Dependence of magnetization on crystal fields and exchange interactions in magnetite

Journal of Magnetism and Magnetic Materials 394 (2015) 217–222

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Dependence of magnetization on crystal fields and exchange interactions in magnetite Mohamed Ouaissa a,n, Abdelilah Benyoussef b, Gavin S. Abo c, Samia Ouaissa a, Mustapha Hafid a, Mohammed Belaiche d a

Laboratoire de Génie Physique et Environnement, Faculté des Sciences, Université Ibn Tofail, Campus Universitaire n BP 133, Kénitra 14000, Morocco Laboratory of Magnetism and Physics of High Energy, Faculty of Science, Mohammed V-Agdal University, Rabat, Morocco c Department of Electrical and Computer Engineering and MINT Center, The University of Alabama, Tuscaloosa, AL 35487, USA d Laboratoire de Magnétisme, Matériaux Magnétiques, Microonde et Céramique, Ecole Normale Supérieure, Université Mohammed V-Agdal, B.P. 9235, Océan, Rabat, Morocco b

art ic l e i nf o

a b s t r a c t

Article history: Received 18 April 2014 Received in revised form 20 November 2014 Accepted 15 June 2015 Available online 19 June 2015

In this work, we study the magnetization of magnetite (Fe3O4) with different exchange interactions and crystal fields using variational method based on the Bogoliubov inequality for the Gibbs free energy within the mean field theory. The magnetic behavior was investigated in the absence and presence of crystal fields. The investigations also revealed that the transition temperature depends on the crystal fields of the octahedral and tetrahedral sites. Magnetite exhibits ferrimagnetic phase with second order transition to paramagnetic phase at 850 K. This result is confirmed using the mean field theory within the Heisenberg model. Important factors that can affect the magnetic behavior of the system are exchange interactions and crystal field. Indeed, a new magnetic behavior was observed depending on these parameters. A first order phase transition from ferrimagnetic to ferromagnetic was found at low temperature, and a second order transition from ferromagnetic to paramagnetic was observed at high temperature. & 2015 Elsevier B.V. All rights reserved.

Keywords: Mean field theory Exchange interactions Magnetization Crystal fields

1. Introduction Magnetite (Fe3O4) is of interest for biological hyperthermia applications [1]. It is ferrimagnetic with an inverse spinel structure, where trivalent Fe3 þ ions are equally distributed between tetrahedral (A) and octahedral (B) sites and all divalent Fe2 þ ions occupy the B sites [2]. It has a Verwey transition (Tv ) at 120 K [3]. The magnetization dependence on exchange interactions, crystal fields, and temperature are important factors to understand the behaviors of the magnetization. The moments of the spin ordering of a ferrimagnetic type on the A site are oriented antiparallel to those on the B sites on account of the strong negative exchange interaction between A and B spins, the magnetic exchange integrals Jab, Jaa and Jbb between A and B sites, and the crystal fields of the A and B sites. Various studies on the dependence of the magnetization versus temperature of magnetite have been done experimentally [4–6], and also, theoretically using Monte Carlo simulation [7,8]. To our

knowledge, the critical phenomena of bulk magnetite and the influence of different exchange integrals for magnetite by the mean field theory have not been reported. This motivated the study of magnetite by the mean field approximation to elucidate the magnetic behavior of this system. The magnetic properties of lattice systems have been extensively studied using different techniques including the mean field approximation (MFA) based on Bogoliubov inequality for the Gibbs free energy, which has been used for other lattices [9–12]. In the present work, we investigate the effect of the crystal fields in each sublattice on the magnetization of magnetite by the mean field approximation. These investigations have revealed very rich information about the behavior of our system in the absence and the presence of crystal fields. The variation of the order parameter (M) versus temperature (T) was studied, and it undergoes a deformation below and at the critical temperature.

2. Method of calculation n

Corresponding author. E-mail address: [email protected] (M. Ouaissa).

http://dx.doi.org/10.1016/j.jmmm.2015.06.034 0304-8853/& 2015 Elsevier B.V. All rights reserved.

The Hamiltonian of the system is given by

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Table 1 Ion-to-ion exchange interactions J in K for Fe3O4 (calculated values from [13]). 3 +−Fe3 +

3 +−Fe2 +

3 +−Fe3 +

3 +−Fe2 +

Fe J AA

Fe J AB

Fe J AB

Fe JBB

–14.0

–28.0

–20.2

–11.0

3 +− Fe 3 +

Fe H = − J AA Fe − JBB

3 +− Fe 3 +

Fe − JBB

3 +− Fe 2 +

Fe − JBB

3+

SiFe SiFe ‵

3+

‵

∑ jj

S jFe3+S jFe‵

3+

‵

∑ jj

S jFe3+S jFe‵

2+

‵

∑ii

2 +− Fe 2 +

S jFe‵ 2+S jFe‵

∑ jj s ‵

3 +− Fe 3 +

∑ij

SiFe3+S jFe

3+

3 +− Fe 2 +

∑ij

SiFe3+S jFe

2+

Fe − J AB Fe − J AB

− Δ Fe 3 + (tet)

∑i (S AFe3+,z )2

− Δ Fe 3 + (oct)

∑ j (SBFe3+,z )

3 +−Fe3 +

Fe JBB

–9.0

3 +− Fe 3 +

Fe hII = − J AB

3 +− Fe 2 +

2 +−Fe2 +

(

2+

=±

( )

Fe 3 +− Fe 3 + , JBB

5

2

( ) 3

, ±

(1)

=±

2

Fe 3 +− Fe2 + , JBB

( ) 1

, ±

( ), ± ( ), ± ( ), 5

3

2

2

Fe2 +− Fe2 + , JBB

3+

2+ SBFe

and

1

2

and SBFe

2+

having values 3 +− Fe 3 +

Fe = 0, ± 1, ± 2. J AA

Fe 3 +− Fe 3 + and J AB

(2)

F0(H0) is the free energy of a trial Hamiltonian H0, which depends on variational parameters, and <⋯>0 denotes a thermal average over the ensemble defined by H0. Now, depending on the choice of the trial Hamiltonian, one can construct approximate methods of different accuracy. However, owing to the complexity of the problem, one of the simplest possible choices of H0, the effective Hamiltonian of the system, was considered in this work. N/3

+ hII ∑ SBFe j

i N/3

− ΔFe 3 +(tet)

∑ i

N/3

− ΔFe 2 +

∑ j

hI = −

3+

(

SAFe

(

2+ SBFe , z

Fe 3 +− Fe 3 + JAA Z1mI

−

,z

N/3

+ hIII ∑ SBFe

⎛ ΔFe 3 + (oct ) ⎞ ⎛ h II ⎞ ⎤N /3 ⎟ cosh ⎜ + 2 exp ⎜ ⎟⎥ ⎝ 2k B T ⎠ ⎦ ⎝ 4k B T ⎠ ⎛ ΔFe 2 + ⎞ ⎛ h III ⎞ × [1+2 exp ⎜ ⎟ cosh ⎜ ⎟ ⎝ kB T ⎠ ⎝ 2k B T ⎠ ⎛ 2h III ⎞ ⎤N /3 ⎞⎟ ⎛ 4ΔFe 2 + ⎞ + 2 exp ⎜ ⎟ cosh ⎜ ⎟⎥ ⎟ ⎝ kB T ⎠ ⎝ kB T ⎠ ⎦ ⎠ N Fe 3 +− Fe 3 + N Fe 3 +− Fe 2 + 2N Z1m I2 − JBB Z 3 m II2 − JBB Z 3 m II m III 3 3 3 Fe 2 +− Fe 2 + N 2 Fe 3 +− Fe 3 + 2N − JBB Z 3 m III − J AB Z2 m I m II 3 3 N N N Fe 3 +− Fe 2 + 2N − J AB Z 3 m I m III − h I m I − h II m II − h III m III 3 3 3 3 3 +− Fe 3 +

Fe − J AA

(

− ΔFe 3 +(oct)

∑ j

Fe 3 +, z B

2

mI = −

)

Fe 3 +− Fe 3 + JAB Z2mII

(3)

−

Fe 3 +− Fe 2 + JAB Z2mIII

(4)

( )+ 5hI 2k BT

⎛ Δ Fe3 + (tet) ⎞ ⎟ sinh exp ⎜ 4k BT ⎝ ⎠ ⎛ 25Δ Fe3 + (tet) ⎞ ⎟⎟ cosh exp ⎜⎜ 4k BT ⎝ ⎠

+

2

)

⎛ 25Δ Fe3 + (tet) ⎞ ⎟⎟ sinh exp ⎜⎜ 4k BT ⎝ ⎠

5 2

(S )

(7)

where Z0 = Tr e−βH0 with β = 1/(k B T ) and k B is the Boltzmann constant. The sublattice magnetizations per site m I , m II and m III are defined as follows:

2+

N/3

2

)

(6)

⎛ 9ΔFe 3 + (oct ) ⎞ ⎛ 3h II ⎞ ⎟⎟ cosh ⎜ +2 exp ⎜⎜ ⎟ k T 4 ⎝ 2k B T ⎠ B ⎝ ⎠

j 3+

Z 3 m III

⎡ ⎛ 25ΔFe 3 + (oct ) ⎞ ⎛ 5h II ⎞ ⎟⎟ cosh ⎜ × ⎢2 exp ⎜⎜ ⎟ ⎢⎣ ⎝ 2k B T ⎠ ⎠ ⎝ 4k B T

,

Fe 3 +− Fe2 + J AB

F ≤ F0 = − k B T ln (Z 0 ) + < H − H0 >0

3+

Fe Z 3 m II − JBB

⎛ ΔFe 3 + (tet ) ⎞ ⎛ h I ⎞ ⎤N /3 ⎟ cosh ⎜ +2 exp ⎜ ⎟⎥ ⎝ 2k B T ⎠ ⎥⎦ ⎝ 4k B T ⎠

while those of

2

each sublattice. The Hamiltonian used for the mean field calculations have been done in the absence of an applied magnetic field. In order to treat the model approximately, the variational method based on the Bogoliubov inequality for the Gibbs free energy was employed [14], where F(H) is the free energy of H given by Eq. (1). Therefore, the Gibbs–Bogoliubov inequality for the free energy per site of a system is

N/3

Fe Z2 m I − JBB

⎛ 9ΔFe 3 + (tet ) ⎞ ⎛ 3h I ⎞ ⎟⎟ cosh ⎜ + 2 exp ⎜⎜ ⎟ ⎝ 2k B T ⎠ ⎝ 4k B T ⎠

in Table 1 are the ion-to-ion exchange interactions between nearest neighbors, where the sums in the Hamiltonian are over the nearest neighbors of the site for the pure component Fe3O4. ΔFe3 + (tet), ΔFe3 + (oct) and ΔFe2 + are the crystal fields of each atom in

H0 = hI ∑ SAFe

2 +− Fe 2 +

(5)

⎛⎡ ⎛ 25ΔFe 3 + (tet ) ⎞ ⎛ 5h I ⎞ ⎟⎟ cosh ⎜ F0 = − k B T ln ⎜⎜ ⎢2 exp ⎜⎜ ⎟ ⎢ 4 k T ⎝ 2k B T ⎠ B ⎝ ⎠ ⎝⎣

sublattice B are occupied by spins SBFe 3+ S AFe

Z 3 m III

Z1 is the number of nearest neighbors of the tetrahedral site, Z2 is the number of the nearest neighbors between the tetrahedral site and octahedral site, and Z3 is the number of the nearest neighbors of the octahedral site. mI and mII are the saturation magnetization of Fe3+ in the tetrahedral and octahedral site, respectively, and mIII is the saturation magnetization of Fe2+ in the octahedral site. hI , hII and hIII are the three effective fields. Already at this stage, it is clear that the use of the trial Hamiltonian naturally leads to the mean field approximation for the present model. The approximated free energy is then obtained by minimizing the right hand side of Eq. (2) with respect to the variational parameters. Due to the simplicity of H0, it is easy to evaluate the expressions in Eq. (2) to obtain

2

3+

3 +− Fe 2 +

3 +− Fe 2 +

þ44.0

where the sites of sublattice A are occupied by spins SiFe , which take the values of S AFe

Fe Z 3 m II − JBB

Fe hIII = − J AB

)

2 +,z

3 +− Fe 3 +

Fe JBB

2

− Δ Fe 2 + ∑ j SBFe

Fe Z2 m I − JBB

1 2

3 2

⎛ 9Δ Fe3 + (tet) ⎞ ⎟⎟ sinh exp ⎜⎜ 4k BT ⎝ ⎠

( ) ⎛ ( ) + exp ⎜⎜⎝

⎛ Δ Fe3 + (tet) ⎞ ⎟ cosh + exp ⎜ 4k BT ⎝ ⎠

( ) 3hI 2k BT

hI 2k BT

5hI 2k BT

( ) hI 2k BT

9Δ

Fe 3 + (tet) ⎞ 4k BT

⎟⎟ cosh ⎠

( ) 3hI 2k BT

(8)

M. Ouaissa et al. / Journal of Magnetism and Magnetic Materials 394 (2015) 217–222

5 2

+ m II = −

⎛ 25Δ Fe3 + (oct) ⎞ ⎟⎟ sinh exp ⎜⎜ 4k BT ⎝ ⎠

( )+ 5hII 2k BT

⎛ Δ Fe3 + (oct) ⎞ ⎟ sinh exp ⎜ 4k BT ⎝ ⎠ ⎛ 25Δ Fe3 + (oct) ⎞ ⎟⎟ cosh exp ⎜⎜ 4k BT ⎝ ⎠

⎛ 9Δ Fe3 + (oct) ⎞ ⎟⎟ sinh exp ⎜⎜ 4k BT ⎝ ⎠

( ) ⎛ ( ) + exp ⎜⎜⎝

1 2

⎛ Δ Fe3 + (oct) ⎞ ⎟ cosh + exp ⎜ 4k BT ⎠ ⎝

m III

3 2

( ) 3hII 2k BT

mB mFe3+,oct

9Δ

Fe 3 + (oct) ⎞ 4k BT

⎟⎟ cosh ⎠

( ) 3hII 2k BT

( ) hII 2k BT

(9)

⎛ ΔFe2 + ⎞ ⎛ ΔFe2 + ⎞ hIII hIII 2 exp ⎜ k + 4 exp ⎜4 k ⎟ sinh k ⎟ sinh 2 k BT BT ⎠ BT ⎝ BT ⎠ ⎝ =− ⎛ ΔFe2 + ⎞ ⎛ ΔFe2 + ⎞ hIII hIII 1 + 2 exp ⎜ k + 2 exp ⎜4 k ⎟ cosh k ⎟ cosh 2 k BT BT ⎠ BT ⎝ BT ⎠ ⎝

( ) ( )

M

mA mB

mA

mI

mFe3+(oct)

hII 2k BT

5hII 2k BT

219

( ) ( )

mII

mFe2+(oct) mIII

mFe2+,oct

mB

mII mIII

M

mA

(10)

The following limits are used to find the initial magnetizations:

m01 = lim m I

T

(11)

T→0

m02 = lim m II

(12)

m03 = lim m III

(13)

T→0

T→0

The limit for the magnetization of Fe3+ for tetrahedral site is

⎧ − 5/2, if hI is positive lim m I = ⎨ T→0 ⎩ 5/2, if hI is negative ⎪

⎪

(14)

The limit for magnetization of Fe

3þ

for octahedral site is

⎧ − 5/2, if hII is positive lim m II = ⎨ T→0 ⎩ 5/2, if hII is negative ⎪

⎪

(15)

The limit for the magnetization of Fe2+ for octahedral site is

⎧ − 2, if hIII is positive limm III = ⎨ T→0 ⎩ 2, if hIII is negative ⎪

⎪

(16) 3þ

The magnetic moments of Fe in the tetrahedral site are down, and the magnetic moment of Fe2 þ and Fe3 þ in the octahedral site are up in the ground state [2], which corresponds to the spontaneous magnetization at T ¼0.

Fig. 1. Total and partial magnetizations when crystal field of Fe2+ in the octahedral site is 4,641 K with exchange interactions of Table 1.

magnetization of magnetite can be explained by the magnetic moments of the Fe3 þ atoms in the sublattice of the tetrahedral sites and the magnetic moments of Fe2 þ and Fe3 þ in the sublattice of the octahedral sites, which are aligned ferrimagnetically with increasing temperature until the critical temperature is obtained. The critical temperature is of second order transition with vanishing order parameter. At this temperature, there is a transition from the ferrimagnetic to paramagnetic state. There is only a transition of second order as shown in Fig. 1 accompanied with a broken symmetry [17] at the critical temperature. 3+

2+

3+

3. Results and discussion

M M M M

The crystal field and the exchange interaction of Fe2 þ in octahedral site were considered. The total and partial magnetizations with a crystal field for Fe2 þ in the octahedral site ΔFe2 + (oct ) ≠ 0 ,

(

)

with ΔFe3 + (tet) = 0 and ΔFe (oct) ¼ 0, are shown in Fig. 1. The crystal field of Fe2 þ is stronger than that of Fe3 þ for the calculation of Fig. 1. Theoretically [15] and experimentally [16], the crystal field stabilization energy (CFSE) is different than zero for Fe2 þ and equal zero for Fe3 þ in tetrahedral and octahedral sites that result in an inverse spinel. The presence of Fe2+ in the octahedral sites is more stable [16]. There are two spin states that are aligned parallel and antiparallel at T¼ 0. It can be seen that the spins are parallel, which means that they are ferromagnetically parallel in each sublattice and antiparallel between the two sublattices in the ferrimagnetic state. The partials magnetizations are positive for the atoms in the octahedral sites and negative for the tetrahedral sites. This is because the order parameter is different than zero for the whole evolution of the behavior of the system. The intrinsic

3+

− Fe Fe − Fe , JAB ) is varied, it is If the exchange interaction ( JFe AB remarked from Fig. 2 that the critical temperature is close to the experimental one at around 850 K [18]. There is still a transition of second order, and the order parameter varies accompanied with the variation around the Curie temperature. Magnetite in this case acts as a pure ferrimagnet, which depends on the superposition of the various combinations of the two opposing sublattice magnetizations. Experimental results have shown that the magnetization of magnetite versus temperature is of the ferrimagnetic phase [19]; this magnetic behavior is confirmed by the mean field theory (Heisenberg model). On the other side, a new magnetic behavior was predicted for

T 3 +−Fe2 + Fe3 +−Fe3 + ( JFe , JAB ) AB

Fig. 2. Effect of exchange interactions Fe2+ in the octahedral site is different than zero.

when crystal field of

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M. Ouaissa et al. / Journal of Magnetism and Magnetic Materials 394 (2015) 217–222

magnetite. A phase transition of first order [17] from ferrimagnetic state to ferromagnetic state and a second order phase transition [17] from ferromagnetic state to paramagnetic state were found theoretically, which has not been observed experimentally for the conditions that follow. This new magnetic behavior is named frustrated magnetic structure, which is caused by the existence of two or more types of magnetic cations with the exchange interactions among JAB , JAA and JBB of different magnitudes and signs [20]. Thus, all the A site moments are oriented antiparallel to all the B site moments. Also, it may be related to the effect of geometrical frustration on magnetic properties above Tv [21]. Magnetite reaches the maximum of conductivity, which was found experimentally up to 300 K [22] and theoretically at 383 K [23]. This may be explained by the raising temperature, where magnetite undergoes the hopping of electron between Fe2+ and Fe3+ after it reaches the maximum. This leads to the magnetic frustration of magnetite. It may be related to the effect of geometrical frustration of the pyrochlore lattice [24]. In the absence of crystal fields, Fig. 3, ΔFe3 + (tet) = 0 , ΔFe3 + (oct) ¼ 0 and ΔFe2 + (oct ) ¼ 0. The free Fe2+ion has five degenerate 3d orbitals [25]. As the temperature increases, the partial magnetization of the atom of Fe3 þ in tetrahedral site is negative, and the partial magnetizations of Fe3 þ and Fe2 þ are positive for the octahedral site, which means the system is in the ferrimagnetic state below the temperature of 400 K. As the crystal field of splitting is weak, a transition of the orientation of spins from ferrimagnetic state to ferromagnetic state will occur with increasing temperature [25]. There is a frustrated magnetic structure corresponding to a transition of first order, and all partial magnetizations become positive. It may be explained by the transition of the electrons between t2 and e causing the orientation of the spins; the system is then in the ferromagnetic state. At the Curie temperature with the value of 690 K, which corresponds to the critical point, there is a transition of second order from the ferromagnetic state to the paramagnetic state. The order parameter of the system vanishes in the paramagnetic state. All of these magnetic phenomena depend on the exchange interaction. Then, the symmetry is broken. In the absence of crystal fields, Fig. 4, the effect of the exchange interactions between the tetrahedral site and octahedral site were 3+

2+

3+

3+

− Fe Fe − Fe , JAB ) led to a studied. The variation of the values of ( JFe AB shift in the Curie temperature of the system. The order parameter varied from ferrimagnetic to ferromagnetic for the first transition of first order and from ferromagnetic to paramagnetic for the

M M M M

T 3 +−Fe2 +

Fig. 4. Effect of exchange interactions ( JFe AB crystal fields.

3 +−Fe3 +

Fe , JAB

) in the absence of

second order phase transition. At the second order transition, the symmetry is broken. This curve is called type P [26], because the magnetization of the octahedral sublattice is higher than the magnetization of the sublattice tetrahedral. In Fig. 5, ΔFe3 + (tet) ≠ 0, ΔFe3 + (oct) ¼ 0 and ΔFe2 + (oct ) ¼ 0, which corresponds to the presence of only the crystal field of Fe3 þ in tetrahedral site. Fig. 6, with ΔFe3 + (tet) = 0, ΔFe3 + (oct)≠ 0 and ΔFe2 + (oct ) ¼ 0, shows the presence of the crystal field of Fe3 þ in the octahedral site. The order parameter acts ferrimagnetic until the temperature of 560 K for Fig. 5 and 500 K for Fig. 6 is reached, then it can be remarked that there is a frustrated magnetic structure [20] at the transition of first order from the ferrimagnetic to ferromagnetic state. The partial magnetizations of the system are all positive in the ferromagnetic state. As the temperature increases further, the magnetization decreases until it vanishes at the temperature of 1,070 K for Fig. 5 and 980 K for Fig. 6. This corresponds to the transition of second order from the ferromagnetic state to the paramagnetic state, which is associated with a broken symmetry. When the crystal fields are not zero, Fig. 7, ΔFe3 + (tet) ≠ 0, ΔFe3 + (oct)≠ 0 and ΔFe2 + (oct )≠ 0. Experimental values of the crystal fields of the ions of NiFe2O4 and CoFe2O4 were found, but not for Fe3O4 [27]. There are no experimental values of the

T Fig. 3. Total and partial magnetizations in the absence of crystal fields for Fe3O4 with exchange interactions of Table 1.

Fig. 5. Total and partial magnetizations when crystal field of Fe3+ in the tetrahedral site is 2,320 K with exchange interactions of Table 1.

M. Ouaissa et al. / Journal of Magnetism and Magnetic Materials 394 (2015) 217–222

221

Table 2 Ion-to-ion exchange interactions J in K for Fe3O4 (observed values from [13]). 3 +−Fe3 +

3 +−Fe3 +

3 +−Fe2 +

3 +−Fe2 +

Fe J AA

Fe J AB

Fe J AB

Fe JBB

–21.0

–28.0

–23.8

–13.2

3 +−Fe3 +

2 +−Fe2 +

Fe JBB

–10.0

Fe JBB

þ 48.4

T Fig. 6. Total and partial magnetizations when crystal field of Fe3+ in the octahedral site is 4,641 K with exchange interactions of Table 1.

crystal fields of Fe3O4 to our knowledge. The system has a transition from the ferrimagnetic state to the ferromagnetic state at the temperature of 850 K, and the order parameter is discontinuous at this transition of the first order. At the temperature of 1,470 K, there is a transition of second order, which corresponds to a transition from the ferromagnetic state to the paramagnetic state. Here, the order parameter is continuous across the critical temperature. The Table 2 observed values of the exchange constant of ions of Fe3O4 were used in the absence of crystal field. It is remarked that it is a frustrated magnetic structure caused by negative exchange constant with a first order transition at the temperature of 390 K and a second order transition at the temperature of 780 K in Fig. 8, accompanied with a broken symmetry. In the presence of the crystal field of Fe2+ of octahedral sites, Fig. 9, it can be seen that there is a transition of first order at the temperature of 530 K. Below this transition the system is acting as a ferrimagnetic, but above of this transition, the orientation of the spins changed from up to down to become ferromagnetic. This was not observed in the first calculation with the exchange constant of Table 1 with the presence of the crystal field of Fe2+ in octahedral site. Still, it can be remarked that the effect of the crystal field of Fe2+ in octahedral site and the exchange constants

T Fig. 8. Total and partial magnetizations in the absence of crystal fields for Fe3O4 with exchange interactions of Table 2.

mB

M

mA mB

mA

mI

mFe3+(oct)

mII

mII

mFe2+(oct) mIII mB

mII mIII

mIII mA

M

T Fig. 9. Total and partial magnetizations when crystal field of Fe2+ in the octahedral site is 1,740.6 K with exchange interactions of Table 2.

T Fig. 7. Total and partial magnetizations when crystal fields are present for all sites with the values ΔFe3 + (tet) ¼6,962.71 K, ΔFe3 + (oct) ¼ 13,925.4 K, and ΔFe2 + (oct ) ¼ 13,925.4 K with exchange interactions of Table 1.

by the effect of temperature is the changing of the orientation of the spins to down. This magnetization curve is called type N [26], because the largest sublattice magnetization at 0 K, which is the magnetization of the octahedral sites, decreases more rapidly with increasing temperature than the magnetization of the tetrahedral sites, and a magnetization curve with a compensation temperature will occur. In the presence of crystal field in tetrahedral sites and octahedral sites, Fig. 10, there is a transition of first order at the temperature of 600 K. This is followed by a transition of second order at the temperature of 1,040 K accompanied with a broken symmetry, and the order parameter is different than zero below this critical temperature and vanishes above of this temperature. It is

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M. Ouaissa et al. / Journal of Magnetism and Magnetic Materials 394 (2015) 217–222

sites. This means that the exchange constant should be positive and the orientation of the spins of octahedral sites are opposites to the tetrahedral sites with spin up, corresponding to the positive constant exchange, and the Table 3 shows this. This behavior is indicative of magnetite acting as a pure ferrimagnetic, Fig. 11, with the critical temperature at 850 K followed by a broken symmetry.

4. Conclusions

Fig. 10. Total and partial magnetizations when crystal fields are present for all sites with the values ΔFe3 + (tet) ¼ 1,740.6 K, ΔFe3 + (oct) ¼ 1,740.6 K, and ΔFe2 + (oct ) ¼1,740.6 K with exchange interactions of Table 2.

This study shows clearly that for some values of crystal fields, exchange constant interaction of the nearest neighbors, and distribution of cations (divalent and trivalent) between tetrahedral and octahedral sites, magnetite can act as a pure ferrimagnet in the presence of the crystal field of Fe2 þ in the octahedral site. There is a second order transition accompanied with a broken symmetry at high temperature. A new magnetic behavior has been observed in the presence of the crystal fields of Fe3 þ in the tetrahedral site and octahedral site. Indeed, frustration exhibits a first order transition from ferrimagnetic to ferromagnetic phases, which is followed, at higher temperature, by a critical transition to paramagnetic phase. The behavior of the magnetization of the magnetite depends on the exchanges constant and the crystal fields.

Table 3 Ion-to-ion exchange interactions J in K for Fe3O4.

References

T

3 +−Fe3 +

3 +−Fe3 +

3 +−Fe2 +

3 +−Fe2 +

3 +−Fe3 +

2 +−Fe2 +

Fe J AA

Fe J AB

Fe J AB

Fe JBB

Fe JBB

Fe JBB

þ21.0

–28.0

–23.8

þ 13.2

þ 10.0

þ48.4

T Fig. 11. Total and partial magnetizations in the absence of crystal fields for Fe3O4 with exchange interactions of Table 3.

observed that there is still a frustration and that there is a shifting of the temperature, but we do not remark that there is a pure ferrimagnetic behavior compared to our previous results. To treat this frustration, reference [3] is considered, the orientation of the spins are down in the same direction (i.e., parallel) in tetrahedral

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