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Activation volume and coercivity in aluminum-substituted Pb–M hexaferrites
Journal of Magnetism and Magnetic Materials 222 (2000) 271}276
Activation volume and coercivity in aluminum-substituted Pb}M hexaferrites J.C. Faloh Gandarilla, S. DmH az-Castan oH n*, N. SuaH rez Almodovar Lab. Magnetismo, Facultad de Fn& sica-IMRE, Universidad de La Habana, La Habana, CP 10400, Cuba Received 23 February 2000; received in revised form 23 May 2000
Abstract PbFe Al O samples were obtained by the chemical coprecipitation method with compositions \V V x"0.0, 3.0, 4.0, 6.0. As was expected, signi"cant di!erences in magnetization reversal mechanisms were found at room temperature in these specimens. A magnetic viscosity study was carried out in the temperature range of 75}300 K in order to obtain information related to the mechanism of magnetization behavior. A logarithmic time dependence of the magnetization was observed during a period of 20}1200 s. The magnetic viscosity S and the activation volume < increase as temperature is increased; the samples x"3.0 and 4.0 show the most rapid increase of the activation volume with temperature. Except for the samples x"3.0 and 4.0, the experimental temperature dependencies of the coercive "eld are in agreement with a model due to Givord and co-workers in which the magnetization reversal occurs in a volume equal to the activation volume. 2000 Elsevier Science B.V. All rights reserved. PACS: 75.50.Gg; 75.60.Lr Keywords: Hexaferrites; Aluminum substitution e!ects; Magnetic viscosity; Coercivity mechanisms; Activation volume
1. Introduction The substitution of Fe> ions has been studied intensively for years leading to great variations in the magnetic properties of hexaferrites. In particular, it is well known that Al-substituted M-type specimens show very large values of coercive "eld [1]. This behavior is mainly explained on the basis of the increase of the anisotropy "eld due to the strong lowering of the saturation magnetization [1,2], but the increase of the critical single-domain radius is also involved. It becomes larger when Al
* Corresponding author. Fax: #53-7-33-3758. E-mail address: magnet@!.oc.uh.cu (S. DmH az-Castan oH n).
concentration is increased and an interesting approach to the Stoner}Wohlfarth coherent rotation can be observed at high Al concentrations [1]. In order to obtain quantitative information related to magnetization reversal processes in PbFe Al O samples, we have used magnetic \V V viscosity formalism. It seems useful to calculate, for di!erent Al concentrations the activation volume < and the magnetic viscosity S , parameters lin ked with reversal processes [3}5]. Magnetic viscosity, the change in magnetization over time at constant applied "eld, has been explained due to thermal activation across activation-energy barriers. It has been of great interest in recent years for its importance from the fundamental standpoint and in various applications. The
0304-8853/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 0 4 3 1 - 5
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activation volume < is the volume associated with the initiation of magnetization reversal [4,5], and both S and < are in principle obtained from conventional magnetic viscosity experiment data. However, as the exact physical signi"cance of these magnitudes remains still not completely clear [6], this investigation is also an attempt to understand it. Therefore, in the present work a magnetic viscosity study on PbFe Al O samples with \V V di!erent Al concentrations is presented. The temperature dependencies of S and < are reported in the range of 75}300 K. In connection with these results the temperature dependence of the coercive "eld is discussed.
2. Experimental Aluminum-substituted hexaferrites of composition PbFe Al O (x"0.0, 3.0, 4.0, 6.0) were \V V obtained by a chemical coprecipitation method after sintering the powders for 1 h at 9203C [7]. X-ray powder di!raction patterns were recorded with a Philips PW 1050/25 modi"ed di!ractometer using Co K radiation (j+1.7902 As ). Microstruca tural studies were performed by using a JEOL JSM-530 scanning electron microscope (SEM). Magnetic properties were measured on small disks with an axial ratio of 10 : 1, with the magnetic "eld applied parallel to the plane of the disks. These measurements were performed in the temperature range of 75}300 K by means of a vibrating sample magnetometer Oxford 3001 (H "16 kOe).
The anisotropy "elds (H ) were determined using the singular point detection technique. Viscosity measurement consists of starting with the material in a known state of magnetization and measuring the decay of magnetization with time applying a constant reverse external "eld. Samples were "rst saturated to a positive "eld and then time variation of the magnetization M(t) was recorded for di!erent reverse steady "elds on the demagnetization curve during 1200 s. Data were "tted to an equation of the form M(t)"constant#S ln(t/t ) from which the mag netic viscosity coe$cient S at the applied "eld was determined. The magnetic viscosity S is related to
S and irreversible susceptibility by the expression S"S s , where S and s must be corrected for demagnetization e!ects. Considering that the irreversible susceptibility s can be expressed as s "s !s , after the measurement of M(t) the "eld was reversed by a small amount and returned to the measurement "eld, then the reversible susceptibility s was estimated as the slope of this minor loop assuming that all the processes which occur are reversible. In order to obtain the total susceptibility s from the slope of an adequate demagnetization curve, the (M, H) values were determined from the M "dM/dt"constant condition. First dM/dt of the measured magnetization was plotted versus M and then the intersection of a horizontal line, dM/dt"constant, with the curves corresponding to the di!erent "elds applied gives the (M, H) values. This procedure was proposed by Givord et al. [8] and is compatible with the magnetic equation of state developed by Estrin et al. [9,10].
3. Results and discussion The magneto-structural characterization of our powders is in good agreement with the reported data, and it was published earlier [7,11]. The analysis by X-ray di!raction con"rms the formation of the M-type hexagonal structure; no secondary phases were detected. The polycrystalline specimens were observed by SEM micrographs formed of hexagonal platelets randomly distributed with an average diameter size shown in Table 1. Some important magnetic properties characterizing the samples are also presented in Table 1. As was expected, we encountered di!erent coercive "eld values for the samples at room temperature. Curves of initial magnetization are in relation to this fact, indicating that the samples have di!erent reversal magnetization mechanisms (Fig. 1). The initial curve largely suggests a nucleation mechanism for x"0.0 and a nearly coherent rotation mechanism for x"6.0 sample. A behavior which can be interpreted as a mixture of both mechanisms is presumed for x"3.0 and 4.0 samples. It is worth noticing that the measured coercive "eld for x"0.0 is lower than the expected value for
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273
Table 1 Curie temperature (¹ ), anisotropy "eld (H ), saturation magnetization (p ), coercive "eld (H ) and grain size of PbFe Al O ! ! \V V samples x
¹ (K) !
p (emu/g) ¹"300 K
H (kOe) ¹"300 K
H (kOe) ¹"300 K
Particle diameter d (lm)
0.0 3.0 4.0 6.0
721 673 663 610
58 48 43 24
13.4 16.5 18.2 23.0
1.55 2.30 3.25 11.6 H*"10.3
1.5 0.4 0.3 0.9
Fig. 1. Initial magnetization curves of PbFe Al O at \V V room temperature. x"0.0 (*), x"3.0 (- - -), x"4.0 (*), x"6.0 (䊏).
Fig. 2. Some representative curves of the magnetization dependence of S . x"3.0, ¹"300 K (䊐); x"4.0, ¹"225 K (䢇); x"6.0, ¹"150 K (;). The line drawn through each set of
data represents the average value of S .
a chemical coprecipitated sample as a consequence of the excessive grain growth [11]. The coercive "eld values H shown in Table 1 were determined as the "eld in which dM/dH is maximum. This de"nition gives information about the irreversible demagnetization processes on the hysteresis loop, expressing the "eld for which the maximum rate of these processes occurs [12]. For all the samples and temperatures, except for x"6.0, these measured H values coincide experi mentally with the conventional coercive "eld H* in which M"0 on the demagnetization branch of the loop. According to the above de"nition of H , it is interesting to note that for x"6.0 the viscosity phenomena was detected just around the "eld in which dM/dH was maximum for each temperature, encountering a very weak e!ect on the demagnetization curve around H*.
All the M(t) curves show a time logarithmic dependence in the range from 20 to 1200 s. The observed magnetic "eld dependencies of S and s show maximum values around H for each temperature. S was obtained from the simul taneous evaluation of s and S. Over a wide range on the demagnetization curve, S remains almost constant, as can be seen in Fig. 2 where some representative experimental curves of S versus the speci"c magnetization are presented. At room temperature the relationship between log(S ) and log (H ) "ts the Barbier linear plot [13] fairly well for all the samples, with a slope of about 1.0. In Fig. 3, the increase of S with temperature and the calculated values of < are shown. The activation volume was estimated by the relation < " k¹/S M , where ¹ is the absolute temperature,
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Fig. 3. Temperature dependence of S and the activation vol ume calculated for all the samples. x"0.0 (*), x"3.0 (䊐), x"4.0 (䢇), x"6.0 (;).
Fig. 4. Activation volume as a function of H for all the samples at two temperatures. x"0.0 (*), x"3.0 (䊐), x"4.0 (䢇), x"6.0 (;). The lines are guides to the eyes.
k the Boltzman constant and M the saturation magnetization [5]. Concerning S (¹), a particular behavior is en countered. For x"3.0 and 4.0, S remains nearly independent of temperature from 150 to 300 K. Since the activation volume has been de"ned as < "*m/M , where *m is the moment variation between the minimum states and the maximum energy states reached at the instant that the energy barrier is overcome [5], it means that for x"3.0 and 4.0, the moment variation *m"< M in volved in the magnetization reversal increases with temperature, despite which M decreases. The clear increase of < with temperature is observed for all the samples in Fig. 3, a behavior that is not in agreement with a previous report on hexaferrites in the same temperature range [14]. In spite of this, the < values (0.5}2.5;10} cm) are in agreement with reported measurements on Ba}M and Sr}M ferrites [14}16]. As in most reports, < has been found to be smaller than the physical particle size, even in x"6.0 sample for which coherent rotation of single domain is supposed. It seems however an incongruity that the activation volume increases with temperature since the coercive "eld also increases and the reversal process becomes more di$cult. For di!erent materials at room temperature it has been noticed experi-
mentally (the Barbier plot) that log(S ) increases when log(H ) increases. i.e., small activation vol umes linked with high coercivities have been encountered in the same manner in which high coercivities are related to small-sized particles. In Fig. 4 this kind of behavior is shown in our samples for two temperatures. On the contrary, for a particular sample, we found that both H and < in crease when temperature is increased. In fact, in ferrite magnets it is well known that the anisotropy "eld H "2K /M increases slightly in the range of the temperature studied, due to the temperature dependence of M and the anisotropy constant K . But the critical single-domain radius R &K /M increases also with temperature, then at least for a simple case of single-domain behavior it seems reasonable to "nd the activation volume increasing as the temperature is raised. Two parameters are varying in our set of samples, composition and particle size. In order to make a comparison, we examine the < values in relation with d/D , where d is the particle diameter, as shown in Table 1 and D is an estimated magni tude proportional to the critical single-domain radius, for each sample and temperature. As an example, it can be seen in Fig. 5 (inset) how the measured values of H at 300 K decrease with d/D , as is expected. In Fig. 5, < versus d/D for all the samples is presented, where the lines are connecting
J.C. Faloh Gandarilla et al. / Journal of Magnetism and Magnetic Materials 222 (2000) 271}276
Fig. 5. Activation volume dependence on the parameter d/D for all the samples. x"0.0 (*), x"3.0 (䊐), x"4.0 (䢇), x"6.0 (;). Inset: H and H* dependence on d/D at 300 K for all the samples. The parameter d/D is only an estimation in order to make a comparison among samples, then we emphasize on the behavior not on the absolute values shown in the d/D axis.
the calculated < values for a sample at all the measurement temperatures. It can be seen that as < decreases when d/D is moved towards the multidomain region, i.e., we "nd larger < values while D increases due to the temperature increase in spite of the H increase. The temperature dependence of H was analyzed applying a model due to Givord and co-workers. Fig. 6 shows the experimental temperature dependence of H for x"0.0 and 6.0, and the calculated H values following the model. Givord et al. pro posed an equation to "t the temperature dependence of the coercive "eld, where thermal agitation is accounted and it is considered that the magnetic reversal occurs in a volume equal to the activation volume [17]. Using that model the temperature dependence of H was "tted to the expression H (¹)"(ac)/(M < )!4pbM }25S , where c is the domain wall energy. The values of a and b parameters associated with the best "t are respectively a"2.2, b"2.5 for x"0.0 and a"4.3, b"26.5 for x"6.0. On the other hand, for x"3.0 and 4.0 a good "t is not obtained. It is important, however, to notice the good "t obtained for x"0.0 and 6.0, taking into account the fact that these samples seem to have very di!erent reversal mechanisms.
275
Fig. 6. Temperature dependence of the experimental coercive "eld (H experimental) and the calculated coercive "eld (H "t). Error bars are shown.
4. Conclusions Using magnetic viscosity measurements we have analyzed the coercive processes on Al-substituted hexaferrites of composition PbFe Al O \V V (x"0.0, 3.0, 4.0, 6.0), in the temperature range of 75}300 K. The activation volume encountered was smaller than the physical particle size in all the cases. In spite of the di!erent reversal mechanisms indicated by the magnetization curves, the activation volumes were found to be of the same order of magnitude, di!ering by a factor of less than 5. However, a notable di!erence between the behavior of x"3.0, 4.0 and the rest of the samples has been observed. In particular, the temperature dependence of the coercivity "eld for x"3.0 and 4.0 cannot be "tted to the equation given by Givord and co-workers, contrasting with x"0.0 and 6.0 samples. It was found that the activation volume increases with temperature for all the samples, showing x"3.0 and 4.0 The most rapid increment of < occurs as temperature is raised up to 300 K. Analyzing all the samples at the same measurement temperature, we encounter the smaller activation volumes linked to higher coercivities. But, for a particular sample, when H increases with temperature we "nd that the calculated < values also have an
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increase. This behavior, for the simple case of coherent rotation, must be related to the increase of the critical single-domain radius as temperature is increased. References [1] [2] [3] [4] [5] [6]
K. Haneda, H. Kojima, Jpn. J. Appl. Phys. 12 (1973) 355. G. Albanese, A. Deriu, Ceram. Int. 5 (1979) 3. R. Street, J.C. Woolley, Proc. Phys. Soc. A 62 (1949) 562. E.P. Wohlfarth, J. Phys. F. 14 (1984) L155. P. Gaunt, J. Appl. Phys. 59 (1986) 4129. D.C. Crew, S.H. Farrant, P.G. McCormick, R. Street, J. Magn. Magn. Mater. 163 (1996) 299. [7] F. Leccabue, B.E. Watts, R. Panizzieri, S. DmH az, J.L. SaH nchez, G. Bocelli, M. Ghidini, Proceedings of the fourth European Conference on Advanced Materials and Processes (EUROMAT 95), Padua/Venice, Italy, Associazione Italiana di Metallurgia, 1995, p. 509.
[8] D. Givord, P. Tenaud, T. Viadieu. J. Magn. Magn. Mater. 72 (1988) 247. [9] Y. Estrin, P.G. McCormick, R. Street, J. Phys.:Condens. Matter 1 (1989) 4845. [10] L. Folks, R. Street, J. Appl. Phys. 76 (1994) 6391. [11] S. DmH az-Castan oH n, J.L. SaH nchez Ll., E. Estevez-Rams, F. Leccabue, B.E. Watts, J. Magn. Magn. Mater. 185 (1998) 194. [12] D.W. Taylor, V. Villas-Boas, Q. Lu, M.F. Rossignol, F.P. Missell, D. Givord, S. Hirosawa, J. Magn. Magn. Mater. 130 (1994) 225. [13] J.C. Barbier, Ann. Phys. 9 (1954) 84. [14] D. Givord, P. Tenaud, T. Viadieu, IEEE Trans. Magn. 24 (1988) 1921. [15] F. Carmona, A.M. Blanco, C. Alemany, J. Magn. Magn. Mater. 92 (1991) 417. [16] H. Nishio, H. Taguchi, F. Hirata, T. Takeishi, IEEE Trans. Magn. 29 (1993) 2637. [17] D. Givord, Q. Lu, M.F. Rossignol, P. Tenaud, T. Viadieu, J. Magn. Magn. Mater. 83 (1990) 183.
Activation volume and coercivity in aluminum-substituted Pb}M hexaferrites J.C. Faloh Gandarilla, S. DmH az-Castan oH n*, N. SuaH rez Almodovar Lab. Magnetismo, Facultad de Fn& sica-IMRE, Universidad de La Habana, La Habana, CP 10400, Cuba Received 23 February 2000; received in revised form 23 May 2000
Abstract PbFe Al O samples were obtained by the chemical coprecipitation method with compositions \V V x"0.0, 3.0, 4.0, 6.0. As was expected, signi"cant di!erences in magnetization reversal mechanisms were found at room temperature in these specimens. A magnetic viscosity study was carried out in the temperature range of 75}300 K in order to obtain information related to the mechanism of magnetization behavior. A logarithmic time dependence of the magnetization was observed during a period of 20}1200 s. The magnetic viscosity S and the activation volume < increase as temperature is increased; the samples x"3.0 and 4.0 show the most rapid increase of the activation volume with temperature. Except for the samples x"3.0 and 4.0, the experimental temperature dependencies of the coercive "eld are in agreement with a model due to Givord and co-workers in which the magnetization reversal occurs in a volume equal to the activation volume. 2000 Elsevier Science B.V. All rights reserved. PACS: 75.50.Gg; 75.60.Lr Keywords: Hexaferrites; Aluminum substitution e!ects; Magnetic viscosity; Coercivity mechanisms; Activation volume
1. Introduction The substitution of Fe> ions has been studied intensively for years leading to great variations in the magnetic properties of hexaferrites. In particular, it is well known that Al-substituted M-type specimens show very large values of coercive "eld [1]. This behavior is mainly explained on the basis of the increase of the anisotropy "eld due to the strong lowering of the saturation magnetization [1,2], but the increase of the critical single-domain radius is also involved. It becomes larger when Al
* Corresponding author. Fax: #53-7-33-3758. E-mail address: magnet@!.oc.uh.cu (S. DmH az-Castan oH n).
concentration is increased and an interesting approach to the Stoner}Wohlfarth coherent rotation can be observed at high Al concentrations [1]. In order to obtain quantitative information related to magnetization reversal processes in PbFe Al O samples, we have used magnetic \V V viscosity formalism. It seems useful to calculate, for di!erent Al concentrations the activation volume < and the magnetic viscosity S , parameters lin ked with reversal processes [3}5]. Magnetic viscosity, the change in magnetization over time at constant applied "eld, has been explained due to thermal activation across activation-energy barriers. It has been of great interest in recent years for its importance from the fundamental standpoint and in various applications. The
0304-8853/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 0 4 3 1 - 5
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activation volume < is the volume associated with the initiation of magnetization reversal [4,5], and both S and < are in principle obtained from conventional magnetic viscosity experiment data. However, as the exact physical signi"cance of these magnitudes remains still not completely clear [6], this investigation is also an attempt to understand it. Therefore, in the present work a magnetic viscosity study on PbFe Al O samples with \V V di!erent Al concentrations is presented. The temperature dependencies of S and < are reported in the range of 75}300 K. In connection with these results the temperature dependence of the coercive "eld is discussed.
2. Experimental Aluminum-substituted hexaferrites of composition PbFe Al O (x"0.0, 3.0, 4.0, 6.0) were \V V obtained by a chemical coprecipitation method after sintering the powders for 1 h at 9203C [7]. X-ray powder di!raction patterns were recorded with a Philips PW 1050/25 modi"ed di!ractometer using Co K radiation (j+1.7902 As ). Microstruca tural studies were performed by using a JEOL JSM-530 scanning electron microscope (SEM). Magnetic properties were measured on small disks with an axial ratio of 10 : 1, with the magnetic "eld applied parallel to the plane of the disks. These measurements were performed in the temperature range of 75}300 K by means of a vibrating sample magnetometer Oxford 3001 (H "16 kOe).
The anisotropy "elds (H ) were determined using the singular point detection technique. Viscosity measurement consists of starting with the material in a known state of magnetization and measuring the decay of magnetization with time applying a constant reverse external "eld. Samples were "rst saturated to a positive "eld and then time variation of the magnetization M(t) was recorded for di!erent reverse steady "elds on the demagnetization curve during 1200 s. Data were "tted to an equation of the form M(t)"constant#S ln(t/t ) from which the mag netic viscosity coe$cient S at the applied "eld was determined. The magnetic viscosity S is related to
S and irreversible susceptibility by the expression S"S s , where S and s must be corrected for demagnetization e!ects. Considering that the irreversible susceptibility s can be expressed as s "s !s , after the measurement of M(t) the "eld was reversed by a small amount and returned to the measurement "eld, then the reversible susceptibility s was estimated as the slope of this minor loop assuming that all the processes which occur are reversible. In order to obtain the total susceptibility s from the slope of an adequate demagnetization curve, the (M, H) values were determined from the M "dM/dt"constant condition. First dM/dt of the measured magnetization was plotted versus M and then the intersection of a horizontal line, dM/dt"constant, with the curves corresponding to the di!erent "elds applied gives the (M, H) values. This procedure was proposed by Givord et al. [8] and is compatible with the magnetic equation of state developed by Estrin et al. [9,10].
3. Results and discussion The magneto-structural characterization of our powders is in good agreement with the reported data, and it was published earlier [7,11]. The analysis by X-ray di!raction con"rms the formation of the M-type hexagonal structure; no secondary phases were detected. The polycrystalline specimens were observed by SEM micrographs formed of hexagonal platelets randomly distributed with an average diameter size shown in Table 1. Some important magnetic properties characterizing the samples are also presented in Table 1. As was expected, we encountered di!erent coercive "eld values for the samples at room temperature. Curves of initial magnetization are in relation to this fact, indicating that the samples have di!erent reversal magnetization mechanisms (Fig. 1). The initial curve largely suggests a nucleation mechanism for x"0.0 and a nearly coherent rotation mechanism for x"6.0 sample. A behavior which can be interpreted as a mixture of both mechanisms is presumed for x"3.0 and 4.0 samples. It is worth noticing that the measured coercive "eld for x"0.0 is lower than the expected value for
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273
Table 1 Curie temperature (¹ ), anisotropy "eld (H ), saturation magnetization (p ), coercive "eld (H ) and grain size of PbFe Al O ! ! \V V samples x
¹ (K) !
p (emu/g) ¹"300 K
H (kOe) ¹"300 K
H (kOe) ¹"300 K
Particle diameter d (lm)
0.0 3.0 4.0 6.0
721 673 663 610
58 48 43 24
13.4 16.5 18.2 23.0
1.55 2.30 3.25 11.6 H*"10.3
1.5 0.4 0.3 0.9
Fig. 1. Initial magnetization curves of PbFe Al O at \V V room temperature. x"0.0 (*), x"3.0 (- - -), x"4.0 (*), x"6.0 (䊏).
Fig. 2. Some representative curves of the magnetization dependence of S . x"3.0, ¹"300 K (䊐); x"4.0, ¹"225 K (䢇); x"6.0, ¹"150 K (;). The line drawn through each set of
data represents the average value of S .
a chemical coprecipitated sample as a consequence of the excessive grain growth [11]. The coercive "eld values H shown in Table 1 were determined as the "eld in which dM/dH is maximum. This de"nition gives information about the irreversible demagnetization processes on the hysteresis loop, expressing the "eld for which the maximum rate of these processes occurs [12]. For all the samples and temperatures, except for x"6.0, these measured H values coincide experi mentally with the conventional coercive "eld H* in which M"0 on the demagnetization branch of the loop. According to the above de"nition of H , it is interesting to note that for x"6.0 the viscosity phenomena was detected just around the "eld in which dM/dH was maximum for each temperature, encountering a very weak e!ect on the demagnetization curve around H*.
All the M(t) curves show a time logarithmic dependence in the range from 20 to 1200 s. The observed magnetic "eld dependencies of S and s show maximum values around H for each temperature. S was obtained from the simul taneous evaluation of s and S. Over a wide range on the demagnetization curve, S remains almost constant, as can be seen in Fig. 2 where some representative experimental curves of S versus the speci"c magnetization are presented. At room temperature the relationship between log(S ) and log (H ) "ts the Barbier linear plot [13] fairly well for all the samples, with a slope of about 1.0. In Fig. 3, the increase of S with temperature and the calculated values of < are shown. The activation volume was estimated by the relation < " k¹/S M , where ¹ is the absolute temperature,
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Fig. 3. Temperature dependence of S and the activation vol ume calculated for all the samples. x"0.0 (*), x"3.0 (䊐), x"4.0 (䢇), x"6.0 (;).
Fig. 4. Activation volume as a function of H for all the samples at two temperatures. x"0.0 (*), x"3.0 (䊐), x"4.0 (䢇), x"6.0 (;). The lines are guides to the eyes.
k the Boltzman constant and M the saturation magnetization [5]. Concerning S (¹), a particular behavior is en countered. For x"3.0 and 4.0, S remains nearly independent of temperature from 150 to 300 K. Since the activation volume has been de"ned as < "*m/M , where *m is the moment variation between the minimum states and the maximum energy states reached at the instant that the energy barrier is overcome [5], it means that for x"3.0 and 4.0, the moment variation *m"< M in volved in the magnetization reversal increases with temperature, despite which M decreases. The clear increase of < with temperature is observed for all the samples in Fig. 3, a behavior that is not in agreement with a previous report on hexaferrites in the same temperature range [14]. In spite of this, the < values (0.5}2.5;10} cm) are in agreement with reported measurements on Ba}M and Sr}M ferrites [14}16]. As in most reports, < has been found to be smaller than the physical particle size, even in x"6.0 sample for which coherent rotation of single domain is supposed. It seems however an incongruity that the activation volume increases with temperature since the coercive "eld also increases and the reversal process becomes more di$cult. For di!erent materials at room temperature it has been noticed experi-
mentally (the Barbier plot) that log(S ) increases when log(H ) increases. i.e., small activation vol umes linked with high coercivities have been encountered in the same manner in which high coercivities are related to small-sized particles. In Fig. 4 this kind of behavior is shown in our samples for two temperatures. On the contrary, for a particular sample, we found that both H and < in crease when temperature is increased. In fact, in ferrite magnets it is well known that the anisotropy "eld H "2K /M increases slightly in the range of the temperature studied, due to the temperature dependence of M and the anisotropy constant K . But the critical single-domain radius R &K /M increases also with temperature, then at least for a simple case of single-domain behavior it seems reasonable to "nd the activation volume increasing as the temperature is raised. Two parameters are varying in our set of samples, composition and particle size. In order to make a comparison, we examine the < values in relation with d/D , where d is the particle diameter, as shown in Table 1 and D is an estimated magni tude proportional to the critical single-domain radius, for each sample and temperature. As an example, it can be seen in Fig. 5 (inset) how the measured values of H at 300 K decrease with d/D , as is expected. In Fig. 5, < versus d/D for all the samples is presented, where the lines are connecting
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Fig. 5. Activation volume dependence on the parameter d/D for all the samples. x"0.0 (*), x"3.0 (䊐), x"4.0 (䢇), x"6.0 (;). Inset: H and H* dependence on d/D at 300 K for all the samples. The parameter d/D is only an estimation in order to make a comparison among samples, then we emphasize on the behavior not on the absolute values shown in the d/D axis.
the calculated < values for a sample at all the measurement temperatures. It can be seen that as < decreases when d/D is moved towards the multidomain region, i.e., we "nd larger < values while D increases due to the temperature increase in spite of the H increase. The temperature dependence of H was analyzed applying a model due to Givord and co-workers. Fig. 6 shows the experimental temperature dependence of H for x"0.0 and 6.0, and the calculated H values following the model. Givord et al. pro posed an equation to "t the temperature dependence of the coercive "eld, where thermal agitation is accounted and it is considered that the magnetic reversal occurs in a volume equal to the activation volume [17]. Using that model the temperature dependence of H was "tted to the expression H (¹)"(ac)/(M < )!4pbM }25S , where c is the domain wall energy. The values of a and b parameters associated with the best "t are respectively a"2.2, b"2.5 for x"0.0 and a"4.3, b"26.5 for x"6.0. On the other hand, for x"3.0 and 4.0 a good "t is not obtained. It is important, however, to notice the good "t obtained for x"0.0 and 6.0, taking into account the fact that these samples seem to have very di!erent reversal mechanisms.
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Fig. 6. Temperature dependence of the experimental coercive "eld (H experimental) and the calculated coercive "eld (H "t). Error bars are shown.
4. Conclusions Using magnetic viscosity measurements we have analyzed the coercive processes on Al-substituted hexaferrites of composition PbFe Al O \V V (x"0.0, 3.0, 4.0, 6.0), in the temperature range of 75}300 K. The activation volume encountered was smaller than the physical particle size in all the cases. In spite of the di!erent reversal mechanisms indicated by the magnetization curves, the activation volumes were found to be of the same order of magnitude, di!ering by a factor of less than 5. However, a notable di!erence between the behavior of x"3.0, 4.0 and the rest of the samples has been observed. In particular, the temperature dependence of the coercivity "eld for x"3.0 and 4.0 cannot be "tted to the equation given by Givord and co-workers, contrasting with x"0.0 and 6.0 samples. It was found that the activation volume increases with temperature for all the samples, showing x"3.0 and 4.0 The most rapid increment of < occurs as temperature is raised up to 300 K. Analyzing all the samples at the same measurement temperature, we encounter the smaller activation volumes linked to higher coercivities. But, for a particular sample, when H increases with temperature we "nd that the calculated < values also have an
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increase. This behavior, for the simple case of coherent rotation, must be related to the increase of the critical single-domain radius as temperature is increased. References [1] [2] [3] [4] [5] [6]
K. Haneda, H. Kojima, Jpn. J. Appl. Phys. 12 (1973) 355. G. Albanese, A. Deriu, Ceram. Int. 5 (1979) 3. R. Street, J.C. Woolley, Proc. Phys. Soc. A 62 (1949) 562. E.P. Wohlfarth, J. Phys. F. 14 (1984) L155. P. Gaunt, J. Appl. Phys. 59 (1986) 4129. D.C. Crew, S.H. Farrant, P.G. McCormick, R. Street, J. Magn. Magn. Mater. 163 (1996) 299. [7] F. Leccabue, B.E. Watts, R. Panizzieri, S. DmH az, J.L. SaH nchez, G. Bocelli, M. Ghidini, Proceedings of the fourth European Conference on Advanced Materials and Processes (EUROMAT 95), Padua/Venice, Italy, Associazione Italiana di Metallurgia, 1995, p. 509.
[8] D. Givord, P. Tenaud, T. Viadieu. J. Magn. Magn. Mater. 72 (1988) 247. [9] Y. Estrin, P.G. McCormick, R. Street, J. Phys.:Condens. Matter 1 (1989) 4845. [10] L. Folks, R. Street, J. Appl. Phys. 76 (1994) 6391. [11] S. DmH az-Castan oH n, J.L. SaH nchez Ll., E. Estevez-Rams, F. Leccabue, B.E. Watts, J. Magn. Magn. Mater. 185 (1998) 194. [12] D.W. Taylor, V. Villas-Boas, Q. Lu, M.F. Rossignol, F.P. Missell, D. Givord, S. Hirosawa, J. Magn. Magn. Mater. 130 (1994) 225. [13] J.C. Barbier, Ann. Phys. 9 (1954) 84. [14] D. Givord, P. Tenaud, T. Viadieu, IEEE Trans. Magn. 24 (1988) 1921. [15] F. Carmona, A.M. Blanco, C. Alemany, J. Magn. Magn. Mater. 92 (1991) 417. [16] H. Nishio, H. Taguchi, F. Hirata, T. Takeishi, IEEE Trans. Magn. 29 (1993) 2637. [17] D. Givord, Q. Lu, M.F. Rossignol, P. Tenaud, T. Viadieu, J. Magn. Magn. Mater. 83 (1990) 183.