Critical exponents of a disordered ferromagnetic alloy: Fe
1 x
Al
x
Nelson O. Moreno and F. C. Montenegro
Departamento de Fı ´sica, Universidade Federal de Pernambuco, 50670-901 Recife, Brazil
Received 21 January 1997; revised manuscript received 10 July 1997
The ferromagnetic phase transition of disordered polycrystalline Fe
1-x
Al
x
alloys with x =0.10, 0.20, and
0.30, was investigated by magnetization measurements. We have determined, for each value of x , the magnetic
ordering temperatures and critical exponents 0.41, 1.34, and 4.26 for the spontaneous magnetiza-
tion, the susceptibility, and the isothermal magnetization, respectively. The values obtained for the exponents
satisfy usual scaling relations and remain essentially unchanged in the range 0.10x 0.30. A good scaling is
obtained both for T T
c
and T T
c
, adjusting the values measured for T
c
( H) and those for the critical
exponents within the error bars of the experiment. S0163-18299705945-6
I. INTRODUCTION
The role of disorder on the critical behavior of magnetic
systems has been a subject of numerous theoretical and ex-
perimental investigations over the past two decades. There
have been recent advances on understanding the influence of
quenched randomness
1
and frustration
2
on the magnetic
phase diagram and critical behavior of simple short-ranged
insulating anisotropic antiferromagnets, which can be rep-
resented by Ising models. However, for band magnets
3
with
long-range exchange interactions the situation is still unclear,
partially due to the lack of theoretical results on models that
could be applied to real experimental systems.
In spite of the considerable extent of information that has
been collected on the magnetic and structural properties of
the randomly diluted ferromagnet Fe
1 -x
Al
x
, the magnetism
of these alloys is still not well understood. The magnetic
ordering temperatures, T
c
( x ), reported in the literature
4–7
show a quasilinear decrease with decreasing x down to 15
at. % Al, where a sudden change in the slope of T
c
( x ) vs x
occurs. This anomaly has not been explained as yet. More-
over, the critical behavior of these alloys remain virtually
unexplored. In this paper we investigate the static critical
behavior of disordered Fe
1 -x
Al
x
alloys x =0.10, 0.20, and
0.30 by magnetization measurements. Ferromagnetic phase
transitions occur for all measured samples. The critical ex-
ponents determined are consistent with scaling relations and
are close to the ones found for pure Fe.
II. EXPERIMENTAL RESULTS
The bulk polycrystalline samples of Fe-Al used in this
investigations have been previously used in earlier Mo
¨
ss-
bauer measurements
4
by G. A. Pe
´
rez Alcazar and E. Galva
˜
o
see Ref. 4 for details about preparation of the samples.
According to the Mo
¨
ssbauer measurements, the solid solu-
tion of these alloys shows ferromagnetic properties at room
temperature in the interval 0 x 0.475. Therefore, in order
to study the nature of the critical behavior in these alloys, we
measured the magnetization curves from room temperature
to the Curie point. Magnetic measurements were made using
a vibrating sample magnetometer mated with a high-
temperature furnace. The magnetic field was varied up to 4.5
kOe and the temperature was controlled within 0.05 K. To
prevent oxidation of the samples, the measurements were
made using an atmosphere of He. The sample with 10 at. %
Al has a spherical shape. Flat disks were made with 20 and
30 at. % Al. The field was applied in the easiest direction
found along the diameter of the sphere ( x =0.10) and along
the plane of the disk x =0.20 and 0.30. The internal field
H =H
ext
-NM was obtained for each sample from the ap-
plied magnetic field H
ext
by subtracting the demagnetizing
field NM , where M is the mass magnetization and N the
demagnetization factor. The correction for demagnetizing
field is determined by the initial slope of isothermal magne-
tization curves see inset of Fig. 1.
To determine the critical temperature T
c
, and the expo-
nents and for each x , we used the Kouvel-Fisher
8
method. In this procedure, the spontaneous magnetization,
M
0
( T ), and the inverse initial susceptibility,
0
-1
( T ), are
determined from the intercepts of various isotherms on the
ordinate ( T T
c
) and abscissa ( T T
c
) of the Arrot-Kouvel
M
2
vs H / M plot.
9
The method is illustrated for the sample
with x =0.20 in Fig. 1. This procedure is based on the set of
magnetization isotherms curves M = M ( x , H , T ), for each
sample. In the inset of Fig. 1 we show the magnetization
isotherms at temperatures close to T
c
for x =0.20. Although
these curves clearly indicate a transition from ferromag-
netism to paramagnetism as T increases straight lines are a
good indicative of the paramagnetic region, it is hard to
determine precisely from them the position of the Curie
point. Plotting the quantities
0
-1
( d
0
-1
/ dT )
-1
=( T -T
c
)/
and M
0
( dM
0
/ dT )
-1
=( T -T
c
)/ as functions of T yield
straight lines for T →T
c
, as shown in Fig. 2. These plots
intercepts the T axes at T
c
( x ) and the slopes give 1/ and
1/ , respectively. The exponents and were determined
from a least mean-square fit of the straight lines in Fig. 2.
The values of the entire set of exponents, for each x , are
given in Table I.
The exponent was determined from the critical isotherm
at T =T
c
, by plotting M vs H data in a logarithmic scale, as
shown in Fig. 3. The slope of the straight line obtained for
each x gives =4.280.06, 4.260.01, and 4.230.05 for
x =0.10, 0.20, and 0.30, respectively. As expected, since
only two independent exponents exist, the values of are
PHYSICAL REVIEW B 1 DECEMBER 1997-I VOLUME 56, NUMBER 21
56 0163-1829/97/5621/137084/$10.00 13 708 © 1997 The American Physical Society