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Kinetics of induced uniaxial magnetic anisotropy rotation in some amorphous ferromagnetic alloys
Journal of Magnetism and Magnetic North-Holland Publishing Company
Materials
147
26 (1982) 147- 152
KINETICS OF INDUCED UNIAXIAL MAGNETIC AMORPHOUS FERROMAGNETIC ALLOYS W. CHAMBRON,
F. LANCON
ANISOTROPY
ROTATION
IN SOME
and A. CHAMBEROD
Centre d’Eiudes Nuclkuires de Grenoble, Dkpartement
de Recherehe Fondamentule,
Section de Physique du Solide, 8.5 X-38041
Grenoble Cedex, Frunce
A model is proposed to describe the induced magnetic anisotropy in amorphous alloys by the superimposition of independent “components”, each of them with an exponential kinetic. The experimental results, obtained as a function of temperature, suggest that the time constants of the different components correspond to similar activation energies, but to different preexponential factors.
1. Intraduction
the experimental results, and we show that the first hypothesis does not agree with them.
A number of experiments have shown that ferromagnetic amorphous alloys present, as a result of thermomagnetic treatments (TMT), an uniaxial induced magnetic anisotropy (IMA). For a firstly relaxed alloy, i.e. annealed at a temperature a little lower than the crystallization temperature, the observed IMA is reversible with respect both to the direction of the magnetic field applied during the TMT, and to the temperature of the TMT. It is clear that such a phenomenon is related to atomic rearrangements comparable to the directional order, a well-known feature in crystalline materials [l]. However, while, in the latter, IMA establishment kinetic corresponds to an exponential, it is broadened by time in amorphous materials. Such a kinetics slowing, when the asymptotic value is approached, is also observed for the structural relaxation, i.e. the irreversible evolution occurring in an as-quenched amorphous alloy during the first heating. Two processes have been suggested to explain this slowing [2]: i) a progressive increase of the time constant [3], which would depend on the relaxation degree (in other words, the process would be non-linear [2]); ii) existence of a time constant distribution, an assumption strongly supported by the “cross-over” effect [4]. In this paper, we propose a model based on the second hypothesis, which accounts very well for 0304~8853/82/0000-0000/$02.75
0 1982 North-Holland
2. IMA representation IMA is deduced from torque measurements. A torque is usually expressed in terms of a Fourier series as a function of 8, the angle between the reference direction and the variable position of the applied field during measurement [5]: F(e)=
i
(A,sinnB+B,cosn@).
n=2
In amorphous materials, it is experimentally observed that only the coefficients A, and B2 show significant variations after thermomagnetic treatments, characterizing an uniaxial IMA; the variations of the other coefficients are negligible, more than in polycrystals, clearly because of a better isotropy. The state of a sample can thus be represented in a rectangular axis system by a point P, the coordinates of which being X = -A A, and Y = A B,, as proposed by Pechart [6] and Aubert [7], and developed elsewhere [5]. With such a notation, the state without IMA is represented by X= 0, Y = 0. During an isothermal TMT, the figurative point P moves towards Pm, whose coordinates are X= Kz cos 2~ and Y = Kz sin 2~; KF is the IMA energy density asymptotic value
characteristic of the treatment temperature, TA, and’ cp is the direction of the applied field during the TMT. Hereafter, a TMT will be designated by the temperature TA and the angle cp: (TA, cp).
ature To > TA; this rate increase has been observed later to disappear gradually during a following TMT at TA [5,8]. It must be noticed that the 7, time evolution does not depend on the coordinates xi and yi, i.e. the IMA development. On the contrary, in models like (i) (see introduction), r depends on time uniquely through the channel of the IMA value. The eqs. (2) can be integrated:
3. Basic model Let us note that, while a given state corresponds to one point P, reciprocally, a point P may correspond to several states. Thus, as an example, from an initial state represented by P,,, the motion of P during a TMT ( TA, cp) depends on the way through which P,, has been. obtained: Therefore, the same point P represents different microscopic states of the alloy, not sufficiently depicted by the macroscopic IMA. Such a statement leads to the proposal of a model so that the IMA represented by point P is the superimposition of elementary “components”, referenced by indice i:
x,(r) =xpO -
(3) where x9 and vi0 represent and y, and
1,2 ,..., n,
-xi)
dt/r&),
d.v, = (.Y? -n)
df/ri(r),
of x,
(1) 4. Experiment 1
and
where xp” and yp” point out the asymptotic values of xi and y, during the TMT (TA, cp); they are functions of ‘p and TA. A time-constant dependence on time has been introduced to take into account the structural evolution, which happens at TA. Indeed, it has been shown that the IMA establishment kinetic rate depends, to some extent, on the alloy structural state. In particular, this rate increases if the sample is quenched from a temper-
values
One sees that P is the center of gravity of the n points, whose coordinates are nx, and ny,, each of them moving straightforward to its asymptotic position whose coordinates are nxy and nv,“, each with its own rate. Generally, the trajectory of P is not a straight line. Let us now examine different experimental results on this basis.
with the following assumptions: a) During a TMT, each component varies independently from each other b) The variation law of a component i can be written in a simple form: dxi =(x7
the initial
19;(t) =/,t dt/r;(t).
and i=
exp[-8,(t)],
Yi(r)=~pO-((yim-_YP)exp[-Bj(t)]~
i
Y(t)=xyi(t), i
(xp” -xp)
(2)
After the expressions (3), the coordinates x, are supposed to vary during a TMT following a law independent of the values of yi; after (l), X(f) is a function independent of Y(t) too. The experimental test has been made on the alloy Fe,,,Ni,P,,$ (Metglas 2826 purchased from Allied Chemical Corporation). A sample has first been annealed 30 min at 360°C to be stabilized (structurally relaxed). In a first step (la), this sample was annealed 40 min at 32O’C under a magnetic field directed to the direction cp = 0, then cooled at constant rate, still under field. At this temperature, 32O”C, being above the Curie temperature (290°C) any previous IMA was wiped out. However, during the cooling, an IMA appeared, represented by the point P, (fig. l), which lies on the X axis owing to symmetry. From this initial state, a TMT (180°C
W. Chumhron et ul. /
Induced mugnetic urtisotrop.y in omorphow
149
ullqvs
associated to a component (-xy, -JJ;‘) of the same 8,(t), in (lb). Thus, this third sum is constituted, both in (1 a) and (1 b), of the same terms, although in a different succession. Finally, the experimental fact that the function Y(t) is independent of the values of X(t) justifies the assumption, made a priori, that the differential equations (2) are independent of each other. 0
X Urn*)
5o
Fig. 1. Successive positions of the representative point of the IMA. during a TMT (IRO’C, cp=20°). from the two positions Pa and P,. representing two different initial states. We observe that Y(t) is the same for both experiments (a) and (b). The time interval between successive points was 20 min.
cp= 20')was achieved; fig. 1 shows the successive positions of the representative point, each 20 min. In a second step (1 b), a similar experiment was carried out on the same sample except that the initial state Pi, was obtained in a field oriented to the direction cp’= 90”. Fig. 1 shows that during the TMT (180°C q = 20’) the function Y(t) was exactly the same in both experiments (la) and (lb), as provided by the model: from eqs. (1) and (3) it appears: Y(r) =xy,”
--X_Y;~exp(--0,)
The first sum equals K; sin40”; the second one does not depend on the initial state (it is the same in la and 1b); the third one is zero, because the amorphous alloys is supposed to be isotropic macroscopically, for the sake of balancing the local configurations: indeed, for each component in the initial state (xp,_~~~‘), a corresponding component ( xp? -y(O)>, with the same e,(t), can be found, and the terms counterbalance each other in the sum. Fig. 1 shows also that, if we take the (X,, Y,) axis system as reference, the Y,(t) values, which are opposite in the initial state, remain like that during the TMT. This agrees with what is expected from (4) when the equations are written in the new axis system. Indeed, the two first sums are zero, owing to symmetry with respect to the asymptotic direction, ‘p = 0. As concerns the third sum, any initial component (x,e,y,‘) in (la) can be
5. Experiment 2 A sample, similar to the one used in experiment 1, was stabilized in te same conditions (30, min, 36O’C). The three successive experiments were then made: (2a) TMT 40 min, (320°C, cp = 0), followed by a cooling at constant rate. The corresponding initial IMA, is represented by PO (fig. 2), the same as P, in experiment 1; the ordinate of PO is Y = 0. Then, an isothermal TMT (180°C cp = 40’) was performed. Curve a, on fig. 2, shows the trajectory of P, and curve a, on fig. 3, gives the variations of r,(t). (2b) The same initial state PO, followed by a TMT t, = 80 min (18O”C, cp= 0), so that the figurative point moves to P,. Then, an isothermal TMT, like in (2a) (18O”C, cp = 407, was performed. The trajectory of P and the function Yb(t) appear on curves b, figs. 2 and 3. The curve b of fig. 3 has been translated in b’, to be compared
0 0
50
X (J.m-3)
1013
Fig. 2. Successive positions of the representative point IMA during the 3 experiments described in section 5.
of the
Y,( t ) = K,” sin 80“ 3
- XYP” exP[-~i(f,)--,(t)]~
(8)
lWexP[-4(4].
(9)
k
z 2-
-
50
i
In our experiments, Y.(r) = r,(t)
0 0
100
200
300 Time
Fig. 3. Kinetics of Y (ordinate dashed
curve (calculated
cxpcrimental
from the non-linear
of the point P). from fig. 2. The
from our model)
one (curvcc),
400
(min)
whcrcas
model) disagrees strongly.
- Y&).
(10)
Pig. 3 gives (c,) the calculated curve from (10); it is seen to coincide practically with the experimental one (c). With regard to the difference between the curves (a) and (b’) (fig. 3) we can write, considering again the time constants r,(t):
fits very well the
the curve (d) (calculated
f, = tz, and therefore
I
Y,(t) - Yb.(r) =xyy
-l:‘+‘dr/r,(r)
1
--exp[ -[dr/c(i)]}.(!I) with curvea: the kinetic is observed to be a little slower. (2~) The same initial state Pa followed by the same TMT as (2a) (lSO°C, cp= 40”) but just for tz = 80 min, to the point P2 (figs. 2 and 3). After that, a TMT (18O“C cp = 0) was performed corresponding to the curves (c) (figs. 2 and 3). Let us note yp the values of y, for the initial state P,,, yi” the asymptotic values for the TMT (180°C cp = 40”), and y’” the asymptotic values for the TMT (180°C ‘p = 0). Owing to symmetry, as explained above, it appears clearly that, for any t: ?I;”
exp[ -6(t)]
=O
(5)
It means a small increase (on an average), of the time constants as a function of time. This phenomenon is connected to the structural state variations, which are reversible as a function of temperature [5,8]. Thus, in the initial state P,, the values of T, are, on an average, a little smaller than the asymptotic ones in the 180°C anneal. As a conclusion, these experiments are well accounted for by a model with a set of time constants. Now, we can ask the question of what happens if we try to acount for the experimental curve (c) by a non-linear model (see (i) in the introduction). i.e. a model with a unique time constant T varying as a function of the IMA degree. Y. This model leads to the differential equation:
and dY= ZypO exp[ -0;(t)]
=O.
(6)
Using (3) yi( t) can be calculated in the three cases (a), (b) and (c), and taking into account (5) and (6), we find:
Y,(t)=K,”
sin80°-~ypa
exp[-r$(f)],
(7)
(Y”
- Y) dt/r(Y).
Estimating Y”, at 180°C to be equal to 360 J rnp3 [8], T(Y) is calculated from the curve (a); then the above equation is integrated using Y” = 0, to describe the curve (c); such a calculation leads to the curve (d), in total disagreement with the experiment, and this model of a unique time constant must be set aside.
151
6. Case where the initial state is an asymptotic one
Clearly,
Let us remember that an asymptotic state is one obtained by a TMT (T,, cpO)of a sufficient time so that the figurative point has reached an equilibrium position; such a state can be obtained in a reasonable time only if the temperature is high enough. Then, the experiment shows that [5,7], from such a state, the point P follows a straight line during a second TMT (T,, cp,) (that is not the case in fig. 2). We will now test that a calculation, based on the above model, agrees with this result. To simplify the calculations, we choose ‘pO= 45” and ‘p, = 0. At the end ofthe first TMT (To,45’), the figurative point is P, (fig. 4) with coordinates:
x (x”/K,)
x0
=x:x; =o,
y, =&;=Ku(To)=Ko, and at the end of the second figurative point is P,:
X,=Xx; Y,
(T,,O'),the
TMT
=K,(T,)=K,,
=&,‘=o.
After (1) and (3), X(t) and Y(t) of P during the second TMT can be calculated, and the relationship is established:
X(f) + --= y(r)
-
Ko
K,
1 +F
$
T
5
-
if T, = To,K, = K, too, and cxp[-fl,(r)]
’ = z (x,j/K,) i
exp[-$(t)];
(13)
indeed, to each component, which was asymptotically in (~~,y,~) for cp= 45”, (point pp fig. 4), corresponds symmetrically to another component, which is asymptotically in (x~,y~), for ‘p = 0 (point pj), with x; =_vp,
J$ =xg.
The components i and j have the same time constant, because of the macroscopic isotropy of the material, and therefore the same function O(r). As a result, both members of (13) are identical, and, after (12), the trajectory of P is the straight line POP,. Nevertheless, the point p), asymptotic of pi, for ‘p = 0, can be different of pj, because the isotropy of each individual component has not been assumed. When T, # To,then K, ZK,; but as it has been experimentally observed, the trajectory of P is a straight line [5,8], and therefore the relationship (13) should be carried out. This suggests that the relationship yto/K,=x:/K, is still valid, i.e. the elementary components are proportional to the macroscopic IMA energy, K,, when the temperature is varied.
exp[--B,(t)] 7. Application to the study of atomic mobility exp[-0,(t)].
(12)
b
X
Fig. 4. Schematic moving of the figurative point P (macroscopic IMA) and of the figurative point p, (one of the components): when the initial and final states are asymptotic ones, the point P moves along a straight line.
AS announced in the introduction, IMA results from atomic rearrangements. The mean time constant, TV, of the establishment kinetic is inversely proportional to the atomic mobility. However, in amorphous alloys, the time constant distribution is very broad. As a result, it is rather difficult to obtain in an acceptable time an estimate of the IMA asymptotic value. Yet, this value is absolutely necessary to obtain T,,, from a limited part of the kinetic curve. Fortunately, experiments like (2a) give a good solution [9]. In ref. [9], a sample of Fe,Ni40P,4B6, firstly stabilized by an anneal of 10 min at 330°C, has
been treated as fcillows: a) anneal 40 min, 300°C, with the magnetic field directed along cp= 40’; one obtains the initial state P,(X,, Y,) corresponding to the IMA induced during cooling. This state is very reproducible. b) TMT (7”, cp = 0): Y(t) decreases from Y, to the asymptotic value 0. By performing again the operationsa and b at different temperatures, TA, a series of curves Y(T,, t) is obtained. To analyze the experimental results, the variation of T, as a function of i can be neglected because it is weak, and we will assume that each 7, is characterized by an activation energy E, and a pre-exponential term (Y,: T, = (Y~exp( E,/kT,). Using (1) and (3), it appears:
whose ?; are small, are favoured. For long times, when Y(r) approaches zero, exp( - r/T) has decreased more rapidly for the small ~;‘s., and the terms with large 7,‘s are preponderant. Experiments have shown [9] that E, is practically constant ((1.74 ‘-c 0.04) eV for the Fe,,Ni,P,,$ alloy), whatever the part of the curve Y(r) is used (indicated B(r) in ref. [9]). It is concluded that the scattering of the 7; values results essentially from the scattering of the ai terms, rather than of the activation energies, E,. The large dispersion experimentally observed [9] (ratio of about 3000 of the extreme values of the 7,‘s) certainly results from the inhomogeneity of the amorphous structure; this inhomogeneity could be at the scale of the local atomic configurations [lo]; but it is then surprising that the dispersion does not concern the potential barriers rather than vibration frequencies.
dY(T,,t)=zypexp i
(14) At a different temperature T,, a different time t(Y) is necessary to obtain a given value of Y. An apparent activation energy, E,, is defined as E,
=dlnt(Y)/d(l/kT,);
the derivative
2
(15)
is made at constant
[Y?xP(--t/dhlE,
Em = ‘x[y:exp(-rt/-r,)/T,] Thus, E,,
f
(16)
the measured apparent activation energy, value of E;‘s, weighted by
is the mean
(17)
.$ exp( - t/q Wi. At
Y, after (14):
r = 0, it equals
yp/~,,
and
the components,
References S. Chikazumi and C.D. Graham. in: Magnetism and Metallurgy. rds. A.E. Berkowitz and E. Kneller (Academic Press. New-York. 1972). and R.W. Hopper. in: Metallic Glasses PI D.R. Uhlmann (The American Society for Metals. Metals Park. OH 44073. 197X) p. 149. [31 T. Egami. J. Appl. Phys. 50 ( 1979) 1564. [41 A.L. Greer and J.A. Leake. J. Non-Cryt. Solids 33 (1979) 291. and A, Chamherod. Solid State Commun. [51 W. Chamhron 35 (1980) 61. 161 R. Pechart, Thesis. Grenoble. France (1972). (71 G. Auhert. J. Magn. Magn. Mat. 19 (1980) 396. and A. Chamherod. J. Phys. 41 (1980) PI W. Chamhron cg-710. and A. Chamherod. Solid State Commun. PI W. Chamhron 33 (1980) 157. [lOI T. Egami. K. Macda and V. Vitek. Phil. Mag. A 41 ( 1980) 883.
Materials
147
26 (1982) 147- 152
KINETICS OF INDUCED UNIAXIAL MAGNETIC AMORPHOUS FERROMAGNETIC ALLOYS W. CHAMBRON,
F. LANCON
ANISOTROPY
ROTATION
IN SOME
and A. CHAMBEROD
Centre d’Eiudes Nuclkuires de Grenoble, Dkpartement
de Recherehe Fondamentule,
Section de Physique du Solide, 8.5 X-38041
Grenoble Cedex, Frunce
A model is proposed to describe the induced magnetic anisotropy in amorphous alloys by the superimposition of independent “components”, each of them with an exponential kinetic. The experimental results, obtained as a function of temperature, suggest that the time constants of the different components correspond to similar activation energies, but to different preexponential factors.
1. Intraduction
the experimental results, and we show that the first hypothesis does not agree with them.
A number of experiments have shown that ferromagnetic amorphous alloys present, as a result of thermomagnetic treatments (TMT), an uniaxial induced magnetic anisotropy (IMA). For a firstly relaxed alloy, i.e. annealed at a temperature a little lower than the crystallization temperature, the observed IMA is reversible with respect both to the direction of the magnetic field applied during the TMT, and to the temperature of the TMT. It is clear that such a phenomenon is related to atomic rearrangements comparable to the directional order, a well-known feature in crystalline materials [l]. However, while, in the latter, IMA establishment kinetic corresponds to an exponential, it is broadened by time in amorphous materials. Such a kinetics slowing, when the asymptotic value is approached, is also observed for the structural relaxation, i.e. the irreversible evolution occurring in an as-quenched amorphous alloy during the first heating. Two processes have been suggested to explain this slowing [2]: i) a progressive increase of the time constant [3], which would depend on the relaxation degree (in other words, the process would be non-linear [2]); ii) existence of a time constant distribution, an assumption strongly supported by the “cross-over” effect [4]. In this paper, we propose a model based on the second hypothesis, which accounts very well for 0304~8853/82/0000-0000/$02.75
0 1982 North-Holland
2. IMA representation IMA is deduced from torque measurements. A torque is usually expressed in terms of a Fourier series as a function of 8, the angle between the reference direction and the variable position of the applied field during measurement [5]: F(e)=
i
(A,sinnB+B,cosn@).
n=2
In amorphous materials, it is experimentally observed that only the coefficients A, and B2 show significant variations after thermomagnetic treatments, characterizing an uniaxial IMA; the variations of the other coefficients are negligible, more than in polycrystals, clearly because of a better isotropy. The state of a sample can thus be represented in a rectangular axis system by a point P, the coordinates of which being X = -A A, and Y = A B,, as proposed by Pechart [6] and Aubert [7], and developed elsewhere [5]. With such a notation, the state without IMA is represented by X= 0, Y = 0. During an isothermal TMT, the figurative point P moves towards Pm, whose coordinates are X= Kz cos 2~ and Y = Kz sin 2~; KF is the IMA energy density asymptotic value
characteristic of the treatment temperature, TA, and’ cp is the direction of the applied field during the TMT. Hereafter, a TMT will be designated by the temperature TA and the angle cp: (TA, cp).
ature To > TA; this rate increase has been observed later to disappear gradually during a following TMT at TA [5,8]. It must be noticed that the 7, time evolution does not depend on the coordinates xi and yi, i.e. the IMA development. On the contrary, in models like (i) (see introduction), r depends on time uniquely through the channel of the IMA value. The eqs. (2) can be integrated:
3. Basic model Let us note that, while a given state corresponds to one point P, reciprocally, a point P may correspond to several states. Thus, as an example, from an initial state represented by P,,, the motion of P during a TMT ( TA, cp) depends on the way through which P,, has been. obtained: Therefore, the same point P represents different microscopic states of the alloy, not sufficiently depicted by the macroscopic IMA. Such a statement leads to the proposal of a model so that the IMA represented by point P is the superimposition of elementary “components”, referenced by indice i:
x,(r) =xpO -
(3) where x9 and vi0 represent and y, and
1,2 ,..., n,
-xi)
dt/r&),
d.v, = (.Y? -n)
df/ri(r),
of x,
(1) 4. Experiment 1
and
where xp” and yp” point out the asymptotic values of xi and y, during the TMT (TA, cp); they are functions of ‘p and TA. A time-constant dependence on time has been introduced to take into account the structural evolution, which happens at TA. Indeed, it has been shown that the IMA establishment kinetic rate depends, to some extent, on the alloy structural state. In particular, this rate increases if the sample is quenched from a temper-
values
One sees that P is the center of gravity of the n points, whose coordinates are nx, and ny,, each of them moving straightforward to its asymptotic position whose coordinates are nxy and nv,“, each with its own rate. Generally, the trajectory of P is not a straight line. Let us now examine different experimental results on this basis.
with the following assumptions: a) During a TMT, each component varies independently from each other b) The variation law of a component i can be written in a simple form: dxi =(x7
the initial
19;(t) =/,t dt/r;(t).
and i=
exp[-8,(t)],
Yi(r)=~pO-((yim-_YP)exp[-Bj(t)]~
i
Y(t)=xyi(t), i
(xp” -xp)
(2)
After the expressions (3), the coordinates x, are supposed to vary during a TMT following a law independent of the values of yi; after (l), X(f) is a function independent of Y(t) too. The experimental test has been made on the alloy Fe,,,Ni,P,,$ (Metglas 2826 purchased from Allied Chemical Corporation). A sample has first been annealed 30 min at 360°C to be stabilized (structurally relaxed). In a first step (la), this sample was annealed 40 min at 32O’C under a magnetic field directed to the direction cp = 0, then cooled at constant rate, still under field. At this temperature, 32O”C, being above the Curie temperature (290°C) any previous IMA was wiped out. However, during the cooling, an IMA appeared, represented by the point P, (fig. l), which lies on the X axis owing to symmetry. From this initial state, a TMT (180°C
W. Chumhron et ul. /
Induced mugnetic urtisotrop.y in omorphow
149
ullqvs
associated to a component (-xy, -JJ;‘) of the same 8,(t), in (lb). Thus, this third sum is constituted, both in (1 a) and (1 b), of the same terms, although in a different succession. Finally, the experimental fact that the function Y(t) is independent of the values of X(t) justifies the assumption, made a priori, that the differential equations (2) are independent of each other. 0
X Urn*)
5o
Fig. 1. Successive positions of the representative point of the IMA. during a TMT (IRO’C, cp=20°). from the two positions Pa and P,. representing two different initial states. We observe that Y(t) is the same for both experiments (a) and (b). The time interval between successive points was 20 min.
cp= 20')was achieved; fig. 1 shows the successive positions of the representative point, each 20 min. In a second step (1 b), a similar experiment was carried out on the same sample except that the initial state Pi, was obtained in a field oriented to the direction cp’= 90”. Fig. 1 shows that during the TMT (180°C q = 20’) the function Y(t) was exactly the same in both experiments (la) and (lb), as provided by the model: from eqs. (1) and (3) it appears: Y(r) =xy,”
--X_Y;~exp(--0,)
The first sum equals K; sin40”; the second one does not depend on the initial state (it is the same in la and 1b); the third one is zero, because the amorphous alloys is supposed to be isotropic macroscopically, for the sake of balancing the local configurations: indeed, for each component in the initial state (xp,_~~~‘), a corresponding component ( xp? -y(O)>, with the same e,(t), can be found, and the terms counterbalance each other in the sum. Fig. 1 shows also that, if we take the (X,, Y,) axis system as reference, the Y,(t) values, which are opposite in the initial state, remain like that during the TMT. This agrees with what is expected from (4) when the equations are written in the new axis system. Indeed, the two first sums are zero, owing to symmetry with respect to the asymptotic direction, ‘p = 0. As concerns the third sum, any initial component (x,e,y,‘) in (la) can be
5. Experiment 2 A sample, similar to the one used in experiment 1, was stabilized in te same conditions (30, min, 36O’C). The three successive experiments were then made: (2a) TMT 40 min, (320°C, cp = 0), followed by a cooling at constant rate. The corresponding initial IMA, is represented by PO (fig. 2), the same as P, in experiment 1; the ordinate of PO is Y = 0. Then, an isothermal TMT (180°C cp = 40’) was performed. Curve a, on fig. 2, shows the trajectory of P, and curve a, on fig. 3, gives the variations of r,(t). (2b) The same initial state PO, followed by a TMT t, = 80 min (18O”C, cp= 0), so that the figurative point moves to P,. Then, an isothermal TMT, like in (2a) (18O”C, cp = 407, was performed. The trajectory of P and the function Yb(t) appear on curves b, figs. 2 and 3. The curve b of fig. 3 has been translated in b’, to be compared
0 0
50
X (J.m-3)
1013
Fig. 2. Successive positions of the representative point IMA during the 3 experiments described in section 5.
of the
Y,( t ) = K,” sin 80“ 3
- XYP” exP[-~i(f,)--,(t)]~
(8)
lWexP[-4(4].
(9)
k
z 2-
-
50
i
In our experiments, Y.(r) = r,(t)
0 0
100
200
300 Time
Fig. 3. Kinetics of Y (ordinate dashed
curve (calculated
cxpcrimental
from the non-linear
of the point P). from fig. 2. The
from our model)
one (curvcc),
400
(min)
whcrcas
model) disagrees strongly.
- Y&).
(10)
Pig. 3 gives (c,) the calculated curve from (10); it is seen to coincide practically with the experimental one (c). With regard to the difference between the curves (a) and (b’) (fig. 3) we can write, considering again the time constants r,(t):
fits very well the
the curve (d) (calculated
f, = tz, and therefore
I
Y,(t) - Yb.(r) =xyy
-l:‘+‘dr/r,(r)
1
--exp[ -[dr/c(i)]}.(!I) with curvea: the kinetic is observed to be a little slower. (2~) The same initial state Pa followed by the same TMT as (2a) (lSO°C, cp= 40”) but just for tz = 80 min, to the point P2 (figs. 2 and 3). After that, a TMT (18O“C cp = 0) was performed corresponding to the curves (c) (figs. 2 and 3). Let us note yp the values of y, for the initial state P,,, yi” the asymptotic values for the TMT (180°C cp = 40”), and y’” the asymptotic values for the TMT (180°C ‘p = 0). Owing to symmetry, as explained above, it appears clearly that, for any t: ?I;”
exp[ -6(t)]
=O
(5)
It means a small increase (on an average), of the time constants as a function of time. This phenomenon is connected to the structural state variations, which are reversible as a function of temperature [5,8]. Thus, in the initial state P,, the values of T, are, on an average, a little smaller than the asymptotic ones in the 180°C anneal. As a conclusion, these experiments are well accounted for by a model with a set of time constants. Now, we can ask the question of what happens if we try to acount for the experimental curve (c) by a non-linear model (see (i) in the introduction). i.e. a model with a unique time constant T varying as a function of the IMA degree. Y. This model leads to the differential equation:
and dY= ZypO exp[ -0;(t)]
=O.
(6)
Using (3) yi( t) can be calculated in the three cases (a), (b) and (c), and taking into account (5) and (6), we find:
Y,(t)=K,”
sin80°-~ypa
exp[-r$(f)],
(7)
(Y”
- Y) dt/r(Y).
Estimating Y”, at 180°C to be equal to 360 J rnp3 [8], T(Y) is calculated from the curve (a); then the above equation is integrated using Y” = 0, to describe the curve (c); such a calculation leads to the curve (d), in total disagreement with the experiment, and this model of a unique time constant must be set aside.
151
6. Case where the initial state is an asymptotic one
Clearly,
Let us remember that an asymptotic state is one obtained by a TMT (T,, cpO)of a sufficient time so that the figurative point has reached an equilibrium position; such a state can be obtained in a reasonable time only if the temperature is high enough. Then, the experiment shows that [5,7], from such a state, the point P follows a straight line during a second TMT (T,, cp,) (that is not the case in fig. 2). We will now test that a calculation, based on the above model, agrees with this result. To simplify the calculations, we choose ‘pO= 45” and ‘p, = 0. At the end ofthe first TMT (To,45’), the figurative point is P, (fig. 4) with coordinates:
x (x”/K,)
x0
=x:x; =o,
y, =&;=Ku(To)=Ko, and at the end of the second figurative point is P,:
X,=Xx; Y,
(T,,O'),the
TMT
=K,(T,)=K,,
=&,‘=o.
After (1) and (3), X(t) and Y(t) of P during the second TMT can be calculated, and the relationship is established:
X(f) + --= y(r)
-
Ko
K,
1 +F
$
T
5
-
if T, = To,K, = K, too, and cxp[-fl,(r)]
’ = z (x,j/K,) i
exp[-$(t)];
(13)
indeed, to each component, which was asymptotically in (~~,y,~) for cp= 45”, (point pp fig. 4), corresponds symmetrically to another component, which is asymptotically in (x~,y~), for ‘p = 0 (point pj), with x; =_vp,
J$ =xg.
The components i and j have the same time constant, because of the macroscopic isotropy of the material, and therefore the same function O(r). As a result, both members of (13) are identical, and, after (12), the trajectory of P is the straight line POP,. Nevertheless, the point p), asymptotic of pi, for ‘p = 0, can be different of pj, because the isotropy of each individual component has not been assumed. When T, # To,then K, ZK,; but as it has been experimentally observed, the trajectory of P is a straight line [5,8], and therefore the relationship (13) should be carried out. This suggests that the relationship yto/K,=x:/K, is still valid, i.e. the elementary components are proportional to the macroscopic IMA energy, K,, when the temperature is varied.
exp[--B,(t)] 7. Application to the study of atomic mobility exp[-0,(t)].
(12)
b
X
Fig. 4. Schematic moving of the figurative point P (macroscopic IMA) and of the figurative point p, (one of the components): when the initial and final states are asymptotic ones, the point P moves along a straight line.
AS announced in the introduction, IMA results from atomic rearrangements. The mean time constant, TV, of the establishment kinetic is inversely proportional to the atomic mobility. However, in amorphous alloys, the time constant distribution is very broad. As a result, it is rather difficult to obtain in an acceptable time an estimate of the IMA asymptotic value. Yet, this value is absolutely necessary to obtain T,,, from a limited part of the kinetic curve. Fortunately, experiments like (2a) give a good solution [9]. In ref. [9], a sample of Fe,Ni40P,4B6, firstly stabilized by an anneal of 10 min at 330°C, has
been treated as fcillows: a) anneal 40 min, 300°C, with the magnetic field directed along cp= 40’; one obtains the initial state P,(X,, Y,) corresponding to the IMA induced during cooling. This state is very reproducible. b) TMT (7”, cp = 0): Y(t) decreases from Y, to the asymptotic value 0. By performing again the operationsa and b at different temperatures, TA, a series of curves Y(T,, t) is obtained. To analyze the experimental results, the variation of T, as a function of i can be neglected because it is weak, and we will assume that each 7, is characterized by an activation energy E, and a pre-exponential term (Y,: T, = (Y~exp( E,/kT,). Using (1) and (3), it appears:
whose ?; are small, are favoured. For long times, when Y(r) approaches zero, exp( - r/T) has decreased more rapidly for the small ~;‘s., and the terms with large 7,‘s are preponderant. Experiments have shown [9] that E, is practically constant ((1.74 ‘-c 0.04) eV for the Fe,,Ni,P,,$ alloy), whatever the part of the curve Y(r) is used (indicated B(r) in ref. [9]). It is concluded that the scattering of the 7; values results essentially from the scattering of the ai terms, rather than of the activation energies, E,. The large dispersion experimentally observed [9] (ratio of about 3000 of the extreme values of the 7,‘s) certainly results from the inhomogeneity of the amorphous structure; this inhomogeneity could be at the scale of the local atomic configurations [lo]; but it is then surprising that the dispersion does not concern the potential barriers rather than vibration frequencies.
dY(T,,t)=zypexp i
(14) At a different temperature T,, a different time t(Y) is necessary to obtain a given value of Y. An apparent activation energy, E,, is defined as E,
=dlnt(Y)/d(l/kT,);
the derivative
2
(15)
is made at constant
[Y?xP(--t/dhlE,
Em = ‘x[y:exp(-rt/-r,)/T,] Thus, E,,
f
(16)
the measured apparent activation energy, value of E;‘s, weighted by
is the mean
(17)
.$ exp( - t/q Wi. At
Y, after (14):
r = 0, it equals
yp/~,,
and
the components,
References S. Chikazumi and C.D. Graham. in: Magnetism and Metallurgy. rds. A.E. Berkowitz and E. Kneller (Academic Press. New-York. 1972). and R.W. Hopper. in: Metallic Glasses PI D.R. Uhlmann (The American Society for Metals. Metals Park. OH 44073. 197X) p. 149. [31 T. Egami. J. Appl. Phys. 50 ( 1979) 1564. [41 A.L. Greer and J.A. Leake. J. Non-Cryt. Solids 33 (1979) 291. and A, Chamherod. Solid State Commun. [51 W. Chamhron 35 (1980) 61. 161 R. Pechart, Thesis. Grenoble. France (1972). (71 G. Auhert. J. Magn. Magn. Mat. 19 (1980) 396. and A. Chamherod. J. Phys. 41 (1980) PI W. Chamhron cg-710. and A. Chamherod. Solid State Commun. PI W. Chamhron 33 (1980) 157. [lOI T. Egami. K. Macda and V. Vitek. Phil. Mag. A 41 ( 1980) 883.