- Email: [email protected]
Compensation behaviors of mixed Ising model with transverse crystal field in an external magnetic field
Journal of Magnetism and Magnetic Materials 345 (2013) 261–265
Contents lists available at SciVerse ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Compensation behaviors of mixed Ising model with transverse crystal field in an external magnetic field C.Q. Xu a,b, S.L. Yan a,b,c,n a
Department of Physics, Soochow University, Suzhou 215006, PR China Jiangsu Key Loboratory of Film Materials, Soochow University, Suzhou 215006, PR China c CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, PR China b
art ic l e i nf o
a b s t r a c t
Article history: Received 5 February 2013 Received in revised form 22 May 2013 Available online 29 June 2013
Within the effective field theory (EFT), the compensation behaviors of mixed spin-1/2 and spin-1 Ising model with transverse crystal field in an external magnetic field are studied on a simple cubic lattice. The model exhibits two types of temperature and magnetic compensation point (T k and hk ) in MT and Mh space, respectively. At zero or nonzero magnetic field, the single or double T k is induced by positive transverse crystal field (þDx =J), but not or only the single T k in negative crystal field (Dx =J). The single hk is discovered and adjusted by 7 Dx =J. A complex oscillation of magnetization curves at low magnetic field and large 7 Dx =J can be observed. The ranges of 7 Dx =J which can induce the hk take on an opposite tendency and the oscillation phenomena become weak when the temperature is up. In particular, the influence of transverse crystal field on the temperature and magnetic compensation point is primarily proposed in this work. & 2013 Elsevier B.V. All rights reserved.
Keywords: Mixed Ising model Transverse crystal field External magnetic field Compensation behavior
1. Introduction In past decades, people have observed macro-uniaxial anisotropy in-plane for RE–TM films and Co–M (M¼Zr, Nb, Ti) films in experiments [1,2]. This is powerful evidence that there is transverse crystal field. The transverse crystal field represents the direction of the hard magnetization x-axis, while traditional longitudinal crystal field is the direction of the easy magnetization z-axis. The different uniaxial crystal fields play a distinct role in determination of phase transition features. Many theoretical researches have dealt with the effect of transverse crystal field. Eddeqaqi et al. studied the ferromagnetic spin-1 Ising model with a transverse crystal field [3]. Xu et al. studied the critical behaviors of the spin-1 Ising model with a transverse random crystal field [4]. Ling et al. studied the critical properties of a random transverse crystal field Ising model with bond dilution [5]. Ferrimagnetic property is another remarkable problem. In ferrimagnet, there are single or multi-compensation points at which the resultant magnetization disappears below the Curie temperature [6–8]. Recently, Deviren et al. showed dynamic phase transitions and compensation temperatures in a mixed spin-3/2 and spin-5/2 Ising system [9]. Kis-Cam et al. proposed compensation temperature of 3d mixed ferro-ferrimagnetic ternary alloy [10]. Kaneyoshi discussed
n Corresponding author at: Department of Physics, Soochow University, Suzhou 215006, PR China. Tel.: +86 512 67166225; fax: +86 512 67161060. E-mail address: [email protected] (S.L. Yan).
0304-8853/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmmm.2013.06.013
compensation temperature of cylindrical Ising nanowire and transverse Ising nanoparticles with negative core–shell coupling [11,12]. At the same time, one of the present authors gave a novel magnetic compensation behavior of the Blume–Capel model with a unique magnetic field [13]. Some further works were given [14–16]. On the other hand, Jiang et al. studied a ferromagnetic or the ferrimagnetic bilayer Ising system with a transverse crystal field [17]. Htoutou et al. investigated a mixed spin bilayer system with a transverse crystal field [18]. Ekiz et al. considered mixed spin-1/2 and spin-1 Ising models with uniaxial and biaxial single-ion anisotropy on the Bethe lattice [19]. In this work, The model exhibits two types of temperature and magnetic compensation point (T k and hk ) in MT and Mh space, respectively. At zero or nonzero magnetic field, the single or double T k is induced by positive transverse crystal field (þDx =J), but not or only the single T k in negative crystal field (Dx =J). The single hk is discovered and adjusted by 7Dx =J. A complex oscillation of magnetization curves at low magnetic field and large 7 Dx =J can be observed. The ranges of 7Dx =J which can induce the hk take on an opposite tendency and the oscillation phenomena become weak when the temperature is up. As far as we known, above-mentioned results have not been reported. The theoretical procedure is based on the effective field theory (EFT). The outline of the work is as follows. Section 2 covers the definition and technical essential of mixed spin-1/2 and spin-1 Ising model with transverse crystal field in an external magnetic field. The detailed numerical results and discussions are provided in Section 3. Finally, a brief summary is given in Section 4.
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C.Q. Xu, S.L. Yan / Journal of Magnetism and Magnetic Materials 345 (2013) 261–265
2. Theory The Hamiltonian of the mixed spin-1/2 and spin-1 transverse crystal field Ising model with an external magnetic field can be defined as follows: H ¼ J ∑ szi Szj Dx ∑ ðSxj Þ2 h∑ szi h∑ Szj : 〈i;j〉
j
i
ð1Þ
j
The lattice is composed of two interpenetrating sublattices A and B, where szi is spin-1/2 operator at site i in sublattice A, Szj and Sxj are spin-1 operators at site j in sublattice B. J is the exchange interaction, assumed to be J o 0. The first summation only runs over all nearest-neighbor pairs of spins. The second and the forth summation sums over all sites of sublattice B, the third summation sums over all sites of sublattice A. Dx represents the transverse crystal field. h is an external magnetic field parallel to the (z) axis. On the basis of the EFT, the average magnetizations are given by z
s ¼ 〈szi 〉 ¼ 〈 ∏ ½ðSzj Þ2 coshðJ∇Þ þ Szj sinhðJ∇Þ þ 1ðSzj Þ2 〉FðxÞjx ¼ 0 ; j¼1
* m ¼ 〈Szj 〉 ¼
z
∏
i¼1
+ 1 1 J∇ þ 2szi sinh J∇ GðxÞjx ¼ 0 : cosh 2 2
The quadrupolar moment q is given by * + z 1 1 J∇ þ 2szi sinh J∇ LðxÞjx ¼ 0 ; q ¼ 〈ðSzj Þ2 〉 ¼ ∏ cosh 2 2 i¼1
ð2Þ
ð3Þ
ð4Þ
where ∇ ¼ ∂=∂x, z is the lattice coordination number. The 〈⋯〉 denotes the canonical thermal average. The function FðxÞ, GðxÞ and LðxÞ can be described as 1 1 ð5Þ FðxÞ ¼ tanh βðx þ hÞ ; 2 2 3 2βDx GðxÞ ¼ ∑ exp 2βCE1 ðnÞ þ 3 n¼1 " ## 2ðx þ hÞ ðx þ hÞ3 Dx E1 ðnÞ þ E2 ðnÞ Z 1 ; 3C 27BC 3 2βDx LðxÞ ¼ ∑ exp 2βCE1 ðnÞ þ 3 n¼1 " ## Dx 4ðx þ hÞ4 þ ðx þ hÞ2 D2x 2 E1 ðnÞ þ E2 ðnÞ þ Z 1 ; 3 9C 54BC where β ¼ 1=kB T, the partition function Z is 3 ðn1Þ⋅2π þ θ 2βDx þ ; Z ¼ ∑ exp 2βC cos 3 3 n¼1
ð6Þ
ð7Þ
ð8Þ
and E1 ðnÞ ¼ cos
ðn1Þ2π þ θ ; 3
ðn1Þ2π þ θ ; 3 A θ ¼ arccos 3 ; C E2 ðnÞ ¼ sin
ð9Þ ð10Þ ð11Þ
1 A ¼ 27 D3x 16ðx þ hÞ2 Dx ;
ð12Þ
h i1=2 ; B ¼ 19 3ðx þ hÞ6 þ 34ðx þ hÞ4 D2x
ð13Þ
h i1=2 : C ¼ 13 3ðx þ hÞ2 þ D2x
ð14Þ
If we try to treat the spin–spin correlation in Eqs. (2)–(4) exactly, the problem becomes mathematically intractable. A cutting approximation is usually adopted: 〈szi szj ⋯szl 〉≈〈szi 〉〈szj 〉⋯〈szl 〉;
ð15Þ
〈Szj ðSzk Þ2 ⋯Szm 〉≈〈Szj 〉〈ðSzk Þ2 〉⋯〈Szm 〉
ð16Þ
for i≠j≠⋯≠m. Then the Eqs. (2)–(4) can be written as s ¼ ½q coshðJ∇Þ þ m sinhðJ∇Þ þ 1qz FðxÞjx ¼ 0 ;
ð17Þ
z m ¼ cosh 12J∇ þ 2s sinh 12J∇ GðxÞjx ¼ 0 ;
ð18Þ
z q ¼ cosh 12J∇ þ 2s sinh 12J∇ LðxÞjx ¼ 0 :
ð19Þ
The magnetizations s and m in sublattices A and B are obtained from solutions of the set of Eqs. (17)–(19) numerically. The averaged magnetization per site is given by M 1 ¼ 2ðs þ mÞ; N
ð20Þ
where N is the number of magnetic atoms. As a result of J o0, the magnetizations s and m in sublattices A and B have opposing sign in a ferrimagnet. For this reason, the magnetizations s and m in sublattices A and B can compensate each other completely in the compensation points in MT or Mh space. Say it in another way, the total magnetization reaches zero when s≠0 and m≠0. Here, we select a simple cubic lattice (z¼ 6) for a three-dimensional example, because three-dimensional system is the most important dimension in experiment.
3. Results and discussions Fig. 1(a) and (b) shows the magnetization dependences of temperature for h=J ¼ 0 when (a) positive and (b) negative crystal field values are changed. In Fig. 1(a), the magnetization curve reduces rapidly with the influence of þDx =J. From strict numerical calculations, the temperature compensation point ðT k Þ does not exist in the range of 0≤Dx =J≤6:591. The magnetization curve takes on single T k for 6:591 o Dx =J o 11:238. Moreover, the T k shifts toward the direction of low temperature with increasing þDx =J. It is clear that the Dx =J acts only on sublattice B. When the þDx =J is large enough, the magnetic ordering m in sublattice B can be suppressed sufficiently. This leads to the magnetizations s and m can compensate each other. The larger þDx =J is the faster magnetic ordering m in sublattice B decreases. It is inevitable that the T k shifts toward the direction of low temperature. The þDx =J has an effective impact on the presence of the T k . Compared with Fig. 1(a), the magnetization curve becomes very simple, the magnetization region is distensible, and the T k does not exist for all Dx =J in Fig. 1(b). We infer that the Dx =J for suppression of the magnetic ordering is weak. The difference of the both is visible. Fig. 2(a) and (b) shows the magnetization diagrams with h=J ¼ 1:0 in MT space when (a) positive and (b) negative crystal field values are changed. Compared with Fig. 1(a), we do not find the compensation point for 0≤Dx =J≤9:87 in Fig. 2(a). Obviously, the external magnetic field strengthens the magnetic ordering. The appearance of the T k needs a larger þDx =J in sublattice B. The magnetization curve takes on double T k for 9:87 oDx =J≤14:64. With increasing þDx =J, the double T k show a shift in the opposite direction. The T k on the left side tends to the direction of low temperature. It is consistent with Fig. 1(a). Another T k tends to the direction of high temperature due to the existence of external magnetic field. The double T k vanish and there is single T k for Dx =J 414:64. Fig. 2(b) shows that the T k does not exist for 14:64≤Dx =J≤0. However, the single T k can occur in the range of
C.Q. Xu, S.L. Yan / Journal of Magnetism and Magnetic Materials 345 (2013) 261–265
263
Fig. 1. Magnetization dependences of temperature with h=J ¼ 0 are plotted. The numbers on the curves are the values of (a) positive and (b) negative transverse crystal field.
Fig. 2. Magnetization dependences of temperature with h=J ¼ 1:0 are plotted. The numbers on the curves are the values of (a) positive and (b) negative transverse crystal field.
Fig. 3. Magnetization dependences of external magnetic field with kB T=J ¼ 0:1 are plotted. The numbers on the curves are the values of (a) positive and (b) negative crystal field.
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C.Q. Xu, S.L. Yan / Journal of Magnetism and Magnetic Materials 345 (2013) 261–265
Fig. 4. Magnetization dependences of external magnetic field with kB T=J ¼ 0:3 are plotted. The numbers on the curves are the values of (a) positive and (b) negative transverse crystal field.
Dx =J o 14:64. The T k shifts toward the direction of high temperature with increasing Dx =J. In principle, the T k shall tend to infinite with increasing 7 Dx =J. Thus, the existence of external magnetic field affects apparently the temperature compensation behavior. Here, the fact that the T k can be induced by 7 Dx =J is primarily proposed in this work although the role of 7 Dx =J has some differences. Fig. 3(a) and (b) expresses the magnetization behaviors in Mh space for a given kB T=J ¼ 0:1, when (a) positive and (b)negative transverse crystal field values are changed. In Fig. 3(a), the magnetic compensation point ðhk Þ does not exist in the range of 0≤Dx =J≤10:32. The magnetization curve takes on single hk for Dx =J 4 10:32. With increasing þDx =J, the hk shifts toward the direction of high magnetic field. On the other hand, the magnetization curve takes place a complex oscillation when the þDx =J is large and the h is low. This may come from the fact that the magnetic ordering m in sublattice B is subjected to strong suppression owing to large þDx =J. The system can overcome the impact when the h becomes large. So the oscillation disappears gradually. Finally, the magnetization curve tends to saturation with increasing h. In Fig. 3(b), the effect of Dx =J on the oscillation and the hk is similar to Fig. 3(a). The single hk is found in the range of Dx =J o 11:43. It is exciting to obtain the hk under the 7 Dx =J conditions. This is since the hk can be only induced by a negative longitudinal crystal field in previous work [13]. The magnetization behaviors with kB T=J ¼ 0:3 in Mh space are plotted in Fig. 4(a, b), when (a) positive and (b) negative transverse crystal field values are changed. In Fig. 4(a), the hk does not exist in the range of 0≤Dx =J≤8:92. The single hk appears for Dx =J 4 8:92. Compared with Fig. 3(a), the smaller þDx =J value can produce the single hk . It is comprehensible that the result is the effect of thermal perturbation due to a higher temperature. In addition, we note that the rise of temperature makes for the disappearance of magnetization oscillation at small magnetic field. Fig. 4(b) displays the single hk for Dx =J o 11:89. The system needs a larger Dx =J to induce the hk at higher temperature. In the other hand, we see that the magnetization oscillation at low magnetic field does not disappear completely and retains a weak status for larger Dx =J. Thus, the Dx =J is propitious to existence of the oscillation. Hence, the role of 7 Dx =J about the hk and magnetization oscillation is different for a given temperature. Comparing Fig. 4 with Fig. 3, although the low or high temperature does not influence on magnetic compensation behavior, the change of 7 Dx =J is opposite
tendency. Here some physical mechanisms need still further explorations.
4. Summary In summary, we have studied the compensation behaviors of mixed spin-1/2 and spin-1 Ising model with transverse crystal field in an external magnetic field. In MT space, we find the single T k in þDx =J, but not in Dx =J at zero magnetic field. There exists the single or double T k for þDx =J, while the single T k is induced by Dx =J at h ¼ 1. The increase of 7 Dx =J and h can affect the T k . In Mh space, the single hk is discovered and adjusted by 7 Dx =J. Magnetization curves occur complex oscillation when the 7 Dx =J is large and the h is small at low temperature. The rise of temperature makes for the disappearance of magnetization oscillation, brings about opposite changes for the scope of 7 Dx =J which can induce the hk . The increase of 7 Dx =J can drive the hk shift toward the direction of high magnetic field. The effect of transverse crystal field on the temperature and magnetic compensation point is primarily proposed in this work.
Acknowledgments This work was supported by a project funded by 2010 the Priority Academic Program Development of Jiangsu Higher Education. References [1] G. Suran, K. Ounadjela, F. Machizaud, Physical Review Letters 57 (1986) 3109. [2] H.J. Witt, C.M.M. Witmer, F.W.A. Dirne, IEEE Transactions on Magnetics 23 (1987) 2123. [3] N.C. Eddeqaqi, M. Saber, A. El-Atri, M. Kerouad, Physica A 272 (1999) 144. [4] L. Xu, S.L. Yan, Acta Physica Sinica 56 (2007) 1691. [5] W. Ling, S.L. Yan, Central European Journal of Physics 9 (2011) 1084. [6] C. Mathoniere, C.J. Nuttall, S.G. Carling, P. Day, Inorganic Chemistry 35 (1996) 1201. [7] R. Carey, L. Dieu, D.M. Newman, Journal of Magnetism and Magnetic Materials 175 (1997) 99. [8] T. Kaneyoshi, Journal of Physical Society of Japan 70 (2001) 884. [9] B. Deviren, M. Keskin, Journal of Statistical Physics 140 (2010) 934. [10] E. Kis-Cam, E. Aydiner, Journal of Magnetism and Magnetic Materials 322 (2010) 1706.
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[11] [12] [13] [14] [15] [16]
T. Kaneyoshi, Physica A 390 (2011) 3697. T. Kaneyoshi, Solid State Communications 152 (2012) 883. S.L. Yan, L. Liu, Journal of Magnetism and Magnetic Materials 312 (2007) 285. Y.F. Zhang, S.L. Yan, Solid State Communications 146 (2008) 478. Y.F. Zhang, S.L. Yan, Physics Letters A 372 (2008) 2696. D.R. Li, S.L. Yan, Solid State Communications 150 (2010) 2186.
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[17] W. Jiang, G.Z. Wei, A. Du, Journal of Magnetism and Magnetic Materials 250 (2002) 49. [18] K. Htoutou, A. Ainane, M. Saber, Journal of Magnetism and Magnetic Materials 269 (2004) 245. [19] C. Ekiz, J. Strecka, M. Jascur, Central European Journal of Physics 7 (2009) 509.
Contents lists available at SciVerse ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Compensation behaviors of mixed Ising model with transverse crystal field in an external magnetic field C.Q. Xu a,b, S.L. Yan a,b,c,n a
Department of Physics, Soochow University, Suzhou 215006, PR China Jiangsu Key Loboratory of Film Materials, Soochow University, Suzhou 215006, PR China c CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, PR China b
art ic l e i nf o
a b s t r a c t
Article history: Received 5 February 2013 Received in revised form 22 May 2013 Available online 29 June 2013
Within the effective field theory (EFT), the compensation behaviors of mixed spin-1/2 and spin-1 Ising model with transverse crystal field in an external magnetic field are studied on a simple cubic lattice. The model exhibits two types of temperature and magnetic compensation point (T k and hk ) in MT and Mh space, respectively. At zero or nonzero magnetic field, the single or double T k is induced by positive transverse crystal field (þDx =J), but not or only the single T k in negative crystal field (Dx =J). The single hk is discovered and adjusted by 7 Dx =J. A complex oscillation of magnetization curves at low magnetic field and large 7 Dx =J can be observed. The ranges of 7 Dx =J which can induce the hk take on an opposite tendency and the oscillation phenomena become weak when the temperature is up. In particular, the influence of transverse crystal field on the temperature and magnetic compensation point is primarily proposed in this work. & 2013 Elsevier B.V. All rights reserved.
Keywords: Mixed Ising model Transverse crystal field External magnetic field Compensation behavior
1. Introduction In past decades, people have observed macro-uniaxial anisotropy in-plane for RE–TM films and Co–M (M¼Zr, Nb, Ti) films in experiments [1,2]. This is powerful evidence that there is transverse crystal field. The transverse crystal field represents the direction of the hard magnetization x-axis, while traditional longitudinal crystal field is the direction of the easy magnetization z-axis. The different uniaxial crystal fields play a distinct role in determination of phase transition features. Many theoretical researches have dealt with the effect of transverse crystal field. Eddeqaqi et al. studied the ferromagnetic spin-1 Ising model with a transverse crystal field [3]. Xu et al. studied the critical behaviors of the spin-1 Ising model with a transverse random crystal field [4]. Ling et al. studied the critical properties of a random transverse crystal field Ising model with bond dilution [5]. Ferrimagnetic property is another remarkable problem. In ferrimagnet, there are single or multi-compensation points at which the resultant magnetization disappears below the Curie temperature [6–8]. Recently, Deviren et al. showed dynamic phase transitions and compensation temperatures in a mixed spin-3/2 and spin-5/2 Ising system [9]. Kis-Cam et al. proposed compensation temperature of 3d mixed ferro-ferrimagnetic ternary alloy [10]. Kaneyoshi discussed
n Corresponding author at: Department of Physics, Soochow University, Suzhou 215006, PR China. Tel.: +86 512 67166225; fax: +86 512 67161060. E-mail address: [email protected] (S.L. Yan).
0304-8853/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmmm.2013.06.013
compensation temperature of cylindrical Ising nanowire and transverse Ising nanoparticles with negative core–shell coupling [11,12]. At the same time, one of the present authors gave a novel magnetic compensation behavior of the Blume–Capel model with a unique magnetic field [13]. Some further works were given [14–16]. On the other hand, Jiang et al. studied a ferromagnetic or the ferrimagnetic bilayer Ising system with a transverse crystal field [17]. Htoutou et al. investigated a mixed spin bilayer system with a transverse crystal field [18]. Ekiz et al. considered mixed spin-1/2 and spin-1 Ising models with uniaxial and biaxial single-ion anisotropy on the Bethe lattice [19]. In this work, The model exhibits two types of temperature and magnetic compensation point (T k and hk ) in MT and Mh space, respectively. At zero or nonzero magnetic field, the single or double T k is induced by positive transverse crystal field (þDx =J), but not or only the single T k in negative crystal field (Dx =J). The single hk is discovered and adjusted by 7Dx =J. A complex oscillation of magnetization curves at low magnetic field and large 7 Dx =J can be observed. The ranges of 7Dx =J which can induce the hk take on an opposite tendency and the oscillation phenomena become weak when the temperature is up. As far as we known, above-mentioned results have not been reported. The theoretical procedure is based on the effective field theory (EFT). The outline of the work is as follows. Section 2 covers the definition and technical essential of mixed spin-1/2 and spin-1 Ising model with transverse crystal field in an external magnetic field. The detailed numerical results and discussions are provided in Section 3. Finally, a brief summary is given in Section 4.
262
C.Q. Xu, S.L. Yan / Journal of Magnetism and Magnetic Materials 345 (2013) 261–265
2. Theory The Hamiltonian of the mixed spin-1/2 and spin-1 transverse crystal field Ising model with an external magnetic field can be defined as follows: H ¼ J ∑ szi Szj Dx ∑ ðSxj Þ2 h∑ szi h∑ Szj : 〈i;j〉
j
i
ð1Þ
j
The lattice is composed of two interpenetrating sublattices A and B, where szi is spin-1/2 operator at site i in sublattice A, Szj and Sxj are spin-1 operators at site j in sublattice B. J is the exchange interaction, assumed to be J o 0. The first summation only runs over all nearest-neighbor pairs of spins. The second and the forth summation sums over all sites of sublattice B, the third summation sums over all sites of sublattice A. Dx represents the transverse crystal field. h is an external magnetic field parallel to the (z) axis. On the basis of the EFT, the average magnetizations are given by z
s ¼ 〈szi 〉 ¼ 〈 ∏ ½ðSzj Þ2 coshðJ∇Þ þ Szj sinhðJ∇Þ þ 1ðSzj Þ2 〉FðxÞjx ¼ 0 ; j¼1
* m ¼ 〈Szj 〉 ¼
z
∏
i¼1
+ 1 1 J∇ þ 2szi sinh J∇ GðxÞjx ¼ 0 : cosh 2 2
The quadrupolar moment q is given by * + z 1 1 J∇ þ 2szi sinh J∇ LðxÞjx ¼ 0 ; q ¼ 〈ðSzj Þ2 〉 ¼ ∏ cosh 2 2 i¼1
ð2Þ
ð3Þ
ð4Þ
where ∇ ¼ ∂=∂x, z is the lattice coordination number. The 〈⋯〉 denotes the canonical thermal average. The function FðxÞ, GðxÞ and LðxÞ can be described as 1 1 ð5Þ FðxÞ ¼ tanh βðx þ hÞ ; 2 2 3 2βDx GðxÞ ¼ ∑ exp 2βCE1 ðnÞ þ 3 n¼1 " ## 2ðx þ hÞ ðx þ hÞ3 Dx E1 ðnÞ þ E2 ðnÞ Z 1 ; 3C 27BC 3 2βDx LðxÞ ¼ ∑ exp 2βCE1 ðnÞ þ 3 n¼1 " ## Dx 4ðx þ hÞ4 þ ðx þ hÞ2 D2x 2 E1 ðnÞ þ E2 ðnÞ þ Z 1 ; 3 9C 54BC where β ¼ 1=kB T, the partition function Z is 3 ðn1Þ⋅2π þ θ 2βDx þ ; Z ¼ ∑ exp 2βC cos 3 3 n¼1
ð6Þ
ð7Þ
ð8Þ
and E1 ðnÞ ¼ cos
ðn1Þ2π þ θ ; 3
ðn1Þ2π þ θ ; 3 A θ ¼ arccos 3 ; C E2 ðnÞ ¼ sin
ð9Þ ð10Þ ð11Þ
1 A ¼ 27 D3x 16ðx þ hÞ2 Dx ;
ð12Þ
h i1=2 ; B ¼ 19 3ðx þ hÞ6 þ 34ðx þ hÞ4 D2x
ð13Þ
h i1=2 : C ¼ 13 3ðx þ hÞ2 þ D2x
ð14Þ
If we try to treat the spin–spin correlation in Eqs. (2)–(4) exactly, the problem becomes mathematically intractable. A cutting approximation is usually adopted: 〈szi szj ⋯szl 〉≈〈szi 〉〈szj 〉⋯〈szl 〉;
ð15Þ
〈Szj ðSzk Þ2 ⋯Szm 〉≈〈Szj 〉〈ðSzk Þ2 〉⋯〈Szm 〉
ð16Þ
for i≠j≠⋯≠m. Then the Eqs. (2)–(4) can be written as s ¼ ½q coshðJ∇Þ þ m sinhðJ∇Þ þ 1qz FðxÞjx ¼ 0 ;
ð17Þ
z m ¼ cosh 12J∇ þ 2s sinh 12J∇ GðxÞjx ¼ 0 ;
ð18Þ
z q ¼ cosh 12J∇ þ 2s sinh 12J∇ LðxÞjx ¼ 0 :
ð19Þ
The magnetizations s and m in sublattices A and B are obtained from solutions of the set of Eqs. (17)–(19) numerically. The averaged magnetization per site is given by M 1 ¼ 2ðs þ mÞ; N
ð20Þ
where N is the number of magnetic atoms. As a result of J o0, the magnetizations s and m in sublattices A and B have opposing sign in a ferrimagnet. For this reason, the magnetizations s and m in sublattices A and B can compensate each other completely in the compensation points in MT or Mh space. Say it in another way, the total magnetization reaches zero when s≠0 and m≠0. Here, we select a simple cubic lattice (z¼ 6) for a three-dimensional example, because three-dimensional system is the most important dimension in experiment.
3. Results and discussions Fig. 1(a) and (b) shows the magnetization dependences of temperature for h=J ¼ 0 when (a) positive and (b) negative crystal field values are changed. In Fig. 1(a), the magnetization curve reduces rapidly with the influence of þDx =J. From strict numerical calculations, the temperature compensation point ðT k Þ does not exist in the range of 0≤Dx =J≤6:591. The magnetization curve takes on single T k for 6:591 o Dx =J o 11:238. Moreover, the T k shifts toward the direction of low temperature with increasing þDx =J. It is clear that the Dx =J acts only on sublattice B. When the þDx =J is large enough, the magnetic ordering m in sublattice B can be suppressed sufficiently. This leads to the magnetizations s and m can compensate each other. The larger þDx =J is the faster magnetic ordering m in sublattice B decreases. It is inevitable that the T k shifts toward the direction of low temperature. The þDx =J has an effective impact on the presence of the T k . Compared with Fig. 1(a), the magnetization curve becomes very simple, the magnetization region is distensible, and the T k does not exist for all Dx =J in Fig. 1(b). We infer that the Dx =J for suppression of the magnetic ordering is weak. The difference of the both is visible. Fig. 2(a) and (b) shows the magnetization diagrams with h=J ¼ 1:0 in MT space when (a) positive and (b) negative crystal field values are changed. Compared with Fig. 1(a), we do not find the compensation point for 0≤Dx =J≤9:87 in Fig. 2(a). Obviously, the external magnetic field strengthens the magnetic ordering. The appearance of the T k needs a larger þDx =J in sublattice B. The magnetization curve takes on double T k for 9:87 oDx =J≤14:64. With increasing þDx =J, the double T k show a shift in the opposite direction. The T k on the left side tends to the direction of low temperature. It is consistent with Fig. 1(a). Another T k tends to the direction of high temperature due to the existence of external magnetic field. The double T k vanish and there is single T k for Dx =J 414:64. Fig. 2(b) shows that the T k does not exist for 14:64≤Dx =J≤0. However, the single T k can occur in the range of
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Fig. 1. Magnetization dependences of temperature with h=J ¼ 0 are plotted. The numbers on the curves are the values of (a) positive and (b) negative transverse crystal field.
Fig. 2. Magnetization dependences of temperature with h=J ¼ 1:0 are plotted. The numbers on the curves are the values of (a) positive and (b) negative transverse crystal field.
Fig. 3. Magnetization dependences of external magnetic field with kB T=J ¼ 0:1 are plotted. The numbers on the curves are the values of (a) positive and (b) negative crystal field.
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Fig. 4. Magnetization dependences of external magnetic field with kB T=J ¼ 0:3 are plotted. The numbers on the curves are the values of (a) positive and (b) negative transverse crystal field.
Dx =J o 14:64. The T k shifts toward the direction of high temperature with increasing Dx =J. In principle, the T k shall tend to infinite with increasing 7 Dx =J. Thus, the existence of external magnetic field affects apparently the temperature compensation behavior. Here, the fact that the T k can be induced by 7 Dx =J is primarily proposed in this work although the role of 7 Dx =J has some differences. Fig. 3(a) and (b) expresses the magnetization behaviors in Mh space for a given kB T=J ¼ 0:1, when (a) positive and (b)negative transverse crystal field values are changed. In Fig. 3(a), the magnetic compensation point ðhk Þ does not exist in the range of 0≤Dx =J≤10:32. The magnetization curve takes on single hk for Dx =J 4 10:32. With increasing þDx =J, the hk shifts toward the direction of high magnetic field. On the other hand, the magnetization curve takes place a complex oscillation when the þDx =J is large and the h is low. This may come from the fact that the magnetic ordering m in sublattice B is subjected to strong suppression owing to large þDx =J. The system can overcome the impact when the h becomes large. So the oscillation disappears gradually. Finally, the magnetization curve tends to saturation with increasing h. In Fig. 3(b), the effect of Dx =J on the oscillation and the hk is similar to Fig. 3(a). The single hk is found in the range of Dx =J o 11:43. It is exciting to obtain the hk under the 7 Dx =J conditions. This is since the hk can be only induced by a negative longitudinal crystal field in previous work [13]. The magnetization behaviors with kB T=J ¼ 0:3 in Mh space are plotted in Fig. 4(a, b), when (a) positive and (b) negative transverse crystal field values are changed. In Fig. 4(a), the hk does not exist in the range of 0≤Dx =J≤8:92. The single hk appears for Dx =J 4 8:92. Compared with Fig. 3(a), the smaller þDx =J value can produce the single hk . It is comprehensible that the result is the effect of thermal perturbation due to a higher temperature. In addition, we note that the rise of temperature makes for the disappearance of magnetization oscillation at small magnetic field. Fig. 4(b) displays the single hk for Dx =J o 11:89. The system needs a larger Dx =J to induce the hk at higher temperature. In the other hand, we see that the magnetization oscillation at low magnetic field does not disappear completely and retains a weak status for larger Dx =J. Thus, the Dx =J is propitious to existence of the oscillation. Hence, the role of 7 Dx =J about the hk and magnetization oscillation is different for a given temperature. Comparing Fig. 4 with Fig. 3, although the low or high temperature does not influence on magnetic compensation behavior, the change of 7 Dx =J is opposite
tendency. Here some physical mechanisms need still further explorations.
4. Summary In summary, we have studied the compensation behaviors of mixed spin-1/2 and spin-1 Ising model with transverse crystal field in an external magnetic field. In MT space, we find the single T k in þDx =J, but not in Dx =J at zero magnetic field. There exists the single or double T k for þDx =J, while the single T k is induced by Dx =J at h ¼ 1. The increase of 7 Dx =J and h can affect the T k . In Mh space, the single hk is discovered and adjusted by 7 Dx =J. Magnetization curves occur complex oscillation when the 7 Dx =J is large and the h is small at low temperature. The rise of temperature makes for the disappearance of magnetization oscillation, brings about opposite changes for the scope of 7 Dx =J which can induce the hk . The increase of 7 Dx =J can drive the hk shift toward the direction of high magnetic field. The effect of transverse crystal field on the temperature and magnetic compensation point is primarily proposed in this work.
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