Advances in Condensed Matter Physics

Volume 2008, Article ID 703479, 7 pages

http://dx.doi.org/10.1155/2008/703479

## Preparation and Characterization of Manganese Ferrite Aluminates

^{1}P. G. Department of Applied Physics, S.D. College, Ambala Cantt 133 001, India^{2}Department of Physics, Maharshi Dayanand University, Rohtak 124 001, India^{3}UGC-DAE, Consortium for Scientific Research, Khandwa Road, Indore 452 017, India

Received 13 June 2008; Revised 30 September 2008; Accepted 31 December 2008

Academic Editor: R. N. P. Choudhary

Copyright © 2008 R. L. Dhiman et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Aluminum doped manganese ferrites
with
have been prepared by the double ceramic route. The formation of mixed spinel phase has been confirmed by X-ray diffraction analysis. The unit cell parameter `' is found to decrease linearly with aluminum concentration due to smaller ionic radius of aluminum. The cation distributions were estimated from X-ray diffraction intensities of various planes. The theoretical lattice parameter, X-ray density, oxygen positional parameter, ionic radii, jump length, and bonds and edges lengths of the tetrahedral (A) and octahedral (B) sites were determined. ^{57}Fe Mössbauer spectra recorded at room temperature were fitted with two sextets corresponding to Fe^{3+} ions at A- and B-sites. In the present ferrite system, the area ratio of Fe^{3+}
ions at the A- and B-sites determined from the spectral analysis of Mössbauer spectra gives evidence that Al^{3+} ions replace iron ions at B-sites.
This change in the site preference reflects an abrupt change in magnetic hyperfine fields at A- and B-sites as aluminum concentration increases, which has been explained on the basis of supertransferred hyperfine field. On the basis of estimated cation distribution, it is concluded that aluminum doped manganese ferrites exhibit a 55% normal spinel structure.

#### 1. Introduction

Spinel
ferrites have been the subject of great interest for the past five decades, because of their wide
range of applications in transformers,
inductors, choke
coils, noise filters magnetic recording heads, and so forth [1]. These ferrites
possessing cubic
close-packed structure of oxygen ions, are described by the formula , where (A) and [B] represent tetrahedral and
octahedral sites, respectively. The site occupancy is often depicted in the
chemical formula as , where round and square brackets denote the A-
and B-sites, respectively, M represents a metal cation, and ‘’
is the inversion parameter. The degree of inversion ‘’ for spinel
ferrites is defined as the fraction of tetrahedral (A)-sites occupied by
trivalent cations. Accordingly, for a normal spinel and for a
completely inverse spinel, . The magnetic and the electronic
properties of such a ferrite system depend upon the type of metal cations and their distribution
among the two interstitial sites, that is, A- and B-sites. Therefore, the
knowledge of cation distribution is essential to understand the magnetic
behavior of spinel ferrites. Manganese ferrite is early known to be a mixed
inverse spinel, and the degree of inversion mainly depends upon the method of
preparation. The presence of nonmagnetic ions in these spinel ferrites is found
to alter their magnetic and electronic properties. The addition of metal
cations such as trivalent or tetravalent influences the electronic and magnetic
properties of the ferrite system [2–6]. Various studies
showed that heating might change the distribution of metal cations at the A-
and B-sites of . It has been reported that by
using neutron diffraction technique, the degree of inversion, that is, the
distribution of the cationic ions between the tetrahedral and octahedral sites
of prepared by usual ceramic route was determined
81% normal [7]. However, this value reduced to 33%, when was prepared by wet chemical method [8]. Thus, the method of preparation may
play a crucial role in order to obtain the desired electronic and magnetic
properties. Extensive investigations regarding the substitution of metal
cations, for example, Cu^{2+}, Zn^{2+}, Ti^{2+}, Co^{3+},
and Ni^{2+} in manganese ferrites
have been reported, giving useful
information about the influence of such metal cations [9–13]. However, no
systematic results regarding the estimation of cation distribution on
substitution of aluminum ion in was available. In the present studies, the
cation distribution in tetrahedral and octahedral sites of aluminum substituted
manganese ferrites synthesized via double ceramic route has been determined by X-ray diffraction
and Mössbauer spectroscopic measurements.

#### 2. Experimental

Samples of the mixed spinel ferrites for , and 1.0 were synthesized by usual double ceramic processing technique.
The starting materials were high-purity
analytical reagent grade oxides, , MnO, and .
The required compositions were weighed and mixed in a mortar and pestle. The
mixed powders were presintered at for 10 hours in the air and allowed to cool to room
temperature at the rate of about /min. In the final sintering
process, the samples were placed in a furnace at for 10 hours
in the air and then
cooled slowly to room temperature at the rate of /min. The
finally sintered materials were well grounded. To ensure their single-phase
nature, the powder X-ray diffraction studies were made on Regaku X-ray
diffractometer by using Cu-K_{α} radiation of
wave length 1.54060 Å. Mössbauer absorption spectra were
recorded in transmission geometry at room temperature using a multichannel
analyzer with a drive in constant acceleration mode. A
source with initial activity of 20 mCi was used. The spectrometer was
periodically calibrated using a natural iron foil as a standard.

#### 3. Results and Discussion

##### 3.1. X-Ray Diffraction Analysis

The X-ray diffraction patterns of mixed spinel ferrites ( for , and 1.0) are shown in Figure 1.

The positions of diffraction
peaks from various planes were identified using JCPDS file no. 74-2403. It is
evident from Figure 1 that each ferrite sample exhibits single-phase cubic
spinel structure with Fd-3m (227) space group. The value of the lattice
constant ‘*a _{O}*’ for all the
samples was determined from the position of principal (311) peak using where

*h, k*, and

*l*are the miller indices.

The observed values of lattice
constant ‘*a _{O}*’ listed in
Table 1 are slightly smaller than the JCPDS table value of 8.518 Å. The lattice
constant ‘

*a*’ is found to decrease linearly with aluminum concentration (

_{O}*x*) as shown in Figure 2(a), thereby obeying Vegard’s law [14].

The decrease in lattice constant is
attributed to the fact that the Pauling ionic radius of Al^{3+} (0.50 Å) is smaller than that of Fe^{3+} (0.64 Å), which causes the shrinking in the unit cell dimensions. The decrease in ‘*a _{O}*’ and the shift of reflections toward higher angle
with the increasing aluminum concentration (

*x*) show that aluminum atoms have been incorporated into the spinel structure [15].

The X-ray density ‘*d _{x}*’
was calculated using the formula [16] where ‘

*M*’ is the molecular weight, ‘

*N*’ is the Avogadro's number, and ‘

*a*’ is the lattice constant of the spinel ferrite. The calculated values of X-ray density are listed in Table 1. The X-ray density decreases with increasing aluminum concentration (

_{O}*x*), as shown in Figure 2(b). The decrease in X-ray density is due to the decrease in mass, which overtakes the decrease in volume of the unit cell.

The cation distribution in the
various spinel ferrite systems has been estimated from X-ray diffraction [5, 6], Mössbauer’s
effect [17, 18], and magnetization
measurements [19, 20]. It has been
reported [21, 22] that the best
information in estimation of cation distribution can be achieved by comparing
the experimental and theoretical intensity ratios for reflections (220), (422),
and (400). However, the intensities of (220), (422), and (400) planes are more
sensitive to cations on A- and B-sites [23, 24]. The X-ray
diffraction intensity of the respective planes was calculated using the formula [25] where is the relative integral
intensity; is the
structure factor; *P* is the
multiplicity factor; *Lp* is the Lorentz factor. The structural factors
were calculated by using the equation suggested by Porta and Furuhashi et al.
[26, 27]. The
multiplicity factor and the Lorentz factors were taken from the literature
[16]. The ionic scattering factor reported in the international tables for
X-ray crystallography [28] is used for the calculation of structural factor. It
is well established that the intensity ratios / and / are considered to be sensitive
to cation distribution [29]. Therefore in the present ferrite system, intensity
ratios of these planes have been used in estimation of cation distribution. The
intensity ratios of these planes were calculated for various cation distributions
using the following expression suggested by Bertaut [21]: For all the samples, the calculated values,
those closest to the experimental observed values, are given in Table 2. The
theoretical lattice constant ‘*a _{t}*’
for all composition was calculated on the basis of estimated cation
distribution by using the relation [30] where and are the radii of
the - and the -sites, respectively, and
is the radius of the oxygen ion O

^{2−}(1.48 Å). The calculated values ‘

*a*’ are nearly equal to the experimental observed value ‘

_{t}*a*’ which confirms the estimated cation distribution (see Table 2). The site radii and used above were determined using the following: The calculated values of and are listed in Table 3. The value of decreases slowly; however the value of decreases noticeably with increasing aluminum concentration. This is due to the replacement of larger ionic radii () with smaller ionic radii () and their distribution amongst the - and -sites. The value of the oxygen positional parameter ‘

_{O}*u*’ was calculated by using the following relation: The determined values of ‘

*u*’ are listed in Table 3. The values of the tetrahedral (), octahedral bond length (), tetrahedral edge length (), and shared () and unshared octahedral edge lengths () were calculated by using the experimental values of lattice constant ‘

*a*’ and oxygen positional parameter ‘

_{O}*u*’ from the following [30, 31]: Various calculated X-ray parameters are given in Table 3. It is observed that , , , , and decrease with increasing aluminum concentration (

*x*). This is due to the substitution process, that is, replacement of larger ionic radii (Fe

^{3+}) by smaller ionic radii (Al

^{3+}) and their distribution among the - and -sites. These results are in consistent with the reported data [32]. It has been reported that the jump length ‘

*L*’ (the distance between the magnetic ions) of electrons influences the physical properties of the ferrite system [33]. Electrons those are hopping between - and -sites are less probable compared to that between - and -sites, because the distance between the two metal ions placed in -sites is smaller than if they were placed one in -sites and the other in -sites [34]. ‘

*L*’ of the - and -sites is determined from the following relations [35]: It is observed that ‘

*L*’ of - and -sites decreases with increasing aluminum concentration (

*x*) as shown in Figure 3.

The decrease in
jump length is due to the decrease in the distance between the magnetic ions by
the substitution of smaller Al^{3+} ions at the -sites and is similar
to those reported earlier [4, 32].

##### 3.2. Mössbauer Analysis

^{57}Fe Mössbauer absorption
spectra of mixed spinel ferrite system for , and 1.0 recorded at room temperature
are displayed in Figure 4. The experimental data were fitted using least
square-fitting (NORMOS/SITE) program [36]. Each spectrum exhibits a
superposition of two Zeeman sextets, one sextet corresponding to a higher
magnetic field is attributed to Fe^{3+} ions on the -site, and the
other sextet corresponding to lower magnetic field is attributed to Fe^{3+} ions on the -site. The refined values of the hyperfine parameters computed
from the Mössbauer spectra are listed in Table 4. In the present ferrite
system, it is observed that on increasing Al^{3+} ions concentration,
the values of isomer shift (*δ*) of tetrahedral -sites
show almost negligible change, indicating that aluminum ions do not enter in
-sites. The isomer shift of -sites is greater than -site and is in agreement
with the reported data [11]. Furthermore, the observed values of isomer shift (*δ*) are significantly less than the expected
value, 0.5 mm/s for the Fe^{2+} ions [20]. Hence, the presence of Fe^{2+} ions in the present ferrite system is ruled out. Thus the electron exchange interaction
() does not occur, and hence
the oxidation state of Fe^{3+} remains unchanged during synthesis
process. The hyperfine field values at - and -sites show a gradual decrease with increasing Al^{3+} concentration (*x*). This can be explained on the basis of
supertransferred hyperfine field at the central cation that originates from the
magnetic moments of the nearest-neighbor cations, that is, from the intra-sublattice
contributions and and the inter-sublattice contributions and . In the present ferrite
system, the intra-sublattice contributions and are predominant. It has been reported that the intensities corresponding
to (200) and (422) reflections are most sensitive to cations on A-sites
[23, 24]. The X-ray diffraction patterns of the present ferrite system indicate
that the intensity of (220) and (422) reflections remains almost constant as
compared to (311) reflection, suggesting that Al^{3+} ions do not enter
in the A-sites. The value of isomer shift (*δ*)
of -sites remains invariant on substitution of aluminum ions suggesting that Al^{3+} ions do not replace
Fe^{3+} ions from -sites. The introduction of Al^{3+} ions
that replaces Fe^{3+} ions from -sites decreases intra-sublattice
contributions, which in turn decreases the hyperfine field values. As nonmagnetic Al^{3+} ions replace Fe^{3+} ions, the correct amount of Fe^{3+} ions occupying - and -sites is estimated by determining the area under the Mössbauer absorption spectra through the least square
fitting program. The Fe^{3+}(B)/Fe^{3+}(A) ratio obtained
from the Mössbauer spectra is
in good agreement with those calculated from X-ray intensities. It is observed
that this ratio decreases with increasing aluminum concentration (*x*) suggesting a decrease in
ferrimagnetic behavior.

#### 4. Conclusion

Aluminum substituted manganese ferrites for , and 1.0 have been prepared by double ceramic processing
technique. The unit cell parameter decreases linearly with the increase of
aluminum concentration (*x*) due to its
small ionic radius. The cation distribution estimated from X-ray intensity
ratios has been verified by comparing the theoretical and experimental lattice parameters. It is observed that the correct amount of Fe^{3+} ions occupying -
and -sites obtained from Mössbauer spectra is in good agreement with those calculated from
X-ray intensity calculations. The hyperfine magnetic field obtained from the
Mössbauer absorption spectra decreases with increasing aluminum concentration
suggesting the decrease in ferrimagnetic behavior and has been explained on the
basis of supertransferred hyperfine field mechanism. The X-ray determined
parameters, for example, lattice constant, X-ray density, ionic radius, bond
length, jump length of the - and -sites, oxygen positional parameter, -site
edge length, and shared and unshared -site edge lengths were determined and
found affected by Al^{3+} ions substitution. On the basis of estimated
cation distribution, it is concluded that the present ferrite system exhibits a
55% normal spinel structure.

#### Acknowledgments

One of the authors (R. L. Dhiman) is grateful to Dr. Alok Banerjee and Dr. R. J. Chaudhary, Scientists, UGC-DAE, Consortium for Scientific Research, University Campus, Khandwa Road, Indore (MP), India, for providing experimental facilities.

#### References

- H. Igarashi and M. Okazaki, “Effects of porosity and grain size on the magnetic properties of NiZn ferrite,”
*Journal of the American Ceramic Society*, vol. 60, no. 1-2, pp. 51–54, 1977. View at Publisher · View at Google Scholar - R. Lal, Suman, N. D. Sharma, S. P. Taneja, and V. R. Reddy, “Structural and magnetic properties of zinc ferrite aluminates,”
*Indian Journal of Pure and Applied Physics*, vol. 45, no. 3, pp. 231–237, 2007. View at Google Scholar - S. Singhal, S. K. Barthwal, and K. Chandra, “Structural, magnetic and Mössbauer spectral studies of nanosize aluminum substituted nickel zinc ferrites,”
*Journal of Magnetism and Magnetic Materials*, vol. 296, no. 2, pp. 94–103, 2006. View at Publisher · View at Google Scholar - A. A. Pandit, A. R. Shitre, D. R. Shengule, and K. M. Jadhav, “Magnetic and dielectric properties of ${\text{Mg}}_{1+x}{\text{Mn}}_{x}{\text{Fe}}_{2-2x}{\text{O}}_{4}$ ferrite system,”
*Journal of Materials Science*, vol. 40, no. 2, pp. 423–428, 2005. View at Publisher · View at Google Scholar - S. Singhal, J. Singh, S. K. Barthwal, and K. Chandra, “Preparation and characterization of nanosize nickel-substituted cobalt ferrites $({\text{Co}}_{1-x}{\text{Ni}}_{x}{\text{Fe}}_{2}{\text{O}}_{4})$,”
*Journal of Solid State Chemistry*, vol. 178, no. 10, pp. 3183–3189, 2005. View at Publisher · View at Google Scholar - R. Justin Joseyphus, A. Narayanasamy, K. Shinoda, B. Jeyadevan, and K. Tohji, “Synthesis and magnetic properties of the size-controlled Mn-Zn ferrite nanoparticles by oxidation method,”
*Journal of Physics and Chemistry of Solids*, vol. 67, no. 7, pp. 1510–1517, 2006. View at Publisher · View at Google Scholar - J. M. Hastings and L. M. Corliss, “Neutron diffraction study of manganese ferrite,”
*Physical Review*, vol. 104, no. 2, pp. 328–331, 1956. View at Publisher · View at Google Scholar - J. Sakurai and T. Shinjo, “Neutron diffraction of manganese ferrite prepared from aqueous solution,”
*Journal of the Physical Society of Japan*, vol. 23, no. 6, p. 1426, 1967. View at Publisher · View at Google Scholar - M. U. Rana, M. U. Islam, and T. Abas, “Cation distribution in Cu-substituted manganese ferrites,”
*Materials Letters*, vol. 41, no. 2, pp. 52–56, 1999. View at Publisher · View at Google Scholar - J. Feng, L.-Q. Guo, X. Xu, S.-Y. Qi, and M.-L. Zhang, “Hydrothermal synthesis and characterization of ${\text{Mn}}_{1-x}{\text{Zn}}_{x}{\text{Fe}}_{2}{\text{O}}_{4}$ nanoparticles,”
*Physica B*, vol. 394, no. 1, pp. 100–103, 2007. View at Publisher · View at Google Scholar - S. Mishra, T. K. Kundu, K. C. Barick, D. Bahadur, and D. Chakravorty, “Preparation of nanocrystalline ${\text{MnFe}}_{2}{\text{O}}_{4}$ by doping with ${\text{Ti}}^{4+}$ ions using solid-state reaction route,”
*Journal of Magnetism and Magnetic Materials*, vol. 307, no. 2, pp. 222–226, 2006. View at Publisher · View at Google Scholar - M. K. Fayek, F. M. Sayed Ahmed, S. S. Ata-Allah, M. K. Elnimer, and M. F. Mostafa, “Crystal, magnetic and electric behaviour of ${\text{CoMn}}_{x}{\text{Fe}}_{2-x}{\text{O}}_{4}$ cubic ferrites,”
*Journal of Materials Science*, vol. 27, no. 17, pp. 4813–4817, 1992. View at Publisher · View at Google Scholar - Q.-M. Wei, J.-B. Li, and Y.-J. Chen, “Cation distribution and infrared properties of ${\text{Ni}}_{x}{\text{Mn}}_{1-x}{\text{Fe}}_{2}{\text{O}}_{4}$ ferrites,”
*Journal of Materials Science*, vol. 36, no. 21, pp. 5115–5118, 2001. View at Publisher · View at Google Scholar - C. G. Whinfrey, D. W. Eckart, and A. Tauber, “Preparation and X-ray diffraction data for some rare earth stannates,”
*Journal of the American Chemical Society*, vol. 82, no. 11, pp. 2695–2697, 1960. View at Publisher · View at Google Scholar - J. A. Toledo, M. A. Valenzuela, P. Bosch et al., “Effect of ${\text{AI}}^{3+}$ introduction into hydrothermally prepared ${\text{ZnFe}}_{2}{\text{O}}_{4}$,”
*Applied Catalysis A*, vol. 198, no. 1-2, pp. 235–245, 2000. View at Publisher · View at Google Scholar - B. D. Cullity,
*Elements of X-Ray Diffraction*, Addison-Wesley, Reading, Mass, USA, 1959. - S. S. Ata-Allah and M. Kaiser, “Cation distribution, hyperfine parameters and conduction mechanism in the ferrimagnetic system ${\text{Cu}}_{0.5}{\text{Co}}_{0.5}{\text{Ga}}_{x}{\text{Fe}}_{2-x}{\text{O}}_{4}$,”
*Physica Status Solidi B*, vol. 242, no. 6, pp. 1324–1335, 2005. View at Publisher · View at Google Scholar - A. Rais, A. M. Gismelseed, and I. A. Al-Omari, “Cation distribution and magnetic properties of nickel-chromium ferrites
${\text{NiCr}}_{x}{\text{Fe}}_{2-x}{\text{O}}_{4}(0\le x\le 1.4)$,”
*Physica Status Solidi B*, vol. 242, no. 7, pp. 1497–1503, 2005. View at Publisher · View at Google Scholar - S. A. Jadhav, “Magnetic properties of Zn-substituted Li-Cu ferrites,”
*Journal of Magnetism and Magnetic Materials*, vol. 224, no. 2, pp. 167–172, 2001. View at Publisher · View at Google Scholar - K. P. Thummer, M. C. Chhantbar, K. B. Modi, G. J. Baldha, and H. H. Joshi, “Localized canted spin behaviour in ${\text{Zn}}_{x}{\text{Mg}}_{1.5-x}{\text{Mn}}_{0.5}{\text{FeO}}_{4}$ spinel ferrite system,”
*Journal of Magnetism and Magnetic Materials*, vol. 280, no. 1, pp. 23–30, 2004. View at Publisher · View at Google Scholar - E. F. Bertaut, “Etude de la nature des ferrites spinelles,”
*Comptes Rendus Hebdomadaires des Séances de l'Academie des Sciences*, vol. 230, pp. 213–215, 1950. View at Google Scholar - L. Weil, E. F. Bertaut, and L. Bochirol, “Propriétés magnétiques et structure de la phase quadratique du ferrite de cuivre,”
*Journal de Physique et Le Radium*, vol. 11, no. 5, pp. 208–212, 1950. View at Publisher · View at Google Scholar - E. Eoiska and W. Woiski, “The evidence of ${\text{Cd}}_{x}{\text{\hspace{0.17em}}}^{\text{2+}}{\text{Fe}}_{1-x}{\text{\hspace{0.17em}}}^{\text{3+}}[{\text{Ni}}_{1-x}{\text{\hspace{0.17em}}}^{\text{2+}}{\text{Fe}}_{1+x}{\text{\hspace{0.17em}}}^{\text{3+}}]{\text{O}}_{4}$ cation distribution based on X-ray and Mössbauer data,”
*Physica Status Solidi A*, vol. 132, no. 1, pp. K51–K56, 1992. View at Publisher · View at Google Scholar - B. P. Ladgaonkar and A. S. Vaingankar, “X-ray diffraction investigation of cation distribution in ${\text{Cd}}_{x}{\text{Cu}}_{1-x}{\text{Fe}}_{2}{\text{O}}_{4}$ ferrite system,”
*Materials Chemistry and Physics*, vol. 56, no. 3, pp. 280–283, 1998. View at Publisher · View at Google Scholar - M. G. Buerger,
*Crystal Structure Analysis*, John Wiley & Sons, New York, NY, USA, 1960. - P. Porta, F. S. Stone, and R. G. Turner, “The distribution of nickel ions among octahedral and tetrahedral sites in ${\text{NiAl}}_{2}{\text{O}}_{4}{\text{-MgAl}}_{2}{\text{O}}_{4}$ solid solutions,”
*Journal of Solid State Chemistry*, vol. 11, no. 2, pp. 135–147, 1974. View at Publisher · View at Google Scholar - H. Furuhashi, M. Inagaki, and S. Naka, “Determination of cation distribution in spinels by X-ray diffraction method,”
*Journal of Inorganic and Nuclear Chemistry*, vol. 35, no. 8, pp. 3009–3014, 1973. View at Publisher · View at Google Scholar - C. H. MacGillavry, G. D. Rieck, and K. Lonsdale,
*Physical and Chemical Tables*, International Tables for X-Ray Crystallography, Volume III, Kynoch Press, Birmingham, UK, 1968. - H. Ohnishi and T. Teranishi, “Crystal distortion in copper ferrite-chromite series,”
*Journal of the Physical Society of Japan*, vol. 16, no. 1, pp. 35–43, 1961. View at Publisher · View at Google Scholar - A. A. Yousif, M. E. Elzain, S. A. Mazen, H. H. Sutherland, M. H. Abdalla, and S. F. Masour, “Mössbauer and X-ray diffraction investigation of Li-Ti ferrites,”
*Journal of Physics: Condensed Matter*, vol. 6, no. 29, pp. 5717–5724, 1994. View at Publisher · View at Google Scholar - M. A. Amer, “${}^{57}\text{F}\text{e}$ Mössbauer, infrared and X-ray studies of the system ${\text{Zn}}_{1-x}{\text{Cu}}_{x}{\text{Cr}}_{0.8}{\text{Fe}}_{1.2}{\text{O}}_{4}$,”
*Physica Status Solidi A*, vol. 181, no. 2, pp. 539–550, 2000. View at Publisher · View at Google Scholar - M. A. Amer, “Mössbauer, infrared, and X-ray studies of Ti-doped ${\text{CoCr}}_{1.2}{\text{Fe}}_{0.8}{\text{O}}_{4}$ ferrites,”
*Physica Status Solidi B*, vol. 237, no. 2, pp. 459–471, 2003. View at Publisher · View at Google Scholar - M. El-Saadawy and M. M. Barakat, “Effect of jump length of electrons on the physical properties of Mn-doped ${\text{Co}}_{0.6}{\text{Zn}}_{0.4}{\text{Fe}}_{2}{\text{O}}_{4}$ ferrite,”
*Journal of Magnetism and Magnetic Materials*, vol. 213, no. 3, pp. 309–311, 2000. View at Publisher · View at Google Scholar - K. H. Rao, S. B. Raju, K. Aggarwal, and R. G. Mendiratta, “Effect of Cr impurity on the dc resistivity of Mn-Zn ferrites,”
*Journal of Applied Physics*, vol. 52, no. 3, pp. 1376–1379, 1981. View at Publisher · View at Google Scholar - A. Globus, H. Pascard, and V. Cagan, “Distance between magnetic ions and fundamental properties in ferrites,”
*Journal de Physique*, vol. 38, no. C1, pp. 163–168, 1977. View at Publisher · View at Google Scholar - R. A. Brand, Laboratorium für Angewandte Physik, Universität Duisburg, Lotharstr 1, D-4100 Duisburg 1, Germany.