Critical behavior in the La0.6Ca0.4−xSrxMnO3 nano-particle compounds for x = 0, 0.05 and 0.4

The critical behavior associated with the magnetic phase transition has been investigated by magnetization isotherms in La0.6Ca0.4−xSrxMnO3 nano-particle compounds for x = 0 (S0C1), 0.05 (SC. 4-4) and 0.4 (S1C0). The critical exponents are estimated by various techniques, such as the Modified Arrott plot, Kouvel-Fisher plot and critical isotherm techniques. Thus, the average values of the critical exponent and the critical temperature obtained by the different methods are (βmoy=0.47;γmoy=1.02;δmoy=3.00;TC=230.6K) for S0C1, (βmoy=0.47;γmoy=1.00;δmoy=3.21;TC=361.9K) for S1C0, and (βmoy=0.26;γmoy=1.02;δmoy=4.92;TC=286.4K) for SC. 4-4. These values are in good agreement with those given by the theoretical models: Mean-Field model (β = 0.5, γ = 1 and δ = 3) for S0C1 and S1C0 compounds; and Tricritical mean-Field model (β = 0.25, γ = 1 and δ = 5) for SC. 4-4 one. The reliability of the critical exponent values was confirmed by the Widom scaling relation and the universal scaling hypothesis.