- Email: [email protected]
Monte Carlo calculations of the magnetocaloric effect in (Gd0.6Tb0.4)5Si4
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 310 (2007) 2805–2807 www.elsevier.com/locate/jmmm
Monte Carlo calculations of the magnetocaloric effect in ðGd0:6 Tb0:4Þ5Si4 E.P. Nobregaa,, N.A. de Oliveirab, P.J. von Rankeb, A. Tropera,b a
Centro Brasileiro de Pesquisas Fı´sicas, Rua Xavier Sigaud 150, Rio de Janeiro, 22290-180, RJ, Brazil Universidade do Estado do Rio de Janeiro, Rua Sa˜o Francisco Xavier 524, Rio de Janeiro, 20550-013, RJ, Brazil
b
Available online 27 November 2006
Abstract In this work we calculate the magnetocaloric effect in the compound ðGd0:6 Tb0:4 Þ5 Si4 . We use a model Hamiltonian of interacting 4f spins and treat the 4f spin–spin interaction in the Monte Carlo simulation. The theoretically calculated isothermal entropy changes upon variations of the magnetic field are in good agreement with the available experimental data. r 2006 Published by Elsevier B.V. PACS: 75.30.Sg; 75.10.Dg; 75.20.En Keywords: Magnetocaloric effect; Monte Carlo calculation
The magnetocaloric effect [1–4] has great importance in the technology of magnetic refrigeration. The magnetocaloric effect in the compounds ðGdx Tb1x Þ5 Si4 has been measured in a wide range of Gd concentration [5]. These compounds exhibit orthorhombic phase in all range of temperature and undergo a second order magnetic phase transition in the whole range of Gd concentration. Experimental data [5] show that the adiabatic temperature changes upon magnetic field variations in this series of compounds are comparable with the ones found in metallic Gadolinium. In the theoretical description of the magnetocaloric effect in this compound we should go beyond the molecular field theory in order to take into account the type of ion ( Gd or Tb) occupying a given site of the crystalline lattice. In this paper, we use the Monte Carlo simulations [6–9] to calculate the magnetocaloric effect in the compounds ðGdx Tb1x Þ5 Si4 . To this end, we start with the following energy: X X X lil Jai Jdl þ zi ðri :Jai Þ2 gmB Jai :H, (1) E¼ il
i
i
Corresponding author. Centro Brasileiro de Pesquisas Fı´ sicas, Rua Xavier Sigaud 150, Rio de Janeiro, 22290-180, RJ, Brazil. Tel.: +5521 21417285. E-mail address: [email protected] (E.P. Nobrega).
0304-8853/$ - see front matter r 2006 Published by Elsevier B.V. doi:10.1016/j.jmmm.2006.10.1057
where lil is the effective exchange interaction parameter, Jai ðJdl Þ (a; d ¼ Gd or Tb) is the total angular momentum of rare earth ions and H is the applied magnetic field. The term zi ðri :Ji Þ2 represents the single ion anisotropy, where ri is the direction of the anisotropy and zi is the anisotropic coefficient. The single ion anisotropic term somehow mimics the effect of the crystalline electrical field on the rare earth ions. In order to calculate the mean energy given in Eq. (1) via the Monte Carlo method, we consider the z components of total angular momentum as quantum quantities, which can assume discrete values in the range JpJ z pJ. For a given J z , the transverse components J x and J y were randomly chosen under the condition ðJ x Þ2 þ ðJ y Þ2 ¼ J 2 ðJ z Þ2 . Within this approach we restrict the number of available states so that the upper limit of the magnetic entropy S mag ¼ R lnð2J þ 1Þ is reproduced, where R is the gas constant. More details about the numerical procedures on this Monte Carlo calculation can be found in Refs. [6–9]. The magnetic part of the heat capacity is calculated by C mag ðT; HÞ ¼
E 2 hE i2 kB T 2
,
(2)
ARTICLE IN PRESS E.P. Nobrega et al. / Journal of Magnetism and Magnetic Materials 310 (2007) 2805–2807
2806
where kB is the Boltzmann constant and the mean values hE a i ða ¼ 1 or 2Þ are given by:
8 7
0
where ‘‘i’’ represents a given Monte Carlo step; N c represents the total number of Monte Carlo steps and N 0 is the number of Monte Carlo steps used for thermalization. The total heat capacity is: C ¼ C el þ C lat þ C mag : Here C el ¼ gT is the contribution from the conduction electrons where g is the Sommerfeld coefficient. C lat is the contribution from the crystalline lattice, which for the sake of simplicity is taken in the Debye approximation. The total entropy of the compound is given by: S ¼ S lat þ S mag þ Sel ; where S el ¼ gT; is the contribution from the electron gas and S lat is the lattice contribution, which is also taken in the Debye approximation. S mag is the contribution from the magnetic ions calculated from Z T C mag ðT; HÞ dT. (4) S mag ðT; HÞ ¼ T 0 The magnetocaloric potentials DS and DT ad are calculated, respectively, from DS ¼ SðT; Ha0ÞSðT; H ¼ 0Þ and DT ad ¼ T 2 T 1 under the adiabatic condition SðT 2 ; Ha0Þ ¼ SðT 1 ; H ¼ 0Þ: In this work we apply the present model to calculate the magnetocaloric effect in the compound ðGd0:6 Tb0:4 Þ5 Si4 . The Lande` factor and the total angular momentum were taken as g ¼ 2 and J ¼ 72 for Gd and g ¼ 1:5 and J ¼ 6 for Tb. In order to perform the Monte Carlo calculations in these compounds we use a tridimensional cluster of 5 5 5 orthorhombic unit cells with 8 rare earth atoms (Gd or Tb) per cell, randomly distributed at the lattice sites, according to the Gd concentration. The magnitude of the components of the total angular momentum of the Gd and Tb ions at each crystalline lattice site was randomly chosen [6,7]. The exchange interaction parameters were taken as: l ¼ 1:39 meV for Gd–Gd interaction l ¼ 0:289 meV for Tb–Tb interaction and l ¼ 0:41 meV for Gd–Tb interaction. The anisotropic coefficients for Gd and Tb ions were taken, respectively, as zi ¼ 0:00 meV and zi ¼ 0:0294 meV. The numerical simulation was performed using 5000 Monte Carlo steps from which 2000 were used for thermalization. In order to calculate the total heat capacity and total entropy, we use g ¼ 5:4 mJ=ðmol K2 ) and YD ¼ 350 K. Within these parameters we calculate the thermodynamical properties and the magnetocaloric effect in the compound ðGd0:6 Tb0:4 Þ5 Si4 . In Figs. 1 and 2 we plot the isothermal entropy change and the adiabatic temperature change for the compound ðGd0:6 Tb0:4 Þ5 Si4 upon magnetic field variations from 0 to 2 T and from 0 to 5 T. From these figures we can notice a good agreement between our calculated isothermal entropy change and the available experimental data [5]. Further experimental data are necessary to compare with our theoretically calculated adiabatic temperature change.
-∆ S (J/mol K)
(3)
6 5 4 3 2 1 0 0
50
100
150
200
250
300
350
400
450
500
Temperature (K)
Fig. 1. Isothermal entropy change for ðGd0:6 Tb0:4 Þ5 Si4 upon magnetic field variations from 0 to 2 T (dashed line) and from 0 to 5 T (solid line). Open and solid circles are the corresponding experimental data [5].
(Gd0.6Tb0.4)5Si4
8
6 ∆Tad (K)
NC X 1 hE a i ¼ Ea, ðN c N 0 Þ i4N i
(Gd0.6Tb0.4)5Si4
9
4
2
0 0
50
100
150
200
250
300
350
400
450
500
Temperature (K)
Fig. 2. Adiabatic temperature change for ðGd0:6 Tb0:4 Þ5 Si4 upon magnetic field variations from 0 to 2 T (dashed line) and from 0 to 5 T (solid line).
In conclusion, in this work we report on the theoretical calculations of the magnetocaloric effect in the compound ðGd0:6 Tb0:4 Þ5 Si4 by using a model Hamiltonian of localized spins, where the 4f spin–spin interaction was treated in the Monte Carlo simulation. The obtained results for the magnetocaloric quantity DS are in good agreement with the available experimental data [5]. We acknowledge partial financial support from the Brazilian agencies CNPq and FAPERJ. This work was also financed by PRONEX No. E-26/171.168/2003.
References [1] K.A. Gschneidner Jr., V.K. Pecharsky, Annu. Rev. Mater. Sci. 30 (2000) 387.
ARTICLE IN PRESS E.P. Nobrega et al. / Journal of Magnetism and Magnetic Materials 310 (2007) 2805–2807 [2] A.M. Tishin, in: Handbook of Magnetic Materials, vol. 12, NorthHolland, Amsterdam, 1999 pp. 395–524. [3] A.M. Tishin, Y.I. Spichkin, The Magnetocaloric Effect and Its Applications, first ed., Institute of Physics, Bristol and Philadelphia, 2003. [4] K.A. Gschneidner Jr., V.K. Pecharsky, A.O. Tsokol, Rep. Prog. Phys. 68 (2005) 1479. [5] Y.I. Spichkin, V.K. Pecharsky, K.A. Gschneidner Jr., J. Appl. Phys. 89 (2001) 1738.
2807
[6] E.P. No´brega, N.A. de Oliveira, P.J. von Ranke, A. Troper, Phys. Rev. B 72 (2005) 134426. [7] E.P. No´brega, N.A. de Oliveira, P.J. von Ranke, A. Troper, J. Phys. Condens. Matter 18 (2006) 1275. [8] E.P. No´brega, N.A. de Oliveira, P.J. von Ranke, A. Troper, J. Appl. Phys. 99 (2006) 08Q103. [9] E.P. No´brega, N.A. de Oliveira, P.J. von Ranke, A. Troper, Physica B 378–380 (2006) 716.
Journal of Magnetism and Magnetic Materials 310 (2007) 2805–2807 www.elsevier.com/locate/jmmm
Monte Carlo calculations of the magnetocaloric effect in ðGd0:6 Tb0:4Þ5Si4 E.P. Nobregaa,, N.A. de Oliveirab, P.J. von Rankeb, A. Tropera,b a
Centro Brasileiro de Pesquisas Fı´sicas, Rua Xavier Sigaud 150, Rio de Janeiro, 22290-180, RJ, Brazil Universidade do Estado do Rio de Janeiro, Rua Sa˜o Francisco Xavier 524, Rio de Janeiro, 20550-013, RJ, Brazil
b
Available online 27 November 2006
Abstract In this work we calculate the magnetocaloric effect in the compound ðGd0:6 Tb0:4 Þ5 Si4 . We use a model Hamiltonian of interacting 4f spins and treat the 4f spin–spin interaction in the Monte Carlo simulation. The theoretically calculated isothermal entropy changes upon variations of the magnetic field are in good agreement with the available experimental data. r 2006 Published by Elsevier B.V. PACS: 75.30.Sg; 75.10.Dg; 75.20.En Keywords: Magnetocaloric effect; Monte Carlo calculation
The magnetocaloric effect [1–4] has great importance in the technology of magnetic refrigeration. The magnetocaloric effect in the compounds ðGdx Tb1x Þ5 Si4 has been measured in a wide range of Gd concentration [5]. These compounds exhibit orthorhombic phase in all range of temperature and undergo a second order magnetic phase transition in the whole range of Gd concentration. Experimental data [5] show that the adiabatic temperature changes upon magnetic field variations in this series of compounds are comparable with the ones found in metallic Gadolinium. In the theoretical description of the magnetocaloric effect in this compound we should go beyond the molecular field theory in order to take into account the type of ion ( Gd or Tb) occupying a given site of the crystalline lattice. In this paper, we use the Monte Carlo simulations [6–9] to calculate the magnetocaloric effect in the compounds ðGdx Tb1x Þ5 Si4 . To this end, we start with the following energy: X X X lil Jai Jdl þ zi ðri :Jai Þ2 gmB Jai :H, (1) E¼ il
i
i
Corresponding author. Centro Brasileiro de Pesquisas Fı´ sicas, Rua Xavier Sigaud 150, Rio de Janeiro, 22290-180, RJ, Brazil. Tel.: +5521 21417285. E-mail address: [email protected] (E.P. Nobrega).
0304-8853/$ - see front matter r 2006 Published by Elsevier B.V. doi:10.1016/j.jmmm.2006.10.1057
where lil is the effective exchange interaction parameter, Jai ðJdl Þ (a; d ¼ Gd or Tb) is the total angular momentum of rare earth ions and H is the applied magnetic field. The term zi ðri :Ji Þ2 represents the single ion anisotropy, where ri is the direction of the anisotropy and zi is the anisotropic coefficient. The single ion anisotropic term somehow mimics the effect of the crystalline electrical field on the rare earth ions. In order to calculate the mean energy given in Eq. (1) via the Monte Carlo method, we consider the z components of total angular momentum as quantum quantities, which can assume discrete values in the range JpJ z pJ. For a given J z , the transverse components J x and J y were randomly chosen under the condition ðJ x Þ2 þ ðJ y Þ2 ¼ J 2 ðJ z Þ2 . Within this approach we restrict the number of available states so that the upper limit of the magnetic entropy S mag ¼ R lnð2J þ 1Þ is reproduced, where R is the gas constant. More details about the numerical procedures on this Monte Carlo calculation can be found in Refs. [6–9]. The magnetic part of the heat capacity is calculated by C mag ðT; HÞ ¼
E 2 hE i2 kB T 2
,
(2)
ARTICLE IN PRESS E.P. Nobrega et al. / Journal of Magnetism and Magnetic Materials 310 (2007) 2805–2807
2806
where kB is the Boltzmann constant and the mean values hE a i ða ¼ 1 or 2Þ are given by:
8 7
0
where ‘‘i’’ represents a given Monte Carlo step; N c represents the total number of Monte Carlo steps and N 0 is the number of Monte Carlo steps used for thermalization. The total heat capacity is: C ¼ C el þ C lat þ C mag : Here C el ¼ gT is the contribution from the conduction electrons where g is the Sommerfeld coefficient. C lat is the contribution from the crystalline lattice, which for the sake of simplicity is taken in the Debye approximation. The total entropy of the compound is given by: S ¼ S lat þ S mag þ Sel ; where S el ¼ gT; is the contribution from the electron gas and S lat is the lattice contribution, which is also taken in the Debye approximation. S mag is the contribution from the magnetic ions calculated from Z T C mag ðT; HÞ dT. (4) S mag ðT; HÞ ¼ T 0 The magnetocaloric potentials DS and DT ad are calculated, respectively, from DS ¼ SðT; Ha0ÞSðT; H ¼ 0Þ and DT ad ¼ T 2 T 1 under the adiabatic condition SðT 2 ; Ha0Þ ¼ SðT 1 ; H ¼ 0Þ: In this work we apply the present model to calculate the magnetocaloric effect in the compound ðGd0:6 Tb0:4 Þ5 Si4 . The Lande` factor and the total angular momentum were taken as g ¼ 2 and J ¼ 72 for Gd and g ¼ 1:5 and J ¼ 6 for Tb. In order to perform the Monte Carlo calculations in these compounds we use a tridimensional cluster of 5 5 5 orthorhombic unit cells with 8 rare earth atoms (Gd or Tb) per cell, randomly distributed at the lattice sites, according to the Gd concentration. The magnitude of the components of the total angular momentum of the Gd and Tb ions at each crystalline lattice site was randomly chosen [6,7]. The exchange interaction parameters were taken as: l ¼ 1:39 meV for Gd–Gd interaction l ¼ 0:289 meV for Tb–Tb interaction and l ¼ 0:41 meV for Gd–Tb interaction. The anisotropic coefficients for Gd and Tb ions were taken, respectively, as zi ¼ 0:00 meV and zi ¼ 0:0294 meV. The numerical simulation was performed using 5000 Monte Carlo steps from which 2000 were used for thermalization. In order to calculate the total heat capacity and total entropy, we use g ¼ 5:4 mJ=ðmol K2 ) and YD ¼ 350 K. Within these parameters we calculate the thermodynamical properties and the magnetocaloric effect in the compound ðGd0:6 Tb0:4 Þ5 Si4 . In Figs. 1 and 2 we plot the isothermal entropy change and the adiabatic temperature change for the compound ðGd0:6 Tb0:4 Þ5 Si4 upon magnetic field variations from 0 to 2 T and from 0 to 5 T. From these figures we can notice a good agreement between our calculated isothermal entropy change and the available experimental data [5]. Further experimental data are necessary to compare with our theoretically calculated adiabatic temperature change.
-∆ S (J/mol K)
(3)
6 5 4 3 2 1 0 0
50
100
150
200
250
300
350
400
450
500
Temperature (K)
Fig. 1. Isothermal entropy change for ðGd0:6 Tb0:4 Þ5 Si4 upon magnetic field variations from 0 to 2 T (dashed line) and from 0 to 5 T (solid line). Open and solid circles are the corresponding experimental data [5].
(Gd0.6Tb0.4)5Si4
8
6 ∆Tad (K)
NC X 1 hE a i ¼ Ea, ðN c N 0 Þ i4N i
(Gd0.6Tb0.4)5Si4
9
4
2
0 0
50
100
150
200
250
300
350
400
450
500
Temperature (K)
Fig. 2. Adiabatic temperature change for ðGd0:6 Tb0:4 Þ5 Si4 upon magnetic field variations from 0 to 2 T (dashed line) and from 0 to 5 T (solid line).
In conclusion, in this work we report on the theoretical calculations of the magnetocaloric effect in the compound ðGd0:6 Tb0:4 Þ5 Si4 by using a model Hamiltonian of localized spins, where the 4f spin–spin interaction was treated in the Monte Carlo simulation. The obtained results for the magnetocaloric quantity DS are in good agreement with the available experimental data [5]. We acknowledge partial financial support from the Brazilian agencies CNPq and FAPERJ. This work was also financed by PRONEX No. E-26/171.168/2003.
References [1] K.A. Gschneidner Jr., V.K. Pecharsky, Annu. Rev. Mater. Sci. 30 (2000) 387.
ARTICLE IN PRESS E.P. Nobrega et al. / Journal of Magnetism and Magnetic Materials 310 (2007) 2805–2807 [2] A.M. Tishin, in: Handbook of Magnetic Materials, vol. 12, NorthHolland, Amsterdam, 1999 pp. 395–524. [3] A.M. Tishin, Y.I. Spichkin, The Magnetocaloric Effect and Its Applications, first ed., Institute of Physics, Bristol and Philadelphia, 2003. [4] K.A. Gschneidner Jr., V.K. Pecharsky, A.O. Tsokol, Rep. Prog. Phys. 68 (2005) 1479. [5] Y.I. Spichkin, V.K. Pecharsky, K.A. Gschneidner Jr., J. Appl. Phys. 89 (2001) 1738.
2807
[6] E.P. No´brega, N.A. de Oliveira, P.J. von Ranke, A. Troper, Phys. Rev. B 72 (2005) 134426. [7] E.P. No´brega, N.A. de Oliveira, P.J. von Ranke, A. Troper, J. Phys. Condens. Matter 18 (2006) 1275. [8] E.P. No´brega, N.A. de Oliveira, P.J. von Ranke, A. Troper, J. Appl. Phys. 99 (2006) 08Q103. [9] E.P. No´brega, N.A. de Oliveira, P.J. von Ranke, A. Troper, Physica B 378–380 (2006) 716.