- Email: [email protected]
More accurate calculations of the magnetic entropy changes
ARTICLE IN PRESS Journal of Magnetism and Magnetic Materials 321 (2009) 3221–3224
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
More accurate calculations of the magnetic entropy changes Haishun Liu a,, Xiexing Miao a,b, Peng Wang a, Min Yang a, Wankui Bu a, Duanming Li a, Youwei Du b a b
School of Sciences, China University of Mining and Technology, Xuzhou 221116, China State Key Laboratory for Geomechanics & Deep underground Engineering, China University of Mining and Technology, Xuzhou 221008, China
a r t i c l e in fo
abstract
Article history: Received 10 November 2008 Received in revised form 8 May 2009 Available online 28 May 2009
A series of Ni44xCoxMn45Sn11 (x ¼ 0, 1, 2) alloys were prepared by means of arc melting, magnetization curves under different temperatures were measured using vibrating sample magnetometer. The bicubic interpolations method was employed to calculate the magnetization M(T, H), and the magnetic entropy changes were figured out using the thermodynamic relation of magnetic materials. Results show that the values calculated based on the bicubic interpolation method are more accurate compared with those from conventional method, and the curves obtained are relatively smooth. & 2009 Elsevier B.V. All rights reserved.
PACS: 75.30. Sg Keywords: Ferromagnetic shape memory alloy Magnetization curve Bicubic interpolation Magnetic entropy change
1. Introduction It is well known that the magnetic refrigeration (MR) technology is developed based upon the magnetocaloric effect (MCE), and great progress has been made in this field during past decades [1,2] because of its numerous advantages such as high efficiency and environmentally friendly feature compared with conventional gas compression refrigeration [3]; thus, it has attracted significant attention academically and technically. In the study of the MCE, the magnetic entropy changes DSM is an important parameter to the investigations of the MR ability. However, the magnetic entropy changes DSM obtained in most of the former studies are discrete [2,4–12], by so, more precise information was omitted as a result, more measurements under various temperatures should be done for detailed studies. The development in numerical analysis provides an alternative to the thorough understanding of DSM and detailed investigations of the MCE, which makes it possible to figure out the continuous variation in DSM and avoid the redundant measurement at the same time. However, little literature on continuous variation in DSM has been found. In this study, a kind of numerical analysis method was introduced to the calculation of the magnetic entropy changes, and Ni44xCoxMn45Sn11 (x ¼ 0, 1, 2) alloys were taken as examples; the function M(T, H) was derived using the bicubic
Corresponding author. Tel.: +86 13912006872.
E-mail address: [email protected] (H. Liu). 0304-8853/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2009.05.040
interpolation (BI) based on the magnetization curves measured, and then, the continuous variation in magnetic entropy changes was calculated by using Maxwell relation. The calculated values were compared with those by using conventional method.
2. Experiment Ni44xCoxMn45Sn11 (x ¼ 0, 1, 2) alloys were prepared by arc melting appropriate amounts of Ni, Co, Mn and Sn in the argon atmosphere. The alloys were remelted three times for homogeneity, then the alloys were polished and cut into little pieces and sealed in quartz tubes followed by annealing at 1173 K for 24 h, and finally, they were quenched in cool water. The magnetization curves were measured by a vibrating sample magnetometer (VSM 7300, Lakeshore) under different temperatures.
3. The improvement of the calculation of DSM 3.1. Conventional calculation method of DSM The Maxwell relation for magnetic materials is written as @SM @M ¼ (1) @H T @T H where SM is the magnetic entropy, T is the temperature, M is the magnetization, and H is the magnetic field intensity. It can be
ARTICLE IN PRESS 3222
H. Liu et al. / Journal of Magnetism and Magnetic Materials 321 (2009) 3221–3224
drawn from Eq. (1) that Z H @MðT; HÞ DSM ¼ dH @T 0 H
effects of the 16 sites around it: (2)
f ði þ u; j þ vÞ ¼ AnBnC
(4)
where DSM is the magnetic entropy change. Theoretically, DSM can be calculated using Eq. (2). In fact, the magnetization curves were measured at a series of discrete temperatures and only one curve can be obtained in a certain measurement, the magnetization curves are discrete; as a result, the precise expression of DSM cannot be obtained by this way directly. Generally, the integral in Eq. (2) was changed into the form of cumulative sum (CS), that is
DSM ¼
X M iþ1 ðT iþ1 ; HÞ M i ðT i ; HÞ i
T iþ1 T i
DH
(3)
So, DSM for Ni44xCoxMn45Sn11 (x ¼ 0, 1, 2) can be obtained by using Eq. (3). However, the DSM obtained are at different certain temperatures, the corresponding curve is just the simple connection of the magnetic entropy changes at different temperatures [6]. 3.2. The calculation of DSM based on bicubic interpolation Accurate values of the variation in the magnetic entropy changes should be obtained for the investigations of the magnetocaloric effect in detail; thus, the fitting of M(T, H) is necessary. Numerous studies of the MCE were conducted preliminary, among them, the magnetization M(T, H) and the continuous variation in DSM were ever obtained based on a method called the two-step least square fitting [13]; However, the algorithm of this method is very complicated, and the optimal orthogonal polynomial must be found, otherwise, a great error is inevitable. Fortunately, an important method of numerical analysis, surface fitting plays an important role in academic research and it is widely used, and bicubic interpolation [14] is one of the most popular methods with higher precision and better fitting results among them; moreover, the bicubic interpolation function can be directly called in popular numerical analysis programs such as MatLaboratory Therefore, the method based on bicubic interpolation is more convenient and effective than that based on the two-step least square fitting. The principle of the bicubic interpolation can be described as follows. The value f(i+u, j+v) at the site (i+u, j+v) can be obtained by using the following interpolation formula considering of the
Fig. 1. Magnetization curves for Ni43Co1Mn45Sn11.
Fig. 2. M(T, H) of Ni43Co1Mn45Sn11 based on conventional method.
Fig. 3. M(T, H) of Ni43Co1Mn45Sn11 based on the bicubic interpolation method.
Fig. 4. Comparison of magnetic entropy changes of Ni43Co1Mn45Sn11 calculated by these two methods.
ARTICLE IN PRESS H. Liu et al. / Journal of Magnetism and Magnetic Materials 321 (2009) 3221–3224
3223
Table 1 Comparison of selected DSM values from conventional method and bicubic interpolation method. Temperatures (K) Conventional method DSM (J/kgK) BI method Difference (%)
228 0.65 0.68 4.4
where A ¼ Sðu þ 1Þ Sðu 0Þ Sðu 1Þ Sðu 2Þ 2 f ði 1; j 1Þ f ði 1; j þ 0Þ f ði 1; j þ 1Þ 6 6 f ði þ 0; j 1Þ f ði þ 0; j þ 0Þ f ði þ 0; j þ 1Þ 6 B¼6 6 f ði þ 1; j 1Þ f ði þ 1; j þ 0Þ f ði þ 1; j þ 1Þ 4 f ði þ 2; j 1Þ
and 2
Sðv þ 1Þ
f ði þ 2; j þ 0Þ
f ði þ 2; j þ 1Þ
232 3.55 3.76 5.6
233 6.04 5.89 2.5
f ði 1; j þ 2Þ
3
7 f ði þ 0; j þ 2Þ 7 7 7 f ði þ 1; j þ 2Þ 7 5 f ði þ 2; j þ 2Þ
3
7 6 6 Sðv þ 0Þ 7 7 6 C¼6 7 6 Sðv 1Þ 7 5 4 Sðv 2Þ with 8 2 3 > < 1 2jwj þ jwj SðwÞ ¼ 4 8jwj þ 5jwj2 jwj3 > : 0
0 jwjo1 1 jwjo2 jwj 2
S(w) is the approximation to sin(pw)/w. The magnetic entropy changes were calculated by using bicubic interpolation. Taking Ni43Co1Mn45Sn11 as an example, firstly, the magnetization curves were measured under different temperatures and are shown in Fig. 1; secondly, M(T, H) was obtained by calling the bicubic interpolation function in Matlaboratory and is depicted in Fig. 3; for a comparison, M(T, H) based on conventional method is depicted in Fig. 2; finally, the magnetic entropy changes were calculated by using Eq. (2), as shown in Fig. 4.
4. Results and discussion The M(T, H) surfaces obtained from conventional method and bicubic interpolation method are shown in Figs. 2 and 3, respectively, and the comparison of the magnetic entropy changes calculated based on these two methods is depicted in Fig. 4; meanwhile, some selected magnetization entropy change values obtained from these two methods are listed in Table 1. It can be seen from Figs. 2 and 3 that the variation in M(T, H) obtained from the BI method is more continuous and relatively slow, meanwhile, its surface is smoother than that based on conventional method. It is shown in Fig. 4 that the curves of DSM calculated by these two methods are coincident with each other on the whole, and similar variations in DSM with the temperature are shown, their peaks appear both near 235 K. However, there are some differences between them, the curves based on the BI method are smoother than those based on conventional method, especially near the region DSM changes drastically; moreover, the maximum DSM based on BI method is a little bigger. Table 1 shows some selected DSM calculated from these two methods, the values are almost the same, however, there are also some differences between them, and the maximum difference between magnetic entropy changes is about 7.6%. The reason for the appearance of this difference is that the conventional method is based on the calculation of cumulative sum, while integral
234 9.99 9.93 0.6
235 14.35 14.88 3.6
236 12.71 13.52 6.0
237 6.48 6.17 5.0
238 2.48 2.31 7.4
239 0.85 0.79 7.6
operation is employed in the BI method, and the latter has higher calculation precision. Magnetization entropy changes for Ni44Mn45Sn11 and Ni42Co2Mn45Sn11 were also calculated in the same way and similar phenomena have been obtained. It is known from mathematics that for discrete data, the general trend of them could be predicted, the variations could be discovered, and the local error or fluctuation could be eliminated by using surface fitting. The procedure of fitting based on the BI method is that: divide the integral interval into little ones, then conduct integral operations in the little intervals by substituting simple functions for complex ones, and the integral value could be obtained by accumulating the values of little intervals. It is known that smaller the step is, more accurate the integral value is. The calculations of DSM are based on this thoughts in this study: divide the temperature intervals into little ones, conduct integral operation in little intervals, fit M(T, H), and then the function of DSM is obtained. Generally, the temperature intervals are bigger than 1 K in the measurement of M(H), so the variation in DSM is rapid and sharp near the martensite transition temperature, the curve of it is rough, moreover, the magnetization corresponding to the maximum DSM may not be measured, so the maximum DSM may be a little smaller. Meanwhile, the step is set as 0.05 K in our calculation, much less than that in the measurement, as a result, the curve of DSM is much smoother, and the DSM obtained should be more accurate.
5. Conclusions The magnetic entropy changes for Ni44xCoxMn45Sn11 (x ¼ 0, 1, 2) were calculated by employing bicubic interpolation based on magnetization curves. The procedure of calculating DSM based on the newly proposed method is not complicated because the calling of the analysis program is easy. The calculated DSM values are more accurate and their corresponding curves are smoother than that of the conventional method, although the latter is simple in calculation. This research provides one more choice in the calculation of DSM in magnetic refrigeration studies.
Acknowledgements This work was supported by the National Basic Research Program of China (2007CB209400) and the Research Fund of the State Key Laboratory of Coal Resources and Mine safety, CUMT (08KF06). References [1] A.M. Tishin, in: K.H. Buschow (Ed.), Handbook of Magnetic Materials, vol. 12, North-Holland, Amsterdam, 1999, pp. 395–524. [2] M.H. Phan, S.C. Yu, J. Magn. Magn. Mater. 308 (2007) 325–340. [3] V.K. Pecharsky, K.A. Gschneidner, A.O. Tsokol, Rep. Prog. Phys. 68 (2005) 1479. [4] Z.D. Han, D.H. Wang, C.L. Zhang, S.L. Tang, B.X. Gu, Y.W. Du, Appl. Phys. Lett. 89 (2006) 182507. [5] Z.D. Han, D.H. Wang, C.L. Zhang, B.X. Gu, Y.W. Du, Appl. Phys. Lett. 90 (2007) 042507. [6] H.S. Liu, C.L. Zhang, Z.D. Han, H.C. Xuan, D.H. Wang, Y.W. Du, J. Alloys Compd. 467 (2009) 27–30.
ARTICLE IN PRESS 3224
H. Liu et al. / Journal of Magnetism and Magnetic Materials 321 (2009) 3221–3224
[7] A. Dhahri, J. Dhahri, S. Zemni, M. Oumezzine, H. Vincent, J. Alloys Compd. 420 (2006) 15–19. [8] J. Dhahri, A. Dhahri, M. Oumezzine, E. Dhahri, J. Magn. Magn. Mater. 320 (2008) 2613–2617. [9] M. Balli, D. Fruchart, D. Gignoux, M. Rosca, S. Miraglia, J. Magn. Magn. Mater. 313 (2007) 43–46. [10] H. Fu, H.J. Yu, B.H. Teng, X.Y. Zhang, X.T. Zu, J. Alloys Compd. 474 (2009) 595–597.
[11] D.M. Raj Kumar, M. Manivel Raja, R. Gopalan, R. Balamuralikrishnan, A.K. Singh, V. Chandrasekaran, J. Alloys Compd. 461 (2008) 14–20. [12] I. Babita, M. Manivel Raja, R. Gopalan, V. Chandrasekaran, S. Ram, J. Alloys Compd. 432 (2007) 23–29. [13] Z.T. Li, P.F. Wu, Y.Q. Tao, D.K. Mao, Acta Phys. Sin. 48S (1999) 126–131. [14] S.M. Xiang, Visual C++ Digital Image and Graph Manipulation, Electronics industry press, Beijing, 2002, p. 280–281.
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
More accurate calculations of the magnetic entropy changes Haishun Liu a,, Xiexing Miao a,b, Peng Wang a, Min Yang a, Wankui Bu a, Duanming Li a, Youwei Du b a b
School of Sciences, China University of Mining and Technology, Xuzhou 221116, China State Key Laboratory for Geomechanics & Deep underground Engineering, China University of Mining and Technology, Xuzhou 221008, China
a r t i c l e in fo
abstract
Article history: Received 10 November 2008 Received in revised form 8 May 2009 Available online 28 May 2009
A series of Ni44xCoxMn45Sn11 (x ¼ 0, 1, 2) alloys were prepared by means of arc melting, magnetization curves under different temperatures were measured using vibrating sample magnetometer. The bicubic interpolations method was employed to calculate the magnetization M(T, H), and the magnetic entropy changes were figured out using the thermodynamic relation of magnetic materials. Results show that the values calculated based on the bicubic interpolation method are more accurate compared with those from conventional method, and the curves obtained are relatively smooth. & 2009 Elsevier B.V. All rights reserved.
PACS: 75.30. Sg Keywords: Ferromagnetic shape memory alloy Magnetization curve Bicubic interpolation Magnetic entropy change
1. Introduction It is well known that the magnetic refrigeration (MR) technology is developed based upon the magnetocaloric effect (MCE), and great progress has been made in this field during past decades [1,2] because of its numerous advantages such as high efficiency and environmentally friendly feature compared with conventional gas compression refrigeration [3]; thus, it has attracted significant attention academically and technically. In the study of the MCE, the magnetic entropy changes DSM is an important parameter to the investigations of the MR ability. However, the magnetic entropy changes DSM obtained in most of the former studies are discrete [2,4–12], by so, more precise information was omitted as a result, more measurements under various temperatures should be done for detailed studies. The development in numerical analysis provides an alternative to the thorough understanding of DSM and detailed investigations of the MCE, which makes it possible to figure out the continuous variation in DSM and avoid the redundant measurement at the same time. However, little literature on continuous variation in DSM has been found. In this study, a kind of numerical analysis method was introduced to the calculation of the magnetic entropy changes, and Ni44xCoxMn45Sn11 (x ¼ 0, 1, 2) alloys were taken as examples; the function M(T, H) was derived using the bicubic
Corresponding author. Tel.: +86 13912006872.
E-mail address: [email protected] (H. Liu). 0304-8853/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2009.05.040
interpolation (BI) based on the magnetization curves measured, and then, the continuous variation in magnetic entropy changes was calculated by using Maxwell relation. The calculated values were compared with those by using conventional method.
2. Experiment Ni44xCoxMn45Sn11 (x ¼ 0, 1, 2) alloys were prepared by arc melting appropriate amounts of Ni, Co, Mn and Sn in the argon atmosphere. The alloys were remelted three times for homogeneity, then the alloys were polished and cut into little pieces and sealed in quartz tubes followed by annealing at 1173 K for 24 h, and finally, they were quenched in cool water. The magnetization curves were measured by a vibrating sample magnetometer (VSM 7300, Lakeshore) under different temperatures.
3. The improvement of the calculation of DSM 3.1. Conventional calculation method of DSM The Maxwell relation for magnetic materials is written as @SM @M ¼ (1) @H T @T H where SM is the magnetic entropy, T is the temperature, M is the magnetization, and H is the magnetic field intensity. It can be
ARTICLE IN PRESS 3222
H. Liu et al. / Journal of Magnetism and Magnetic Materials 321 (2009) 3221–3224
drawn from Eq. (1) that Z H @MðT; HÞ DSM ¼ dH @T 0 H
effects of the 16 sites around it: (2)
f ði þ u; j þ vÞ ¼ AnBnC
(4)
where DSM is the magnetic entropy change. Theoretically, DSM can be calculated using Eq. (2). In fact, the magnetization curves were measured at a series of discrete temperatures and only one curve can be obtained in a certain measurement, the magnetization curves are discrete; as a result, the precise expression of DSM cannot be obtained by this way directly. Generally, the integral in Eq. (2) was changed into the form of cumulative sum (CS), that is
DSM ¼
X M iþ1 ðT iþ1 ; HÞ M i ðT i ; HÞ i
T iþ1 T i
DH
(3)
So, DSM for Ni44xCoxMn45Sn11 (x ¼ 0, 1, 2) can be obtained by using Eq. (3). However, the DSM obtained are at different certain temperatures, the corresponding curve is just the simple connection of the magnetic entropy changes at different temperatures [6]. 3.2. The calculation of DSM based on bicubic interpolation Accurate values of the variation in the magnetic entropy changes should be obtained for the investigations of the magnetocaloric effect in detail; thus, the fitting of M(T, H) is necessary. Numerous studies of the MCE were conducted preliminary, among them, the magnetization M(T, H) and the continuous variation in DSM were ever obtained based on a method called the two-step least square fitting [13]; However, the algorithm of this method is very complicated, and the optimal orthogonal polynomial must be found, otherwise, a great error is inevitable. Fortunately, an important method of numerical analysis, surface fitting plays an important role in academic research and it is widely used, and bicubic interpolation [14] is one of the most popular methods with higher precision and better fitting results among them; moreover, the bicubic interpolation function can be directly called in popular numerical analysis programs such as MatLaboratory Therefore, the method based on bicubic interpolation is more convenient and effective than that based on the two-step least square fitting. The principle of the bicubic interpolation can be described as follows. The value f(i+u, j+v) at the site (i+u, j+v) can be obtained by using the following interpolation formula considering of the
Fig. 1. Magnetization curves for Ni43Co1Mn45Sn11.
Fig. 2. M(T, H) of Ni43Co1Mn45Sn11 based on conventional method.
Fig. 3. M(T, H) of Ni43Co1Mn45Sn11 based on the bicubic interpolation method.
Fig. 4. Comparison of magnetic entropy changes of Ni43Co1Mn45Sn11 calculated by these two methods.
ARTICLE IN PRESS H. Liu et al. / Journal of Magnetism and Magnetic Materials 321 (2009) 3221–3224
3223
Table 1 Comparison of selected DSM values from conventional method and bicubic interpolation method. Temperatures (K) Conventional method DSM (J/kgK) BI method Difference (%)
228 0.65 0.68 4.4
where A ¼ Sðu þ 1Þ Sðu 0Þ Sðu 1Þ Sðu 2Þ 2 f ði 1; j 1Þ f ði 1; j þ 0Þ f ði 1; j þ 1Þ 6 6 f ði þ 0; j 1Þ f ði þ 0; j þ 0Þ f ði þ 0; j þ 1Þ 6 B¼6 6 f ði þ 1; j 1Þ f ði þ 1; j þ 0Þ f ði þ 1; j þ 1Þ 4 f ði þ 2; j 1Þ
and 2
Sðv þ 1Þ
f ði þ 2; j þ 0Þ
f ði þ 2; j þ 1Þ
232 3.55 3.76 5.6
233 6.04 5.89 2.5
f ði 1; j þ 2Þ
3
7 f ði þ 0; j þ 2Þ 7 7 7 f ði þ 1; j þ 2Þ 7 5 f ði þ 2; j þ 2Þ
3
7 6 6 Sðv þ 0Þ 7 7 6 C¼6 7 6 Sðv 1Þ 7 5 4 Sðv 2Þ with 8 2 3 > < 1 2jwj þ jwj SðwÞ ¼ 4 8jwj þ 5jwj2 jwj3 > : 0
0 jwjo1 1 jwjo2 jwj 2
S(w) is the approximation to sin(pw)/w. The magnetic entropy changes were calculated by using bicubic interpolation. Taking Ni43Co1Mn45Sn11 as an example, firstly, the magnetization curves were measured under different temperatures and are shown in Fig. 1; secondly, M(T, H) was obtained by calling the bicubic interpolation function in Matlaboratory and is depicted in Fig. 3; for a comparison, M(T, H) based on conventional method is depicted in Fig. 2; finally, the magnetic entropy changes were calculated by using Eq. (2), as shown in Fig. 4.
4. Results and discussion The M(T, H) surfaces obtained from conventional method and bicubic interpolation method are shown in Figs. 2 and 3, respectively, and the comparison of the magnetic entropy changes calculated based on these two methods is depicted in Fig. 4; meanwhile, some selected magnetization entropy change values obtained from these two methods are listed in Table 1. It can be seen from Figs. 2 and 3 that the variation in M(T, H) obtained from the BI method is more continuous and relatively slow, meanwhile, its surface is smoother than that based on conventional method. It is shown in Fig. 4 that the curves of DSM calculated by these two methods are coincident with each other on the whole, and similar variations in DSM with the temperature are shown, their peaks appear both near 235 K. However, there are some differences between them, the curves based on the BI method are smoother than those based on conventional method, especially near the region DSM changes drastically; moreover, the maximum DSM based on BI method is a little bigger. Table 1 shows some selected DSM calculated from these two methods, the values are almost the same, however, there are also some differences between them, and the maximum difference between magnetic entropy changes is about 7.6%. The reason for the appearance of this difference is that the conventional method is based on the calculation of cumulative sum, while integral
234 9.99 9.93 0.6
235 14.35 14.88 3.6
236 12.71 13.52 6.0
237 6.48 6.17 5.0
238 2.48 2.31 7.4
239 0.85 0.79 7.6
operation is employed in the BI method, and the latter has higher calculation precision. Magnetization entropy changes for Ni44Mn45Sn11 and Ni42Co2Mn45Sn11 were also calculated in the same way and similar phenomena have been obtained. It is known from mathematics that for discrete data, the general trend of them could be predicted, the variations could be discovered, and the local error or fluctuation could be eliminated by using surface fitting. The procedure of fitting based on the BI method is that: divide the integral interval into little ones, then conduct integral operations in the little intervals by substituting simple functions for complex ones, and the integral value could be obtained by accumulating the values of little intervals. It is known that smaller the step is, more accurate the integral value is. The calculations of DSM are based on this thoughts in this study: divide the temperature intervals into little ones, conduct integral operation in little intervals, fit M(T, H), and then the function of DSM is obtained. Generally, the temperature intervals are bigger than 1 K in the measurement of M(H), so the variation in DSM is rapid and sharp near the martensite transition temperature, the curve of it is rough, moreover, the magnetization corresponding to the maximum DSM may not be measured, so the maximum DSM may be a little smaller. Meanwhile, the step is set as 0.05 K in our calculation, much less than that in the measurement, as a result, the curve of DSM is much smoother, and the DSM obtained should be more accurate.
5. Conclusions The magnetic entropy changes for Ni44xCoxMn45Sn11 (x ¼ 0, 1, 2) were calculated by employing bicubic interpolation based on magnetization curves. The procedure of calculating DSM based on the newly proposed method is not complicated because the calling of the analysis program is easy. The calculated DSM values are more accurate and their corresponding curves are smoother than that of the conventional method, although the latter is simple in calculation. This research provides one more choice in the calculation of DSM in magnetic refrigeration studies.
Acknowledgements This work was supported by the National Basic Research Program of China (2007CB209400) and the Research Fund of the State Key Laboratory of Coal Resources and Mine safety, CUMT (08KF06). References [1] A.M. Tishin, in: K.H. Buschow (Ed.), Handbook of Magnetic Materials, vol. 12, North-Holland, Amsterdam, 1999, pp. 395–524. [2] M.H. Phan, S.C. Yu, J. Magn. Magn. Mater. 308 (2007) 325–340. [3] V.K. Pecharsky, K.A. Gschneidner, A.O. Tsokol, Rep. Prog. Phys. 68 (2005) 1479. [4] Z.D. Han, D.H. Wang, C.L. Zhang, S.L. Tang, B.X. Gu, Y.W. Du, Appl. Phys. Lett. 89 (2006) 182507. [5] Z.D. Han, D.H. Wang, C.L. Zhang, B.X. Gu, Y.W. Du, Appl. Phys. Lett. 90 (2007) 042507. [6] H.S. Liu, C.L. Zhang, Z.D. Han, H.C. Xuan, D.H. Wang, Y.W. Du, J. Alloys Compd. 467 (2009) 27–30.
ARTICLE IN PRESS 3224
H. Liu et al. / Journal of Magnetism and Magnetic Materials 321 (2009) 3221–3224
[7] A. Dhahri, J. Dhahri, S. Zemni, M. Oumezzine, H. Vincent, J. Alloys Compd. 420 (2006) 15–19. [8] J. Dhahri, A. Dhahri, M. Oumezzine, E. Dhahri, J. Magn. Magn. Mater. 320 (2008) 2613–2617. [9] M. Balli, D. Fruchart, D. Gignoux, M. Rosca, S. Miraglia, J. Magn. Magn. Mater. 313 (2007) 43–46. [10] H. Fu, H.J. Yu, B.H. Teng, X.Y. Zhang, X.T. Zu, J. Alloys Compd. 474 (2009) 595–597.
[11] D.M. Raj Kumar, M. Manivel Raja, R. Gopalan, R. Balamuralikrishnan, A.K. Singh, V. Chandrasekaran, J. Alloys Compd. 461 (2008) 14–20. [12] I. Babita, M. Manivel Raja, R. Gopalan, V. Chandrasekaran, S. Ram, J. Alloys Compd. 432 (2007) 23–29. [13] Z.T. Li, P.F. Wu, Y.Q. Tao, D.K. Mao, Acta Phys. Sin. 48S (1999) 126–131. [14] S.M. Xiang, Visual C++ Digital Image and Graph Manipulation, Electronics industry press, Beijing, 2002, p. 280–281.