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Magnetic anisotropy in fine magnetic particles
Journal of Magnetism and Magnetic Materials 118 (1993) 359-364 North-Holland
Magnetic anisotropy in fine magnetic particles Sami H. Mahmood Physics Department,
Yarmouk
University, Irbid, Jordan
Received 8 April 1992; in revised form 22 June 1992
In this work, the magnetization curves and the anisotropy energy constants K are calculated for three systems of Fe,O, fine particles using a simple model for the particle-size distribution function. The model gives a value for the mean particle diameter which is in good agreement with the value obtained from transmission electron micrographs. Also, the results for the anisotropy constant are in good agreement with those obtained from the temperature dependence of the remanent magnetization. Furthermore, Mijssbauer spectra are used, together with the calculated anisotropy constants, to determine the particle-size distribution.
sample of fine particles has the form of a Langevin function, i.e.,
1. Introduction
Fine magnetic particles (FMP) have received considerable attention due to their technological and scientific importance [ll. FMPs are single +omain particles with diameters typically = 100 A. The magnetic easy axes of the particles are randomly oriented, and the magnetic anisotropy is usually assumed uniaxial with an effective anisotropy constant K [2]. For such particles, the thermal energy is comparable to the magnetic anisotropy energy, and the magnetic moment of the particle could relax during the measuring time giving a zero net magnetization in the absence of an applied magnetic field. Thus, an assembly of such particles exhibits superparamagnetic behavior, and hence, no remanence or coercivity would be observed. The classical theory of paramagnetism is employed in the interpretations of the magnetic properties of fine particle systems. This theory proved successful for such systems. For details of the theory of superparamagnetism the reader is referred to ref. [3]. In an applied magnetic field H, the classical theory predicts that the magnetization A4 of a Correspondence to: Dr. S.H. Mahmood, Physics Department, Yarmouk University, Irbid, Jordan.
0304-8853/93/$06.00
=&[coth(PpW
- I/PcLH],
(1)
where p = M,V’ is the magnetic moment of the particle, I/ is the volume of the particle, MS is the saturation magnetization of the bulk material, p = l/k,T and ‘M, is the saturation magnetization of the sample. However, in real systems, the size of the particles is not constant. To account for the effect of the variability of the particle size, one assumes a convenient distribution function for the particle size. Chantrell et al. [4] employed a log-normal volume distribution, with the median particle diameter 0, and its standard deviation a, as fitting parameters. In this method, the values of 0, obtained are usually smaller than those obtained by electron micrographs. It is usually suggested that the difference is due to the non-magnetic coating layer. Further, the log-normal distribution does not give a good fit to the experimental data over the entire magnetization curve unless corrected for inter-particle interactions [5]. The anisotropy constant K for magnetic systems was calculated by workers using different
0 1993 - Elsevier Science Publishers B.V. All rights reserved
360
S. ri. Mahmood
/ Magnetic anisotropy in fine magnetic particles
methods. The law of approach to saturation [6] requires high magnetic fields and/or low temperatures, especially in studying materials having high anisotropy energies. The thermal decay of the remanence magnetization was used to calculate the anisotropy constants for weakly interacting systems of fine particles suspended in a liquid carrier [7], and for dried powders of fine particles apparently with significant inter-particle interactions [8]. This method requires temperatures I 5 K, in order to obtain a reasonably accurate value for the saturation remanence. Miissbauer spectroscopy was used to study the anisotropy energy of superparamagnetic microcrystals [2,9]. This method requires knowledge of the particle-size distribution function and of the temperature dependence of the distribution of the magnetic hyperfine fields in the Mossbauer absorption spectra. The purpose of this article is to provide analytical solutions for the magnetization curve and for the anisotropy constant of a system of fine particles using a simple model for particle-size distribution. Also, the calculated anisotropy constant is used in the distribution of hyperfine fields to calculate the particle-size distribution.
2. Experimental Three systems of Fe,O, fine particles in powder form are studied using magnetic data at temperatures between 20 and 15O”C, and MSssbauer measurements at liquid nitrogen temperature. The particles in sample (C) are coated with oleic acid and centrifuged while in suspension, while sample (U) was prepared without coating and was not centrifuged. Sample (A) is part of the coated system, annealed in the powder form at 250°C for 18 h to remove the surfactant coating. The preparation and characterization of the samples are presented elsewhere [8]. The magnetization measurements were carried out using a standard vibrating sample magnetometer in fields up to 12.5 kOe. Mossbauer measurements were taken using a constant acceleration spectrometer with 57Co source in a Pd matrix. The magnetic data were obtained at the physics department, UCNW,
Bangor, UK, and the Miissbauer data were collected at the Physics Department, University of Liverpool, Liverpool, UK.
3. Results and discussion 3.1.
(A)
Magnetization
The three samples, at or above room-temperature (RT) show typical superparamagnetic behavior as indicated by the magnetization curves in a previous article [8]. Further, the reduced magnetization M ( = M/M,) versus H/T curves at different temperatures (20°C < T s lSoOC>superimposed. Fig. 1 shows the reduced magnetization of sample C at 20°C and at 150°C. The figure also shows Langevin’s function L(&LH) with a particle diameter of 70 A (the mean value obtained from transmission electron micrographs). The significant misfit between the experimental and theoretical data suggests the presence of a distribution of particle sizes, in agreement with the conclusion of others [41. To account for the deviation from the Langevin-type behavior, a distribution of the particle sizes should be taken into consideration. The flat topped distribution (FTD), i.e., a uniform distribution extending from a minimum volume V, to a maximum volume Vr was found useful for the following reasons: first, it gives an analytical solution that fits the data reasonably
R
LANGEVIN ---FTD
-
“m
50 HIT
(OelK)
Fig. 1. Experimental reduced magnetization M versus for sample C, Langevin’s function and the theoretical using a single m for the particle sizes.
H/T curve
361
S.H. Mahmood / Magnetic anisotropy in fine magnetic particles
well; second, the result can be applied to more general distributions by dividing the range of particle volumes into a large number of subintervals, the distribution through each subinterval being a FTD; and third, it makes it possible to obtain an analytical expression for the anisotropy constant for the system. Using a FTD, the magnetization of the sample is given by
The values of V, and V, are estimated from the low field and the high field regions of the magnetization curve, respectively. This is possible since the low field behavior is dictated by the magnetic response of the large particles, and the high field behavior is determined by the response of the small particles. Using the asymptotic behavior of L@pH), one obtains
a = 1 - l/PM,KH,
/3pH x=-1.
Thus, V, is obtained from the initial reduced susceptibility Xi = da/dH for HI 50 Oe, and V, from a plot of a versus l/H in the high field region. For sam$e C the diameters were found to be: 0, = 24 A, II, = 120 A. The average of these two values (72 A) is in good agreement with the diameter (70 k 10) A obtained from the electron micrograph (EM), and is in qualitative agreement with the median value of 48 A obtained from the log-normal distribution. The calculated magnetization using eq. (2) is shown in fig. 1. Although this formula fits the experimental data better than Langevin’s function (see fig. l), there is still a significant deviation. This is due to the fact that the distribution of particle volumes in the sample is not uniform. A significantly better fit is achieved by a segmented flat topped distribution (SF’TD), which is obtained by dividing the range between V, and V, into three subin-
C
60 1
OOk--T---
H (kOe)
lo
Fig. 2. Experimental magnetization and the theoretical curves based on SFTDs for particle sizes.
tervals, each having a FTD with an appropriate weight. The limits of the subintervals and the corresponding weights, as will as the saturation magnetization are determined from the best fit. Fig. 2 shows the magnetization curves for the three samples C, A and U, each fitted using three subintervals. The saturation magnetizations and the mean particle diameters obtained from the fitting are summarized in table 1. The results presented in the table show a good agreement with the saturation magnetization M,(ext) obtained from extrapolating the straight line of M Table 1 Saturation magnetization M, (emu/g) from SF and from extrapolation, the mean particle diameters D (A) ybtained by different methods, and the median diameter 0, (A) obtained from the log-normal distribution Sample
C A U
M, (emu/g)
D (&
ext.
SFTD
EM (*lo)
SFTD
D,
Moss.
56.5 59 73
54 56.5 71
70 100 125
74+22 67k21 133+so
48 47 43
71+ 6 83k17 93* 17
S.H. Mahmood
362
/ Magnetic anisotropy in fme magnetic particles
versus l/H in the high field region. They also show a good agreement between the mean diameters obtained from the fitting and those from the electron micrographs. The lower value of the mean diameter obtained from fitting the data of the annealed sample is probably due to the partial oxidation of the sample, producing a weakly ferromagnetic phase.
Table 2 Anisotropy constants K (erg/cm31 evaluated from SFTD, and from the temperature dependence of the remanent magnetization (IRM) Sample
K (erg/cm31
SFTD
IRM
C
4.4 x 105 3.7 x 105 6.0 x lo5
5.4 x 10s 4.1 x 105 8.1 x lo5
U A
3.2. Magnetic anisotropy constant It was suggested by others [6] that the magnetization area method is accurate and convenient to calculate K for uniaxial materials. The relation between the anisotropy constant and the magnetization area for a sample consisting of single domain magnetic particles with uniaxial anisotropy and easy axes randomly oriented in space is given by [6] K = $/““H MR
dM,
(3)
where MR is the remanent magnetization of the sample. For a system of superparamagnetic particles, each with volume V, eq. (3) can be rewritten in the form
= qiHo{l
-L(pM,HV)}
dH
(4) where Ho is the saturation field. Since at saturation /3M,VHo B 1, eq. (3) is reduced to the form K=
This factor was found to contribute only = 3% to the anisotropy constant of the samples under investigation. The value of Ho = 20 kOe was found for the three samples. ThisDresult is consistent with the fact that V, = 24 A for the three samples. The anisotropy constant of the sample was then calculated using a FTD, and the result is
3Mo -ln(2j3it4,HoV). 2PWV
(5)
The saturation field is estimated by extrapolating the magnetization data from its value at 12.5 kOe to the value MO assuming a linear relation with a slope equal to the high field susceptibility. This assumption is consistent with neglecting the l/H * dependence of the high field magnetization [6].
X
{ln2(2PM,HoV,)
- ln2(2j3M,HoI/)}.
(6)
The subintervals (and their corresponding weights) that were used to fit the magnetization curves are used in eq. (6) to calculate K. The result of the SFTD are listed in table 2 (the listed values are corrected for the l/H* high field dependence). These results are in good agreement with those obtained from the remanent magnetization data. However, these values are higher than those calculated from the remanent magnetization data of dilute suspensions of Fe,O, fine particles [7]. The higher anisotropy could be attributed to inter-particle interactions in the powder samples of the present study. 3.3. Miissbauer spectroscopy Fig. 3 shows the Mtissbauer spectra for the three samples at 77 K. The spectra are all magnetically split since the 77 K is below the Miissbauer blocking temperatures of all samples under investigation. Actually, the lowest blocking temperature was found to be = 240 K for sample C [8]. The spectra for the other two samples are magnetically split even at RT. The spectra of fig.
S.H. Mahmood / Magnetic anisotropy in fie
3 were each fitted with a distribution of hyperfine fields P(B,.,r), and the results are shown in fig. 4(a). Since the temperature of the measurements
magnetic particles
(b) P(B,,f)
P(D) .4 .2 0 .4
C
.2
0.00
5
0 .4
K B
363
C
JlA-
5 1.00 .2
5 ti
0
E 2.00
Fig. 4. (a) Distribution of hyperfine fields I’@,,) and the corresponding distribution of particle diameters P(D).
3. 00
is much smaller than the blocking temperatures of the samples, and the observed hyperfine fields are < 10% lower than the saturation hyperfine field of bulk Fe,O,, one can use the relation [2]
U z 0. oc 0 E 8 $j 1.50
B,, = B,( 1 - k,T/2zW).
t 8 2
This relation is used to derive the distribution of particle-diameters, P(D), using the calculated values of K (table 2). The resulting distributions for the particle diameters are shown in fig. 4(b). The large diameter tail in the distribution of sample U confirms the presence of large particles due to the preparation without centrifuging, and the tail in the distribution of sample A is an indication of the growth of the particles due to the annealing process. These distributions were used to determine the mean particle’s diameter in each sample, and the results are tabulated in table 1. The results are in good agreement with those obtained from electron micrographs, and those from fitting the data using a SFTD; this indicates that the calculated values of the anisotropy constants are reliable.
3.00
4.50
A g
0. 00
K 8 $
1.50
& Y g
3.00
4. 50
I
-7. 5
4. Conclusions
I
-2.5 VELOCITY
(7)
2.5
7. 5
WI/SEC) Fig. 3. Miissbauer spectra for the samples at T = 77 K.
In this study, the magnetization curves and the anisotropy constants were calculated using a sim-
364
S.H. Mahmood / Magnetic anisotropy in fine magnetic particles
ple flat topped distribution for the particle sizes. The method discussed above gives results in good agreement with the experimental data, and with the results obtained by other methods. The success of this calculation method makes it possible to determine the mean oarticle volume. and the anisotropy constant (and hence, the blocking temperature) of a fine particle systems using room temperature measurements.
Acknowledgements
I am grateful to J. Popplewell, K. O’Grady and to D.P.E. Dickson for making may scientific visit to the UK, and the use of their equipment possible. Special thanks are due to M. El-Hi10 for his assistance in the experimental part.
References [l] K. Haneda, Can. J. Phys. 65 (1987) 1233. [2] S. Morup, J.A. Dumesic and H. Topsoe, in: Applications of Mossbauer Spectroscopy, vol. 1, ed. R.L. Cohen (Academic Press, New York, 1980) p. 1. [31 IS. Jacobs and C.P. Bean, in: Magnetism, vol. 3, eds. G.T. Rado and H. Suhl (Academic Press, New York, 1963). t41 R.W. Chantrell, J. Popplewell and S.W. Charles, IEEE Trans. Magn. MAG-14 (1978) 975. l51A. Tari, R.W. Chantrell, SW. Charles and J. Popplewell, Physica B 97 (1979) 57. [61 G. Hadjipanayis, D.J. Sellmayer and B. Brand& Phys. Rev. B23 (198913349. [71R.W. Chantrell, M. El-Hi10 and K. G’Grady, IEEE Trans. Magn. MAG-27 (1991) 3570. l81 S.H. Mahmood and I. Abu-Aljarayesh, to be published. 191 S. Morup and H. Topsoe, Appl. Phys. 11 (19761 63.
Magnetic anisotropy in fine magnetic particles Sami H. Mahmood Physics Department,
Yarmouk
University, Irbid, Jordan
Received 8 April 1992; in revised form 22 June 1992
In this work, the magnetization curves and the anisotropy energy constants K are calculated for three systems of Fe,O, fine particles using a simple model for the particle-size distribution function. The model gives a value for the mean particle diameter which is in good agreement with the value obtained from transmission electron micrographs. Also, the results for the anisotropy constant are in good agreement with those obtained from the temperature dependence of the remanent magnetization. Furthermore, Mijssbauer spectra are used, together with the calculated anisotropy constants, to determine the particle-size distribution.
sample of fine particles has the form of a Langevin function, i.e.,
1. Introduction
Fine magnetic particles (FMP) have received considerable attention due to their technological and scientific importance [ll. FMPs are single +omain particles with diameters typically = 100 A. The magnetic easy axes of the particles are randomly oriented, and the magnetic anisotropy is usually assumed uniaxial with an effective anisotropy constant K [2]. For such particles, the thermal energy is comparable to the magnetic anisotropy energy, and the magnetic moment of the particle could relax during the measuring time giving a zero net magnetization in the absence of an applied magnetic field. Thus, an assembly of such particles exhibits superparamagnetic behavior, and hence, no remanence or coercivity would be observed. The classical theory of paramagnetism is employed in the interpretations of the magnetic properties of fine particle systems. This theory proved successful for such systems. For details of the theory of superparamagnetism the reader is referred to ref. [3]. In an applied magnetic field H, the classical theory predicts that the magnetization A4 of a Correspondence to: Dr. S.H. Mahmood, Physics Department, Yarmouk University, Irbid, Jordan.
0304-8853/93/$06.00
=&[coth(PpW
- I/PcLH],
(1)
where p = M,V’ is the magnetic moment of the particle, I/ is the volume of the particle, MS is the saturation magnetization of the bulk material, p = l/k,T and ‘M, is the saturation magnetization of the sample. However, in real systems, the size of the particles is not constant. To account for the effect of the variability of the particle size, one assumes a convenient distribution function for the particle size. Chantrell et al. [4] employed a log-normal volume distribution, with the median particle diameter 0, and its standard deviation a, as fitting parameters. In this method, the values of 0, obtained are usually smaller than those obtained by electron micrographs. It is usually suggested that the difference is due to the non-magnetic coating layer. Further, the log-normal distribution does not give a good fit to the experimental data over the entire magnetization curve unless corrected for inter-particle interactions [5]. The anisotropy constant K for magnetic systems was calculated by workers using different
0 1993 - Elsevier Science Publishers B.V. All rights reserved
360
S. ri. Mahmood
/ Magnetic anisotropy in fine magnetic particles
methods. The law of approach to saturation [6] requires high magnetic fields and/or low temperatures, especially in studying materials having high anisotropy energies. The thermal decay of the remanence magnetization was used to calculate the anisotropy constants for weakly interacting systems of fine particles suspended in a liquid carrier [7], and for dried powders of fine particles apparently with significant inter-particle interactions [8]. This method requires temperatures I 5 K, in order to obtain a reasonably accurate value for the saturation remanence. Miissbauer spectroscopy was used to study the anisotropy energy of superparamagnetic microcrystals [2,9]. This method requires knowledge of the particle-size distribution function and of the temperature dependence of the distribution of the magnetic hyperfine fields in the Mossbauer absorption spectra. The purpose of this article is to provide analytical solutions for the magnetization curve and for the anisotropy constant of a system of fine particles using a simple model for particle-size distribution. Also, the calculated anisotropy constant is used in the distribution of hyperfine fields to calculate the particle-size distribution.
2. Experimental Three systems of Fe,O, fine particles in powder form are studied using magnetic data at temperatures between 20 and 15O”C, and MSssbauer measurements at liquid nitrogen temperature. The particles in sample (C) are coated with oleic acid and centrifuged while in suspension, while sample (U) was prepared without coating and was not centrifuged. Sample (A) is part of the coated system, annealed in the powder form at 250°C for 18 h to remove the surfactant coating. The preparation and characterization of the samples are presented elsewhere [8]. The magnetization measurements were carried out using a standard vibrating sample magnetometer in fields up to 12.5 kOe. Mossbauer measurements were taken using a constant acceleration spectrometer with 57Co source in a Pd matrix. The magnetic data were obtained at the physics department, UCNW,
Bangor, UK, and the Miissbauer data were collected at the Physics Department, University of Liverpool, Liverpool, UK.
3. Results and discussion 3.1.
(A)
Magnetization
The three samples, at or above room-temperature (RT) show typical superparamagnetic behavior as indicated by the magnetization curves in a previous article [8]. Further, the reduced magnetization M ( = M/M,) versus H/T curves at different temperatures (20°C < T s lSoOC>superimposed. Fig. 1 shows the reduced magnetization of sample C at 20°C and at 150°C. The figure also shows Langevin’s function L(&LH) with a particle diameter of 70 A (the mean value obtained from transmission electron micrographs). The significant misfit between the experimental and theoretical data suggests the presence of a distribution of particle sizes, in agreement with the conclusion of others [41. To account for the deviation from the Langevin-type behavior, a distribution of the particle sizes should be taken into consideration. The flat topped distribution (FTD), i.e., a uniform distribution extending from a minimum volume V, to a maximum volume Vr was found useful for the following reasons: first, it gives an analytical solution that fits the data reasonably
R
LANGEVIN ---FTD
-
“m
50 HIT
(OelK)
Fig. 1. Experimental reduced magnetization M versus for sample C, Langevin’s function and the theoretical using a single m for the particle sizes.
H/T curve
361
S.H. Mahmood / Magnetic anisotropy in fine magnetic particles
well; second, the result can be applied to more general distributions by dividing the range of particle volumes into a large number of subintervals, the distribution through each subinterval being a FTD; and third, it makes it possible to obtain an analytical expression for the anisotropy constant for the system. Using a FTD, the magnetization of the sample is given by
The values of V, and V, are estimated from the low field and the high field regions of the magnetization curve, respectively. This is possible since the low field behavior is dictated by the magnetic response of the large particles, and the high field behavior is determined by the response of the small particles. Using the asymptotic behavior of L@pH), one obtains
a = 1 - l/PM,KH,
/3pH x=-1.
Thus, V, is obtained from the initial reduced susceptibility Xi = da/dH for HI 50 Oe, and V, from a plot of a versus l/H in the high field region. For sam$e C the diameters were found to be: 0, = 24 A, II, = 120 A. The average of these two values (72 A) is in good agreement with the diameter (70 k 10) A obtained from the electron micrograph (EM), and is in qualitative agreement with the median value of 48 A obtained from the log-normal distribution. The calculated magnetization using eq. (2) is shown in fig. 1. Although this formula fits the experimental data better than Langevin’s function (see fig. l), there is still a significant deviation. This is due to the fact that the distribution of particle volumes in the sample is not uniform. A significantly better fit is achieved by a segmented flat topped distribution (SF’TD), which is obtained by dividing the range between V, and V, into three subin-
C
60 1
OOk--T---
H (kOe)
lo
Fig. 2. Experimental magnetization and the theoretical curves based on SFTDs for particle sizes.
tervals, each having a FTD with an appropriate weight. The limits of the subintervals and the corresponding weights, as will as the saturation magnetization are determined from the best fit. Fig. 2 shows the magnetization curves for the three samples C, A and U, each fitted using three subintervals. The saturation magnetizations and the mean particle diameters obtained from the fitting are summarized in table 1. The results presented in the table show a good agreement with the saturation magnetization M,(ext) obtained from extrapolating the straight line of M Table 1 Saturation magnetization M, (emu/g) from SF and from extrapolation, the mean particle diameters D (A) ybtained by different methods, and the median diameter 0, (A) obtained from the log-normal distribution Sample
C A U
M, (emu/g)
D (&
ext.
SFTD
EM (*lo)
SFTD
D,
Moss.
56.5 59 73
54 56.5 71
70 100 125
74+22 67k21 133+so
48 47 43
71+ 6 83k17 93* 17
S.H. Mahmood
362
/ Magnetic anisotropy in fme magnetic particles
versus l/H in the high field region. They also show a good agreement between the mean diameters obtained from the fitting and those from the electron micrographs. The lower value of the mean diameter obtained from fitting the data of the annealed sample is probably due to the partial oxidation of the sample, producing a weakly ferromagnetic phase.
Table 2 Anisotropy constants K (erg/cm31 evaluated from SFTD, and from the temperature dependence of the remanent magnetization (IRM) Sample
K (erg/cm31
SFTD
IRM
C
4.4 x 105 3.7 x 105 6.0 x lo5
5.4 x 10s 4.1 x 105 8.1 x lo5
U A
3.2. Magnetic anisotropy constant It was suggested by others [6] that the magnetization area method is accurate and convenient to calculate K for uniaxial materials. The relation between the anisotropy constant and the magnetization area for a sample consisting of single domain magnetic particles with uniaxial anisotropy and easy axes randomly oriented in space is given by [6] K = $/““H MR
dM,
(3)
where MR is the remanent magnetization of the sample. For a system of superparamagnetic particles, each with volume V, eq. (3) can be rewritten in the form
= qiHo{l
-L(pM,HV)}
dH
(4) where Ho is the saturation field. Since at saturation /3M,VHo B 1, eq. (3) is reduced to the form K=
This factor was found to contribute only = 3% to the anisotropy constant of the samples under investigation. The value of Ho = 20 kOe was found for the three samples. ThisDresult is consistent with the fact that V, = 24 A for the three samples. The anisotropy constant of the sample was then calculated using a FTD, and the result is
3Mo -ln(2j3it4,HoV). 2PWV
(5)
The saturation field is estimated by extrapolating the magnetization data from its value at 12.5 kOe to the value MO assuming a linear relation with a slope equal to the high field susceptibility. This assumption is consistent with neglecting the l/H * dependence of the high field magnetization [6].
X
{ln2(2PM,HoV,)
- ln2(2j3M,HoI/)}.
(6)
The subintervals (and their corresponding weights) that were used to fit the magnetization curves are used in eq. (6) to calculate K. The result of the SFTD are listed in table 2 (the listed values are corrected for the l/H* high field dependence). These results are in good agreement with those obtained from the remanent magnetization data. However, these values are higher than those calculated from the remanent magnetization data of dilute suspensions of Fe,O, fine particles [7]. The higher anisotropy could be attributed to inter-particle interactions in the powder samples of the present study. 3.3. Miissbauer spectroscopy Fig. 3 shows the Mtissbauer spectra for the three samples at 77 K. The spectra are all magnetically split since the 77 K is below the Miissbauer blocking temperatures of all samples under investigation. Actually, the lowest blocking temperature was found to be = 240 K for sample C [8]. The spectra for the other two samples are magnetically split even at RT. The spectra of fig.
S.H. Mahmood / Magnetic anisotropy in fie
3 were each fitted with a distribution of hyperfine fields P(B,.,r), and the results are shown in fig. 4(a). Since the temperature of the measurements
magnetic particles
(b) P(B,,f)
P(D) .4 .2 0 .4
C
.2
0.00
5
0 .4
K B
363
C
JlA-
5 1.00 .2
5 ti
0
E 2.00
Fig. 4. (a) Distribution of hyperfine fields I’@,,) and the corresponding distribution of particle diameters P(D).
3. 00
is much smaller than the blocking temperatures of the samples, and the observed hyperfine fields are < 10% lower than the saturation hyperfine field of bulk Fe,O,, one can use the relation [2]
U z 0. oc 0 E 8 $j 1.50
B,, = B,( 1 - k,T/2zW).
t 8 2
This relation is used to derive the distribution of particle-diameters, P(D), using the calculated values of K (table 2). The resulting distributions for the particle diameters are shown in fig. 4(b). The large diameter tail in the distribution of sample U confirms the presence of large particles due to the preparation without centrifuging, and the tail in the distribution of sample A is an indication of the growth of the particles due to the annealing process. These distributions were used to determine the mean particle’s diameter in each sample, and the results are tabulated in table 1. The results are in good agreement with those obtained from electron micrographs, and those from fitting the data using a SFTD; this indicates that the calculated values of the anisotropy constants are reliable.
3.00
4.50
A g
0. 00
K 8 $
1.50
& Y g
3.00
4. 50
I
-7. 5
4. Conclusions
I
-2.5 VELOCITY
(7)
2.5
7. 5
WI/SEC) Fig. 3. Miissbauer spectra for the samples at T = 77 K.
In this study, the magnetization curves and the anisotropy constants were calculated using a sim-
364
S.H. Mahmood / Magnetic anisotropy in fine magnetic particles
ple flat topped distribution for the particle sizes. The method discussed above gives results in good agreement with the experimental data, and with the results obtained by other methods. The success of this calculation method makes it possible to determine the mean oarticle volume. and the anisotropy constant (and hence, the blocking temperature) of a fine particle systems using room temperature measurements.
Acknowledgements
I am grateful to J. Popplewell, K. O’Grady and to D.P.E. Dickson for making may scientific visit to the UK, and the use of their equipment possible. Special thanks are due to M. El-Hi10 for his assistance in the experimental part.
References [l] K. Haneda, Can. J. Phys. 65 (1987) 1233. [2] S. Morup, J.A. Dumesic and H. Topsoe, in: Applications of Mossbauer Spectroscopy, vol. 1, ed. R.L. Cohen (Academic Press, New York, 1980) p. 1. [31 IS. Jacobs and C.P. Bean, in: Magnetism, vol. 3, eds. G.T. Rado and H. Suhl (Academic Press, New York, 1963). t41 R.W. Chantrell, J. Popplewell and S.W. Charles, IEEE Trans. Magn. MAG-14 (1978) 975. l51A. Tari, R.W. Chantrell, SW. Charles and J. Popplewell, Physica B 97 (1979) 57. [61 G. Hadjipanayis, D.J. Sellmayer and B. Brand& Phys. Rev. B23 (198913349. [71R.W. Chantrell, M. El-Hi10 and K. G’Grady, IEEE Trans. Magn. MAG-27 (1991) 3570. l81 S.H. Mahmood and I. Abu-Aljarayesh, to be published. 191 S. Morup and H. Topsoe, Appl. Phys. 11 (19761 63.