling 3' modif. Arrott 3' Kouvel-Fisher 3' scaling 3" modif. Arrott 3" scaling 8 In M vs. In H width of crit. regime (% of Kouvel-Fisher, Kouvel-Fisher,

3'

Tc) fl 3'

FeNi 3

Fe 3 Pt disorder

Fe 3 Pt order

FePd 3 disorder

FePd 3 order

872.9 0.40 0.39 0.39 1.30 1.30 1.31 1.30 1.30 4.32

432.1 0.43 0.46 0.46 1.20 1.26 1.25 1.20 1.25 4.19

461.6 0.40 0.41 0.40 1.20 1.24 1.20 1.22 1.18 4.02

526.1 0.43 0.44 0.47 1.10 1.16 1.20 1.13 1.15 3.66

543.4 0.43 0.44 0.44 1.05 1.12 1.10 1.09 1.06 3.50

1.06 0.96

0.57 0.81

0.71 0.65

0.72 0.66

0.68 1.14

and disordered systems. Obviously there is no influence of disorder on the critical behaviour of these spin systems with Heisenberg symmetry, in agreement with the theoretical prediction of the Harris criterion [6]. 5. For the ferromagnetic materials Fe3Pt and FePd 3 we find considerable deviations from the Heisenberg values towards the mean field values. To illustrate this further, fig. 8 represents the modified Arrott plot for the ordered FePd 3 sample by using Heisenberg exponents. Fig. 9 gives a plot with optimized exponent values. In fig. 8 we observe a considerable curvature even for the near-critical isotherms. This result indicates that

FePd]

" 25

.~

tl

-

I . 387

8

-

O, 3 6 5

o A + X o ÷

~o ~x

• Y ×

t5

x(

x O o

10

+ × O ÷ >~

5

534 84 1< 5]5 71 K 5Y3 56 K 537 43 1< 53~K 538 59 K 539 7O 1< 539 g6 K 540 50 I< 541 29K 542 14 K 543 2L K 544 10 K 544 71 g 545 62 1< 54661 1< 54769 K 54869 K 54964 K 550 59 K

Heisenberg 3 d [38]

0.365

1.387

4.803

the Heisenberg values do not lead to a correct description in the critical region of this material. The modification of the exponent values towards the mean field values in fig. 9 may be due to the effect of long-range exchange interactions in Fe3Pt and FePd 3 (see introduction). A first hint to a lowering of the susceptibility exponent value due to long-range interactions may be obtained from the low value of the ratio T p / T o where Tp is the paramagnetic Curie temperature. For isotropic, homogeneous ferromagnets with short-range interactions Tp may be estimated [26] by T p / T c .-~ 7. For the ordered FePt system with composition around Fe3Pt Sumiyama et al. [27] find To~To =

1,2~~,,~ ~

8,

"_ , 0 . 4 03 ~0 0

~

t o

5

~~537,5425~1< ~1<+ O

536%K

×

538691<

)~ z

54O 50 K 541~K L4 K

• x D o t~ + x

5-14 tO K 544 71 K 54562 K 546 61K 54?69 K 5,18 69 K 5,t96,t K

~

55059

>4

551 42 K 552 Ig K

K

^

p. 2

4

fi

fl

10

12 (H/MI1/I

14

16

I1~

20

22

-.-t.

Fig. 8. Modified Arrott plot with Heisenberg exponents for FePd 3 (ordered).

"'~ .........

,'i .....

'~i .......

~o' ......

',7" ......

(H/M1 i/~

'~o .......

~i .....

_),

Fig. 9. Modified Arrott plot with optimized exponents for FePd 3 (ordered).

399

M. Seeger, H. Kronmiiller / Phase transition in ordered and disordered ferromagnets

1.2 in agreement with our result for "/. Fisher et al. [28] performed a renormalization group analysis for the critical exponents for long-range interactions of the form

J ( r ) - l/ra+°,

(9)

where d denotes the dimension of the system and o is a parameter characterizing the range of the exchange interaction. Fisher et al. obtained the following results: 1. For o > 2 the exponents are given by those of the corresponding nearest neighbour model. 2. For o < d/2 the mean field exponents apply. 3. For d/2 < o < 2 the exponents depend on the parameter o. Similar results have been derived by Joyce [29] for the spherical model by exact calculations. It is of interest to note that for 0 < o < 3 / 2 the critical exponents expected are those of the mean field theory. In fact such a tendency is found for FePd 3 (see fig. 9). In this alloy we may assume that part of the exchange coupling results from the R K K Y indirect s - d exchange coupling. As derived by Ruderman and Kittel [30] this type of exchange interaction contains terms decreasing according to 1 / r 3- and 1/ra-laws if the ions are assumed to be &functions. Yosida [31] assumed a finite range of the width of the exchange coupling and obtained exchange terms varying with 1/r 2 and 1/r 3. Here it should be noted that these power laws are only valid for ranges smaller than the mean free electron path. Otherwise exponential terms limit the range of interaction. We may assume that the leading term of the s - d exchange mechanism follows a 1/r3-1aw, i.e. o = 0, and consequently our experimental results for the critical exponents of FePd 3 may be understood on the basis of longrange s - d exchange interaction. It should be noted that our experimental results do not fully agree with those of other authors. Kouvel and Comly [32] also investigated the phase transition in ordered and disordered FePd 3. Whereas for the ordered system their critical exponent values are consistent with ours, they obtained Heisenberg-like values for the disordered sample. For an interpretation it was assumed that for the disordered state the range of the s - d interaction is

severely limited because of scattering processes [33]. Perhaps the difference between our results and those of Kouvel and Comly [32] indicates that our 'disordered' sample exhibits a much larger degree of atomic order. This conjecture would agree with the fact that the difference in the Curie temperatures of ordered and disordered FePd 3 is about 8.5% for the samples of Kouvel and Comly, whereas it is much smaller in our case (about 3%). For the ordered alloy Fev4Pt26 Shen et al. [19] found a r - v a l u e of 0.455 similar to our values, the "/-value, however, was more Heisenberg-like. This situation is quite similar to the one observed by Kouvel and Comly [32] for strongly diluted FePd alloys. Part of the discrepancy between our results and those of Shen et al. [19] may arise from the fact that in Fe3Pt the width of the asymptotic critical regime as found by means of our Kouvel-Fisher plot is smaller than 1% of Tc. Outside the critical regime the effective exponent "/(T) first increases with increasing temperature (fig. 10). Shen et al. have obtained their exponent values from fits in a temperature range up to 1.031 To and therefore they might have obtained an average exponent value larger than the asymptotic one. Because the increase of -/(T) with increasing temperature is even stronger for the disordered Fe3Pt alloy (compare figs. 10 and 11) an 'ordered' material with a long-range order parameter of 0.8

I

6

Fe3PL

T 15

1.4

........

05

i,,,,,,,,,i

.1

.....

15

.2

,,,,I,,,,,,,,,I,,T,,

.25

.3

,,,,i,

.35

.4

45

........

5

IT ....

55

6

65

(T/T c - I ]

Fig. 10. The temperature dependent exponent ~,(T) for Fe3Pt (disordered).

400

M. Seeger, H. Kronmiiller / Phase transition in ordered and disordered ferromagnets

as the one of Shen et al. may exhibit quite large average exponent values in the considered temperature range. Fig. 10 shows the characteristic nonmonotonic temperature dependence for disordered systems [3] with a maximum at t - - 0 . 1 2 and y(T)-values beyond the Heisenberg value ( y ( T ) m a x ~ 1.55). Surprisingly the Kouvel-Fisher exponent 7 ( T ) exhibits a nonmonotonic temperature dependence also for the ordered alloy (fig. 11), in contrast to the usually observed monotonic behaviour of 7 ( T ) for all isotropic homogeneous ferromagnets with short-range exchange interactions [4,26]. In contrast to the disordered case the maximum value of 7 ( T ) for the 'ordered' alloy is below the Heisenberg value. There may be two reasons, for this striking result: 1. It may well be that in a system with long-range exchange interactions, for which the asymptotic value 7 is lowered towards the mean field value, there is an initial increase of y ( T ) with increasing temperature outside the critical regime. A different system, the dipolar ferromagnet, which also exhibits long-range interactions, was studied extensively in the literature. The asymptotic critical behaviour in this system is determined by a dipolar fixed point with exponent values rather similar to their short-range counterparts [34], which dominate the critical behaviour for larger temper-

1 40

Fe3P

t I.35

L

/

~


I 25

128

I

I~

........

05

i i i i ......

1

15

i I I I I I ....

2

25

I ......

3

i i i i i i i i .....

.35

4.

45

I .........

5

I .....

55

(T/T c -

6

t}

65

--')'

Fig. 11. The temperature dependent exponent 7(T) for Fe3Pt (ordered).

FePd

1~ 1.25

3

I 20 • •

•

if • ~

°

°

°

°

o

~

o

@ ooO

~

•

disordered

o

ordered

1

~

o

o

o

o

o

I,l,l,~,l,l,iTl,~,f,l,l,l,,,i,l,l,l,l,l,l,l,l,l,l,i

6

02

04

05

08

I0

,12

14

,16

18

20 IT/T c

22

24

II

Fig. 12. The temperature dependent exponent y(T) for disordered and ordered FePd 3.

atures. However, in the crossover regime a crossover renormalization theory [35] predicts a remarkable dip in the effective exponent 7(T). Indeed, recent Monte Carlo simulations for the dipolar Ising magnet [36] gave evidence for a crossover from the pure Ising exponent 7 to lower values when coming closer to Tc. An experiment which is not able to resolve the asymptotic critical behaviour but at least part of the crossover behaviour would yield a temperature dependence of 7 ( T ) qualitatively similar to the one given in fig. 11. We therefore plan to calculate numerically the temperature dependence of y ( T ) for the longrange spherical model [29]. 2. The nonmonotonic temperature dependence of 7 ( T ) may also arise from the fact that our 'ordered' alloy is not perfectly ordered, but there is a certain amount of structural disorder. Fig. 12 represents the exponent 7 ( T ) for the ordered and for the disordered FePd 3 alloys. Both samples show a nonmonotonic behaviour in y ( T ) as in the case of Fe3Pt. In the latter system the differences of 7 ( T ) between ordered and disordered samples are smaller. However, it should be noted that the ratio 7(T)max/7(Tc) in the ordered alloy is significantly lower as in the disordered alloy, in agreement with the behaviour of the Fe3Pt alloy. The assumption of long-range s - d exchange interactions gives a simple explanation

M. Seeger, H. Kronmiiller / Phase transition in ordered and disordered ferromagnets I 35

-

FeN

i3

1~_

t 30

m m m m 5 B D B m t B D E}D 5 12] B D B B D 0

I

15

-i-i.1-1.1-1.1-1-1-1 .02

04

1.1-1-1-, Of

['1"1"1"1"1"1"1"1"1"1"1 08

10

.12

I'1"1

1"1"1"1"1"1

14

16

(TIT

1"1"1 18

c

-

I

401

5. T h e r e are u n e x p e c t e d features of the K o u v e l F i s h e r e x p o n e n t y ( T ) o u t s i d e the critical regime for these alloys. T o clarify the p h y s i c a l origin for these features, m o r e theoretical a n d e x p e r i m e n t a l w o r k is p e r f o r m e d at present. F o r example, an in p r i n c i p l e exact c a l c u l a t i o n of the K o u v e l - F i s h e r e x p o n e n t "y(T) m a y b e p e r f o r m e d for the longr a n g e spherical m o d e l [29] for all t e m p e r a t u r e s . F u r t h e r m o r e , a r e n o r m a l i z a t i o n c a l c u l a t i o n of y ( T ) close to Tc for a realistic l o n g - r a n g e ferrom a g n e t i c m o d e l m a y be a t t e m p t e d similar to the field theoretical d e s c r i p t i o n of the static crossover in d i p o l a r f e r r o m a g n e t s [37].

20

1 )

Fig. 13. The temperature dependent exponent 3,(T) for FeNi 3.

for this small differences b e t w e e n the y ( T ) - v a l u e s of o r d e r e d a n d d i s o r d e r e d F e P d 3 alloys, w h i c h are e x p e c t e d to vanish for an e x t e n d e d e x c h a n g e mechanism. Fig. 13 shows the results for the ' d i s o r d e r e d ' F e N i 3 alloy. Surprisingly, it exhibits a m o n o t o n i c decrease of 3'(T) as u s u a l l y o b s e r v e d for h o m o g e n e o u s f e r r o m a g n e t s with s h o r t - r a n g e interactions. A s n o t e d in the i n t r o d u c t i o n there is a c o n s i d e r a b l e structural s h o r t - r a n g e o r d e r in this alloy at the m a g n e t i c p h a s e transition. If the m a g n e t i c c o r r e l a t i o n length o u t s i d e the critical regime is a l r e a d y smaller t h a n the range of s h o r t - r a n g e order, the system will a p p e a r m a g n e t i c a l l y as a h o m o g e neous ferromagnet.

5. Summary

In this investigation we have o b t a i n e d the following basic results: 1. T h e critical e x p o n e n t s of F e N i 3, Fe3Pt a n d F e P d 3 satisfy the scaling law. 2. T h e r e is n o influence of structural d i s o r d e r o n the critical e x p o n e n t s of F e 3Pt a n d F e P d 33. T h e ' d i s o r d e r e d ' alloy F e N i 3 exhibits the critical b e h a v i o u r of the 3d H e i s e n b e r g f e r r o m a g n e t . 4. T h e critical e x p o n e n t s of Fe3Pt a n d F e P d 3 are m o d i f i e d t o w a r d s the m e a n field values, p r o b a b l y b e c a u s e of l o n g - r a n g e exchange interactions.

Acknowledgements

T h e a u t h o r s are i n d e b t e d to Dr. M. F~ihnle, D i p l . - P h y s . A. F o r k l a n d D i p l . - P h y s . R. ReiBer for h e l p f u l discussions.

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