- Email: [email protected]
Microcanonical simulation of disordered systems
Journal of Magnetism and Magnetic Materials 226}230 (2001) 606}607
Microcanonical simulation of disordered systems F.G. Brady Moreira*, Marcelo P. Grillon Departamento de Fn& sica, Universidade Federal de Pernambuco, UFPE, 50670}901 Recife, PE, Brazil
Abstract We present the results of microcanonical simulations of the three-dimensional site-diluted antiferromagnetic Ising model, on body-centered cubic lattices of sizes 2;40;40;40 and magnetic concentrations p*0.325. The temperature and concentration dependence of the magnetization, internal energy, and susceptibility are directly calculated from the simulations, whereas the speci"c heat is obtained from the numerical derivative of the energy. Our results for the critical energy and critical temperature as functions of the concentration are compared with the experimental results on the diluted antiferromagnet Co Zn Cs Cl . 2001 Elsevier Science B.V. All rights reserved. N \N Keywords: Dilute antiferromagnet; Phase transitions; Monte Carlo simulation
Experimental results [1] on the Co Zn Cs Cl sysN \N tem in the magnetic concentration range 0.35)p)1.0 indicate that these diluted magnetic compounds are good examples of the quenched site-diluted BCC Ising model with antiferromagnetic nn exchange interactions. Previous Monte Carlo simulations [2}7] were carried out on the related site-diluted Ising model on sc lattices and ferromagnetic interactions. In the present paper we use the multilattice microcanonical simulation technique [8}10], which allows a simultaneous simulation of many model systems at the same energy, to study the thermodynamic properties of the site-diluted BCC Ising model with antiferromagnetic nn interactions, having quenched and randomly distributed magnetic atoms. Our results are compared with the experimental results on the diluted antiferromagnet Co Zn Cs Cl [1]. N \N The simulations were performed on lattices of sizes 2;40 with periodic boundary conditions, and for several values of the magnetic concentration p in the interval [0.325,1.0]. To generate an ensemble of equilibrium con"gurations microcanonically distributed of the system we used the Creutz algorithm [11]. For a given concentration, the con"gurational averages were obtained from the data of simulations on 32 independent samples with
* Corresponding author. Fax: #55-81-2710359. E-mail address: [email protected] (F.G. Brady Moreira).
random distribution of impurities. For each value of energy, we ignored the "rst 200 con"gurations, which represent 10% of the 2000 equilibrium con"gurations per sample considered in the microcanonical averages. Therefore, our "nal results included a total of 6.4;10 equilibrium con"gurations per data point. The estimated error are standard deviations of the mean values. During the simulation, we obtained data for the internal energy, the magnetization, and the mean square magnetization of both sublattices. According to the Creutz's algorithm, the system temperature ¹ is calculated from the spectral distribution of the demon (see Ref. [7] for details of the computational procedure). The susceptibility was then calculated using the #uctuation-dissipation theorem. Fig. 1 shows the temperature and concentration dependence of the speci"c heat, calculated from the numerical derivative of the internal energy. These curves present a well-de"ned maximum even for magnetic concentrations down to p"0.4, indicating that the phase transition remains sharp for all systems with p*0.4. It is worth mentioning that the values for the critical temperature obtained from the position of the peaks in the curves for the susceptibility (not shown) agree with those from the maximum of the speci"c heat. In Fig. 2 we plot our results (open circles) for the reduced critical temperature, ¹ (p)/¹ (1), along with the experimental results (closed symbols) of Ref. [1] on the diluted antiferromagnetic compound Co Zn Cs Cl . N \N
0304-8853/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 0 6 9 5 - 8
F.G.B. Moreira, M.P. Grillon / Journal of Magnetism and Magnetic Materials 226}230 (2001) 606}607
Fig. 1. Temperature dependence of the speci"c heat for values of p"0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0 (curves from left to right).
607
Fig. 3. The reduced critical energy as a function of non-magnetic concentration x. This work (open circles) and the experimental results of Ref. [1] (full circles). The continuous line corresponds to u (x)"0.269/(1!x). A
results of the present simulation (open circles) and the experimental results (closed symbols) of Ref. [1], both follow closely the curve (continuous line) for u (x)" 0.269/(1!x) expected for the site-diluted BCC Ising model. In summary, we have employed the multilattice microcanonical simulation technique to study phase transitions of the body-centered cubic Ising model having quenched (site-diluted) distribution of non-magnetic impurities. The concentration dependence of the critical energy and the critical temperature obtained from simulations on 2;40 lattice spin systems, agree reasonably well with the experimental results of Ref. [1] on the diluted antiferromagnet Co Zn Cs Cl . \V V Fig. 2. The critical temperature as a function of the concentration p of magnetic atoms. The present simulated results (open circles) and the experimental results [1] (full circles) for the diluted antiferromagnet Co Zn Cs Cl . N \N
This work was supported in part by CNPq and FINEP. References
Except for the system with p"0.9, the agreement between the simulated results and the available experimental ones is within error bars. Moreover, for all values of concentration down to p"0.4 we have observed a linear dependence of the critical line with dilution, whereas for the most diluted systems the simulated results indicate a rapid decrease in ¹ (p) (it is expected that ¹ (p )"0, where p +0.269 is the percolation concen tration for the BCC lattice). Fig. 3 shows the reduced critical energy u " E (x)/E (x), where E (x) and E (x) are the system energy at ¹"¹ and 0, respectively, as a function of the con centration x"1!p of non-magnetic impurities. The
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
E. Lagendijk, W.J. Huiskamp, Physica 62 (1972) 444. D.P. Landau, Physica B 86}88 (1977) 731. D.P. Landau, Phys. Rev. B 22 (1980) 2450. A. Labarta, J. Marro, J. Tejada, Physica B 142 (1986) 1. D. Chowdhury, D. Stau!er, J. Stat. Phys. 44 (1986) 203. H.O. Heuer, J. Phys. A 26 (1993) L333. A.A. de Alcantara, A.J.F. de Souza, F.G. Brady Moreira, Phys. Rev. B 49 (1994) 9206. W.G. Wilson, C.A. Vause, Phys. Lett. A 118 (1986) 408. G. Bhanot, D. Duke, R. Salvador, J. Stat. Phys. 44 (1986) 985. G. Bhanot, D. Duke, R. Salvador, Phys. Rev. B 33 (1986) 7841. M. Creutz, Phys. Rev. Lett. 50 (1983) 1411.
Microcanonical simulation of disordered systems F.G. Brady Moreira*, Marcelo P. Grillon Departamento de Fn& sica, Universidade Federal de Pernambuco, UFPE, 50670}901 Recife, PE, Brazil
Abstract We present the results of microcanonical simulations of the three-dimensional site-diluted antiferromagnetic Ising model, on body-centered cubic lattices of sizes 2;40;40;40 and magnetic concentrations p*0.325. The temperature and concentration dependence of the magnetization, internal energy, and susceptibility are directly calculated from the simulations, whereas the speci"c heat is obtained from the numerical derivative of the energy. Our results for the critical energy and critical temperature as functions of the concentration are compared with the experimental results on the diluted antiferromagnet Co Zn Cs Cl . 2001 Elsevier Science B.V. All rights reserved. N \N Keywords: Dilute antiferromagnet; Phase transitions; Monte Carlo simulation
Experimental results [1] on the Co Zn Cs Cl sysN \N tem in the magnetic concentration range 0.35)p)1.0 indicate that these diluted magnetic compounds are good examples of the quenched site-diluted BCC Ising model with antiferromagnetic nn exchange interactions. Previous Monte Carlo simulations [2}7] were carried out on the related site-diluted Ising model on sc lattices and ferromagnetic interactions. In the present paper we use the multilattice microcanonical simulation technique [8}10], which allows a simultaneous simulation of many model systems at the same energy, to study the thermodynamic properties of the site-diluted BCC Ising model with antiferromagnetic nn interactions, having quenched and randomly distributed magnetic atoms. Our results are compared with the experimental results on the diluted antiferromagnet Co Zn Cs Cl [1]. N \N The simulations were performed on lattices of sizes 2;40 with periodic boundary conditions, and for several values of the magnetic concentration p in the interval [0.325,1.0]. To generate an ensemble of equilibrium con"gurations microcanonically distributed of the system we used the Creutz algorithm [11]. For a given concentration, the con"gurational averages were obtained from the data of simulations on 32 independent samples with
* Corresponding author. Fax: #55-81-2710359. E-mail address: [email protected] (F.G. Brady Moreira).
random distribution of impurities. For each value of energy, we ignored the "rst 200 con"gurations, which represent 10% of the 2000 equilibrium con"gurations per sample considered in the microcanonical averages. Therefore, our "nal results included a total of 6.4;10 equilibrium con"gurations per data point. The estimated error are standard deviations of the mean values. During the simulation, we obtained data for the internal energy, the magnetization, and the mean square magnetization of both sublattices. According to the Creutz's algorithm, the system temperature ¹ is calculated from the spectral distribution of the demon (see Ref. [7] for details of the computational procedure). The susceptibility was then calculated using the #uctuation-dissipation theorem. Fig. 1 shows the temperature and concentration dependence of the speci"c heat, calculated from the numerical derivative of the internal energy. These curves present a well-de"ned maximum even for magnetic concentrations down to p"0.4, indicating that the phase transition remains sharp for all systems with p*0.4. It is worth mentioning that the values for the critical temperature obtained from the position of the peaks in the curves for the susceptibility (not shown) agree with those from the maximum of the speci"c heat. In Fig. 2 we plot our results (open circles) for the reduced critical temperature, ¹ (p)/¹ (1), along with the experimental results (closed symbols) of Ref. [1] on the diluted antiferromagnetic compound Co Zn Cs Cl . N \N
0304-8853/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 0 6 9 5 - 8
F.G.B. Moreira, M.P. Grillon / Journal of Magnetism and Magnetic Materials 226}230 (2001) 606}607
Fig. 1. Temperature dependence of the speci"c heat for values of p"0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0 (curves from left to right).
607
Fig. 3. The reduced critical energy as a function of non-magnetic concentration x. This work (open circles) and the experimental results of Ref. [1] (full circles). The continuous line corresponds to u (x)"0.269/(1!x). A
results of the present simulation (open circles) and the experimental results (closed symbols) of Ref. [1], both follow closely the curve (continuous line) for u (x)" 0.269/(1!x) expected for the site-diluted BCC Ising model. In summary, we have employed the multilattice microcanonical simulation technique to study phase transitions of the body-centered cubic Ising model having quenched (site-diluted) distribution of non-magnetic impurities. The concentration dependence of the critical energy and the critical temperature obtained from simulations on 2;40 lattice spin systems, agree reasonably well with the experimental results of Ref. [1] on the diluted antiferromagnet Co Zn Cs Cl . \V V Fig. 2. The critical temperature as a function of the concentration p of magnetic atoms. The present simulated results (open circles) and the experimental results [1] (full circles) for the diluted antiferromagnet Co Zn Cs Cl . N \N
This work was supported in part by CNPq and FINEP. References
Except for the system with p"0.9, the agreement between the simulated results and the available experimental ones is within error bars. Moreover, for all values of concentration down to p"0.4 we have observed a linear dependence of the critical line with dilution, whereas for the most diluted systems the simulated results indicate a rapid decrease in ¹ (p) (it is expected that ¹ (p )"0, where p +0.269 is the percolation concen tration for the BCC lattice). Fig. 3 shows the reduced critical energy u " E (x)/E (x), where E (x) and E (x) are the system energy at ¹"¹ and 0, respectively, as a function of the con centration x"1!p of non-magnetic impurities. The
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
E. Lagendijk, W.J. Huiskamp, Physica 62 (1972) 444. D.P. Landau, Physica B 86}88 (1977) 731. D.P. Landau, Phys. Rev. B 22 (1980) 2450. A. Labarta, J. Marro, J. Tejada, Physica B 142 (1986) 1. D. Chowdhury, D. Stau!er, J. Stat. Phys. 44 (1986) 203. H.O. Heuer, J. Phys. A 26 (1993) L333. A.A. de Alcantara, A.J.F. de Souza, F.G. Brady Moreira, Phys. Rev. B 49 (1994) 9206. W.G. Wilson, C.A. Vause, Phys. Lett. A 118 (1986) 408. G. Bhanot, D. Duke, R. Salvador, J. Stat. Phys. 44 (1986) 985. G. Bhanot, D. Duke, R. Salvador, Phys. Rev. B 33 (1986) 7841. M. Creutz, Phys. Rev. Lett. 50 (1983) 1411.