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The anomalous magnetocaloric effect in HoNi2
Journal of Alloys and Compounds 344 (2002) 145–147
L
www.elsevier.com / locate / jallcom
The anomalous magnetocaloric effect in HoNi 2 a, a b c c P.J. von Ranke *, E.P. Nobrega , I.G. de Oliveira , A.M. Gomes , R.S. Sarthour a
˜ Francisco Xavier 524, 20550 -013, Rio de Janeiro, RJ, Brazil Universidade do Estado do Rio de Janeiro, IF, DEQ , Rua Sao b ´ 2134, 26260, Rio de Janeiro, RJ, Brazil Universidade Iguac¸u, Av. Abilio Augusto Tavora c ´ , Rua Dr. Xavier Sigaud 150, 22290 -180, Rio de Janeiro, RJ, Brazil Centro Brasileiro de Pesquisas Fısicas
Abstract In this work, we report a theoretical investigation of the magnetocaloric effect in the ferromagnet HoNi 2 . To carry out this investigation, we have used a model Hamiltonian that takes into account the crystalline electric field (CEF) and the exchange interaction. Using the proper experimental CEF and exchange parameter, ascribed for HoNi 2 , a change of the easy magnetization direction, from k110l to k001l was predicted at T51.5 K for the critical magnetic field H|2.4 T. The anomalous peak in the isothermal magnetic entropy change and in the adiabatic temperature change with magnetic field was calculated and analysed for the three main crystallographic directions. 2002 Elsevier Science B.V. All rights reserved. PACS: 75.30.Sg; 65.40.1g; 65.50.1m Keywords: Crystal and ligand fields; Thermal analysis
1. Introduction The giant magnetocaloric effect in Gd 5sSi 2 Ge 2d, experimentally discovered, by Pecharsky and Gschneidner Jr. [1] has potential applications, as refrigerant material, to work in cryogenic refrigeration, as well as, in room temperature region. This discovery produced an increasing interest in this research area. The understanding of the microscopic mechanisms that lead magnetic materials to present magnetocaloric effect (MCE) has been our recent interest [2–7]. In this work, we report a theoretical investigation of the magnetocaloric effect in the ferromagnet HoNi 2 . To carry out this investigation, we have used a model Hamiltonian that takes into account the crystalline electric field (CEF) and the exchange interaction. The CEF was treated by the so-called point charge model [8], and the exchange interaction through the use of a molecular field approximation. The two thermodynamic quantities, that characterize the magnetocaloric potential, DSmag (the isothermal magnetic entropy change) and DT ad (the adiabatic temperature change) which are observed upon changes in the external magnetic field were determined for magnetic field change from 0 to 5 T. Using the three-
dimensional mean field theoretical model, as suggested by Bak [9], the change of the easy magnetization direction, from k110l to k001l, at T51.5 K for the critical magnetic field H|2.4 T was predicted. The magnetocaloric potential was investigated when the magnetic field was applied along the three main crystallographic cubic directions. The anomalous peak in the magnetocaloric potential, theoretically predicted [13] in the HoNi 2 , was shown to exist only for the magnetic field applied in the k001l direction.
2. Theoretical model The magnetism of HoNi 2 is described using a magnetic Hamiltonian that includes the CEF, used here in the LLW notation [10] for the cubic symmetry, and the Zeeman exchange interaction terms:
F
X (1 2uXu ) Hˆ 5 W ]sO 04 1 5O 44d 1 ]]]sO 06 2 21O 46d F4 F6 2 gmBsB mx J x 1 B ym J y 1 B zm J zd
G (1)
Where *Corresponding author. E-mail address: [email protected] (P.J. von Ranke).
B xm 5 B cossad 1 lM x
0925-8388 / 02 / $ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S0925-8388( 02 )00354-7
(2)
P. J. von Ranke et al. / Journal of Alloys and Compounds 344 (2002) 145–147
146
B ym 5 B coss bd 1 lM y
(3)
Sel 5 g¯ T
B zm 5 B cossgd 1 lM z
(4)
where T D is the effective Debye temperature and g¯ is the electronic heat capacity coefficient.
Mh 5
e O ke uM ue l FO expS 2 ]] k T DG e 3 expS 2 ]]D k T n
21
n
(8)
h
n
n
n
B
n
(5)
B
In Eq. (5), e n and ue n l are the eigenvalues and eigenvectors of the total Hamiltonian (1), cossad, coss bd and cossgd are the direction cosines along the external magnetic field, and the other symbols have their usual meaning. The component of the magnetization in the direction of the applied magnetic field, M h , is obtained using the three mean field Eqs. (2–4), together with relation (5) using self-consistent standard procedures. The molar magnetic contribution to the entropy is given by:
3 HO
SmagsB,Td 5 R ln
S
en n exp 2 ]] kBT
DJ
K L4
E 1 ]] KT
(6)
KL
where E is the mean energy and R is the universal gas constant. The lattice entropy in the Debye approximation and the electronic entropies are given by:
F
S DG
TD Slatt 5 2 3R ln 1 2 exp ] T TD /T
S DE
T 1 12R ] TD
3
0
x 3 dx ]]] exp(x) 2 1
(7)
3. Results and discussion The model parameters used for HoNi 2 were: (1) CEF parameters W50.021 meV and X5 20.44, taken from Ref. 2 [11]; (2) the exchange parameter, l 5 11.3T / meV, was determined using relation (5) and considering the Curie temperature, T C 515 K [12]; (3) the effective Debye temperature was taken from the nonmagnetic and isostructural systems LaNi 2 and LuNi 2 using the assumptions considered in our previous paper [2]; (4) the electronic heat capacity coefficient, g¯ 5 5.4 mJ mol 21 K 22 , was assumed to be equal to that of the nonmagnetic compound LuNi 2 [2]. Fig. 1 shows the magnetization curves versus magnetic field calculated for the three main crystallographic cubic directions, namely k001l, k111l and k110l at T51.5 K. This calculation was performed considering a three-dimensional mean field theory in order to determine the values of the three components of the magnetization vector. Due to the magnetic anisotropy that comes from the CEF, the magnetization vector does not have the same direction as the applied magnetic field and therefore the curves that appear in Fig. 1 are the components of the magnetization vector along the applied magnetic field. It is worth noticing that the easy magnetization direction, calculated at T51.5 K, changes from k110l to k001l at H|2.4 T. By comparing the values of the free energy, one obtains the lowest value
Fig. 1. The calculated magnetic field dependence of the magnetization in HoNi 2 along the three main crystallographic directions k001l, k111l and k110l at T51.5 K. The inset shows the corresponding magnetic free energies vs. magnetic field calculated at T51.5 K.
P. J. von Ranke et al. / Journal of Alloys and Compounds 344 (2002) 145–147
147
density of states was theoretically predicted and was associated to the high entropy, which leads to the anomalous peak [13]. Fig. 3 shows the adiabatic temperature change for magnetic field change from 0 to 5 T along k001l for HoNi 2 . In addition, the anomalous peak is theoretically observed at about 1.5 K. This anomalous peak disappears when the magnetic field is applied along the other directions.
4. Conclusion
Fig. 2. Temperature dependence of the isothermal entropy change with external magnetic field change from 0 to 5 T, in HoNi 2 , calculated along the three crystallographic directions k001l, k111l and k110l.
for the easy direction of magnetization as displayed in the inset of Fig. 1. Fig. 2 shows the entropy change for an external magnetic field change from 0 to 5 T, in the isothermal process, for the three considered magnetic field directions. In all directions, the maximum MCE effect occurs at the Curie temperature as expected since at this temperature an applied magnetic field has a maximum reduction effect on the magnetic entropy for ferromagnetic systems. A very interesting second anomalous peak is theoretically predicted to be found in HoNi 2 at about T51.5 K when the magnetic field is applied in k001l direction. In part, this anomalous peak can be understood noticing in Fig. 1 that the effect of the magnetic field change, from 0 to 5 T, leads to a maximum magnetization change along the k001l direction compared with the other two directions. The microscopic origin of the anomalous peak was recently investigated by the authors, considering the CEF-levels scheme in the Lea–Leask–Wolf diagram. The CEF high
Fig. 3. The temperature dependence of DT ad for HoNi 2 for a magnetic field change from 0 to 5 T in the k100l direction.
The MCE in HoNi 2 was theoretically investigated considering the CEF anisotropy and the application of magnetic field along the three main crystallographic directions. The existence of a change in the easy magnetization direction from k110l to k001l and an anomalous peak in the magnetocaloric curves at about T51.5 K was theoretically analysed. Single crystal experiments, using HoNi 2 compound, showing the magnetic anisotropy would be particularly important to compare with our theoretical predictions.
Acknowledgements This research was partially supported by CNPq, Faperj, and Capes.
References [1] V.K. Pecharsky, K.A. Gschneidner Jr., Phys. Rev. Lett. 78 (1997) 4494. [2] P.J. von Ranke, V.K. Pecharsky, K.A. Gschneidner Jr., Phys. Rev. B 58 (1998) 12110. [3] P.J. von Ranke, V.K. Pecharsky, K.A. Gschneidner Jr., B.J. Korte, Phys. Rev. B 58 (1998) 14436. [4] I.G. de Oliveira, A. Caldas, E.P. Nobrega, N.A. de Oliveira, P.J. von Ranke, Solid State Commun. 114 (2000) 487. ˜ [5] P.J. von Ranke, I.G. de Oliveira, A.P. Guimaraes, X.A. da Silva, Phys. Rev. B 61 (2000) 447. [6] P.J. von Ranke, N.A. de Oliveira, M.V. Tovar Costa, A. Caldas, I.G. de Oliveira, E.P. Nobrega, J. Magn. Magn. Mater. 226 (2001) 990. [7] P.J. von Ranke, A.L. Lima, E.P. Nobrega, X. da Silva, A.P. ˜ Guimaraes, I.S. Oliveira, Phys. Rev. B 63 (2001) 24422. [8] K.W.H. Stevens, Proc. Phys. Soc. A 65 (1952) 209. [9] P. Bak, J. Phys. C 7 (1974) 4097. [10] K.R. Lea, M.J.M. Leask, W.P. Wolf, J. Phys. Chem. Solids 33 (1962) 1381. [11] A. Andreeff, Th. Frauenheim, E.A. Goremychkin, H. Griessmann, B. Lippold, W. Matz, O.D. Chistyakov, E.M. Savitskii, Phys. Status Solidi (b) 111 (1982) 507. ¨ [12] E.A. Goremychkin, I. Natkaniec, E. Muhle, O.D. Chistyakov, J. Magn. Magn. Mater. 81 (1989) 63. [13] P.J. von Ranke, E.P. Nobrega, I.G. de Oliveira, A.M. Gomes, R.S. Sarthour, Phys. Rev. B 63 (2001) 184406.
L
www.elsevier.com / locate / jallcom
The anomalous magnetocaloric effect in HoNi 2 a, a b c c P.J. von Ranke *, E.P. Nobrega , I.G. de Oliveira , A.M. Gomes , R.S. Sarthour a
˜ Francisco Xavier 524, 20550 -013, Rio de Janeiro, RJ, Brazil Universidade do Estado do Rio de Janeiro, IF, DEQ , Rua Sao b ´ 2134, 26260, Rio de Janeiro, RJ, Brazil Universidade Iguac¸u, Av. Abilio Augusto Tavora c ´ , Rua Dr. Xavier Sigaud 150, 22290 -180, Rio de Janeiro, RJ, Brazil Centro Brasileiro de Pesquisas Fısicas
Abstract In this work, we report a theoretical investigation of the magnetocaloric effect in the ferromagnet HoNi 2 . To carry out this investigation, we have used a model Hamiltonian that takes into account the crystalline electric field (CEF) and the exchange interaction. Using the proper experimental CEF and exchange parameter, ascribed for HoNi 2 , a change of the easy magnetization direction, from k110l to k001l was predicted at T51.5 K for the critical magnetic field H|2.4 T. The anomalous peak in the isothermal magnetic entropy change and in the adiabatic temperature change with magnetic field was calculated and analysed for the three main crystallographic directions. 2002 Elsevier Science B.V. All rights reserved. PACS: 75.30.Sg; 65.40.1g; 65.50.1m Keywords: Crystal and ligand fields; Thermal analysis
1. Introduction The giant magnetocaloric effect in Gd 5sSi 2 Ge 2d, experimentally discovered, by Pecharsky and Gschneidner Jr. [1] has potential applications, as refrigerant material, to work in cryogenic refrigeration, as well as, in room temperature region. This discovery produced an increasing interest in this research area. The understanding of the microscopic mechanisms that lead magnetic materials to present magnetocaloric effect (MCE) has been our recent interest [2–7]. In this work, we report a theoretical investigation of the magnetocaloric effect in the ferromagnet HoNi 2 . To carry out this investigation, we have used a model Hamiltonian that takes into account the crystalline electric field (CEF) and the exchange interaction. The CEF was treated by the so-called point charge model [8], and the exchange interaction through the use of a molecular field approximation. The two thermodynamic quantities, that characterize the magnetocaloric potential, DSmag (the isothermal magnetic entropy change) and DT ad (the adiabatic temperature change) which are observed upon changes in the external magnetic field were determined for magnetic field change from 0 to 5 T. Using the three-
dimensional mean field theoretical model, as suggested by Bak [9], the change of the easy magnetization direction, from k110l to k001l, at T51.5 K for the critical magnetic field H|2.4 T was predicted. The magnetocaloric potential was investigated when the magnetic field was applied along the three main crystallographic cubic directions. The anomalous peak in the magnetocaloric potential, theoretically predicted [13] in the HoNi 2 , was shown to exist only for the magnetic field applied in the k001l direction.
2. Theoretical model The magnetism of HoNi 2 is described using a magnetic Hamiltonian that includes the CEF, used here in the LLW notation [10] for the cubic symmetry, and the Zeeman exchange interaction terms:
F
X (1 2uXu ) Hˆ 5 W ]sO 04 1 5O 44d 1 ]]]sO 06 2 21O 46d F4 F6 2 gmBsB mx J x 1 B ym J y 1 B zm J zd
G (1)
Where *Corresponding author. E-mail address: [email protected] (P.J. von Ranke).
B xm 5 B cossad 1 lM x
0925-8388 / 02 / $ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S0925-8388( 02 )00354-7
(2)
P. J. von Ranke et al. / Journal of Alloys and Compounds 344 (2002) 145–147
146
B ym 5 B coss bd 1 lM y
(3)
Sel 5 g¯ T
B zm 5 B cossgd 1 lM z
(4)
where T D is the effective Debye temperature and g¯ is the electronic heat capacity coefficient.
Mh 5
e O ke uM ue l FO expS 2 ]] k T DG e 3 expS 2 ]]D k T n
21
n
(8)
h
n
n
n
B
n
(5)
B
In Eq. (5), e n and ue n l are the eigenvalues and eigenvectors of the total Hamiltonian (1), cossad, coss bd and cossgd are the direction cosines along the external magnetic field, and the other symbols have their usual meaning. The component of the magnetization in the direction of the applied magnetic field, M h , is obtained using the three mean field Eqs. (2–4), together with relation (5) using self-consistent standard procedures. The molar magnetic contribution to the entropy is given by:
3 HO
SmagsB,Td 5 R ln
S
en n exp 2 ]] kBT
DJ
K L4
E 1 ]] KT
(6)
KL
where E is the mean energy and R is the universal gas constant. The lattice entropy in the Debye approximation and the electronic entropies are given by:
F
S DG
TD Slatt 5 2 3R ln 1 2 exp ] T TD /T
S DE
T 1 12R ] TD
3
0
x 3 dx ]]] exp(x) 2 1
(7)
3. Results and discussion The model parameters used for HoNi 2 were: (1) CEF parameters W50.021 meV and X5 20.44, taken from Ref. 2 [11]; (2) the exchange parameter, l 5 11.3T / meV, was determined using relation (5) and considering the Curie temperature, T C 515 K [12]; (3) the effective Debye temperature was taken from the nonmagnetic and isostructural systems LaNi 2 and LuNi 2 using the assumptions considered in our previous paper [2]; (4) the electronic heat capacity coefficient, g¯ 5 5.4 mJ mol 21 K 22 , was assumed to be equal to that of the nonmagnetic compound LuNi 2 [2]. Fig. 1 shows the magnetization curves versus magnetic field calculated for the three main crystallographic cubic directions, namely k001l, k111l and k110l at T51.5 K. This calculation was performed considering a three-dimensional mean field theory in order to determine the values of the three components of the magnetization vector. Due to the magnetic anisotropy that comes from the CEF, the magnetization vector does not have the same direction as the applied magnetic field and therefore the curves that appear in Fig. 1 are the components of the magnetization vector along the applied magnetic field. It is worth noticing that the easy magnetization direction, calculated at T51.5 K, changes from k110l to k001l at H|2.4 T. By comparing the values of the free energy, one obtains the lowest value
Fig. 1. The calculated magnetic field dependence of the magnetization in HoNi 2 along the three main crystallographic directions k001l, k111l and k110l at T51.5 K. The inset shows the corresponding magnetic free energies vs. magnetic field calculated at T51.5 K.
P. J. von Ranke et al. / Journal of Alloys and Compounds 344 (2002) 145–147
147
density of states was theoretically predicted and was associated to the high entropy, which leads to the anomalous peak [13]. Fig. 3 shows the adiabatic temperature change for magnetic field change from 0 to 5 T along k001l for HoNi 2 . In addition, the anomalous peak is theoretically observed at about 1.5 K. This anomalous peak disappears when the magnetic field is applied along the other directions.
4. Conclusion
Fig. 2. Temperature dependence of the isothermal entropy change with external magnetic field change from 0 to 5 T, in HoNi 2 , calculated along the three crystallographic directions k001l, k111l and k110l.
for the easy direction of magnetization as displayed in the inset of Fig. 1. Fig. 2 shows the entropy change for an external magnetic field change from 0 to 5 T, in the isothermal process, for the three considered magnetic field directions. In all directions, the maximum MCE effect occurs at the Curie temperature as expected since at this temperature an applied magnetic field has a maximum reduction effect on the magnetic entropy for ferromagnetic systems. A very interesting second anomalous peak is theoretically predicted to be found in HoNi 2 at about T51.5 K when the magnetic field is applied in k001l direction. In part, this anomalous peak can be understood noticing in Fig. 1 that the effect of the magnetic field change, from 0 to 5 T, leads to a maximum magnetization change along the k001l direction compared with the other two directions. The microscopic origin of the anomalous peak was recently investigated by the authors, considering the CEF-levels scheme in the Lea–Leask–Wolf diagram. The CEF high
Fig. 3. The temperature dependence of DT ad for HoNi 2 for a magnetic field change from 0 to 5 T in the k100l direction.
The MCE in HoNi 2 was theoretically investigated considering the CEF anisotropy and the application of magnetic field along the three main crystallographic directions. The existence of a change in the easy magnetization direction from k110l to k001l and an anomalous peak in the magnetocaloric curves at about T51.5 K was theoretically analysed. Single crystal experiments, using HoNi 2 compound, showing the magnetic anisotropy would be particularly important to compare with our theoretical predictions.
Acknowledgements This research was partially supported by CNPq, Faperj, and Capes.
References [1] V.K. Pecharsky, K.A. Gschneidner Jr., Phys. Rev. Lett. 78 (1997) 4494. [2] P.J. von Ranke, V.K. Pecharsky, K.A. Gschneidner Jr., Phys. Rev. B 58 (1998) 12110. [3] P.J. von Ranke, V.K. Pecharsky, K.A. Gschneidner Jr., B.J. Korte, Phys. Rev. B 58 (1998) 14436. [4] I.G. de Oliveira, A. Caldas, E.P. Nobrega, N.A. de Oliveira, P.J. von Ranke, Solid State Commun. 114 (2000) 487. ˜ [5] P.J. von Ranke, I.G. de Oliveira, A.P. Guimaraes, X.A. da Silva, Phys. Rev. B 61 (2000) 447. [6] P.J. von Ranke, N.A. de Oliveira, M.V. Tovar Costa, A. Caldas, I.G. de Oliveira, E.P. Nobrega, J. Magn. Magn. Mater. 226 (2001) 990. [7] P.J. von Ranke, A.L. Lima, E.P. Nobrega, X. da Silva, A.P. ˜ Guimaraes, I.S. Oliveira, Phys. Rev. B 63 (2001) 24422. [8] K.W.H. Stevens, Proc. Phys. Soc. A 65 (1952) 209. [9] P. Bak, J. Phys. C 7 (1974) 4097. [10] K.R. Lea, M.J.M. Leask, W.P. Wolf, J. Phys. Chem. Solids 33 (1962) 1381. [11] A. Andreeff, Th. Frauenheim, E.A. Goremychkin, H. Griessmann, B. Lippold, W. Matz, O.D. Chistyakov, E.M. Savitskii, Phys. Status Solidi (b) 111 (1982) 507. ¨ [12] E.A. Goremychkin, I. Natkaniec, E. Muhle, O.D. Chistyakov, J. Magn. Magn. Mater. 81 (1989) 63. [13] P.J. von Ranke, E.P. Nobrega, I.G. de Oliveira, A.M. Gomes, R.S. Sarthour, Phys. Rev. B 63 (2001) 184406.