**3. Scaling analysis**

The following power law relation was cited by the scaling hypothesis near the critical region defined by:

$$\mathcal{M}\_{\varepsilon}(T) = \mathcal{M}\_0(-\varepsilon)^{\beta}, \varepsilon < 0 \tag{1}$$

$$\chi\_0^{-1}(T) = \left(\frac{h\_0}{M\_0}\right)\varepsilon^{\prime}, \varepsilon > 0\tag{2}$$

$$M = DH^{\frac{1}{5}}, \mathbf{e} = \mathbf{0} \tag{3}$$

In which *M*0, *h*<sup>0</sup> and *D* represent the critical amplitudes while *ε* stands for the reduced temperature *ε* ¼ ð Þ *T* � *T*<sup>c</sup> *=T*c. Besides, within the asymptotic critical zone, line with the prediction of the scale formula, the equation of magnetic state is written in the following form:

$$\mathcal{M}(\mu\_0 H, \varepsilon) = \varepsilon^{\beta} \mathfrak{f} \pm \left(\frac{\mu\_0 H}{\varepsilon^{\beta + \gamma}}\right) \tag{4}$$

In this case, *ʄ* � and *ʄ* <sup>þ</sup> represent regular analytical functions and above *T*<sup>c</sup> [20, 21]. This last Eq. (4) shows that for the correct choice of the values of *β*, *γ* and *δ* as well as true scaling relations, the scaled *<sup>M</sup>=*j j *<sup>ε</sup> <sup>β</sup>* plotted as a function of the scaled *<sup>μ</sup>*0*H=*j j *<sup>ε</sup> <sup>β</sup>*þ*<sup>γ</sup>* will fall in two universal curves for *<sup>T</sup>* <sup>&</sup>gt; *<sup>T</sup>*cð Þ *<sup>ε</sup>*<sup>&</sup>gt; <sup>0</sup> and the other for *T* <*T*<sup>c</sup> ð Þ *ε*<0 [22]. This makes possible to have a fairly important critical regime criterion.


#### **Table 1.**

*Critical exponents of the La0.67Ca0.18Sr0.15Mn0.98Ni0.02O3 specimen compared with different theoretical models and previous manganite results.*

More generally, it is possible to point to four models that are based on the critical exponent values; Mean-field model (*β* = 0.5 and *γ* = 1), 3D-Heisenberg model (*β* = 0.365 and *γ* = 1.336), 3D-Ising model (*β* = 0.325 and *γ* = 1.24), in addition to Tricritical mean-field model (*β* = 0.25 and *γ* = 1). The different models can be seen below in **Table 1**.

### **4. Critical behavior**

More generally, it is necessary to understand the way in which the ferromagneticparamagnetic ("FM-PM") phase transition occurs (either first or second order), according to scaling postulate. The magnetic system whose phase transition behavior of second order in the vicinity of the Curie temperature spot is driven by an ensemble of interdependent critical exponents [19] *β* (linked to the spontaneous magnetization *MS*), *γ* (associated with the starting magnetic susceptibility *χ*0), *δ* (linked to the critical magnetization isotherm to *T*c). As already known, it is impossible to find the critical exponents in case of a first-order ferromagnetic phase transition; in fact, the external magnetic field allows this transition to be shifted, leading to a phase depending on the field strength *Tc*ð Þ *H* [27]. Magnetization measurement exponents are mathematically defined and given us by the following relationships:

$$\mathcal{M}\_{\mathfrak{s}}(T) = \mathcal{M}\_0 |\varepsilon|^\beta \quad , \qquad \varepsilon < 0, \qquad T < T\_{\mathfrak{c}} \tag{5}$$

$$\chi\_0^{-1}(T) = \left(\frac{h\_0}{\mathcal{M}\_0}\right)\varepsilon^{\mathcal{I}}, \qquad \varepsilon > 0, \qquad T > T\_\varepsilon \tag{6}$$

$$\mathbf{M} = \mathbf{D} \mathbf{H}^{1/\delta}, \qquad \qquad \varepsilon = \mathbf{0}, \qquad T = T\_{\mathbf{c}} \tag{7}$$

Where ? represents the reduced temperature ð Þ *T* � *T*<sup>c</sup> *=T*c, and *M*0, *h*0*=M*0, and *D*, correspond to the critical amplitudes.

In general, both the critical exponents and the critical temperature may be found easily from the Arott curve. From the Arott-Noakes state equation, ð Þ *<sup>H</sup>=<sup>M</sup>* <sup>1</sup>*=<sup>γ</sup>* <sup>¼</sup> ð Þ *T* � *T*<sup>c</sup> *=T*<sup>c</sup> þ ð Þ *M=M*<sup>1</sup> <sup>1</sup>*=<sup>β</sup>* [28], where *M*<sup>1</sup> represents a material constant, the Normal Arrott has demonstrated that the relationship between *M*<sup>2</sup> as a function of *H*/*M* was essentially a function of the average field model in terms of the critical exponent *β* = 0.5 and *γ* = 1.0. As a result, the *M*<sup>2</sup> versus *H*/*M* curves must exhibit linear behavior around *T*<sup>c</sup> as well as the line at *T* = *T*<sup>c</sup> must exactly get through the origin. In addition, one can determine the magnetic transition order using the slope of those lines, following Banerjee's criterion [29]. From these straight lines, we can identify the magnetic transition order. Given that as a result, there is a positive slope associated with the second-order transition, whereas the negative slope is related to the first-order transition.

**Figure 1** displays the Arrott plot *M*<sup>2</sup> versus *H*/*M* for the La0.67Ca0.18Sr0.15Mn0.98 Ni0.02O3 sample at the temperature range close to *T*c. Clearly, currently, the positive slope of curves *M*<sup>2</sup> versus *H*/*M* curves allows to the conclusion of a second order ferromagnetic phase transition. However, the nonlinearity and the appearance of increasing curvature of all Arrott traces, even at high fields, which means that the average field theory is not, satisfied the current phase transition.

Most of the time, charge effect, differential orbital degrees of freedom and lattice, which exists in high field regions, are removed in a ferromagnetic so that the ordering *Study of the Critical Behavior in La0.67Ca0.18Sr0.15Mn0.98Ni0.02O3 Manganite Oxide DOI: http://dx.doi.org/10.5772/intechopen.105053*

**Figure 1.** *Typical pattern of* M*<sup>2</sup> versus μ*0H*/*M *traces, for La0.67Ca0.18Sr0.15Mn0.98Ni0.02O3 compound.*

parameter is usually confused with macroscopic magnetization. Thus, in order to achieve the correct values *β* and *γ*, a modified Arrott trace requires producing quasilinear lines of the *M*<sup>2</sup> versus *H*/*M* plots. As illustrated by the **Figure 2(a)–(c)**, it was found that three types of test exponents of tricritical mean field (*β* = 0.25, *γ* = 1.0), 3D Ising (*β* = 0.325, *γ* = 1.24) as well as 3D Heisenberg model (*β* = 0.365, *γ* = 1.336) were employed in order to create a modified Arrott plot.

To match these results, the related slopes (RS) were calculated, given by *RS* ¼ *S T*ð Þ*=S T*ð Þ <sup>c</sup> ¼ 319 *K* . Thus, the relative slopes must be kept at 1 apart from the temperatures, whether the modified Arrott graph displays a set which are absolute parallels.

As already seen in **Figure 3**, the mean field RS, the 3D-Heisenberg and the Tricritical mean filed certainly deviate from the RS = 1 straight line; however the RS of the 3D-Ising model approximated it. As a result, Arrott's third plot shows the best result, amongst these three patterns, reporting that the critical properties of the La0.67Ca0.18Sr0.15Mn0.98Ni0.02O3 specimen can be depicted using the 3D-Ising model.

Subsequently, in the **Figure 2(b)**, the linear extrapolation from the high field region two interceptions with axes ð Þ *<sup>H</sup>=<sup>M</sup>* <sup>1</sup>*=<sup>γ</sup>* and ð Þ *<sup>M</sup>* <sup>1</sup>*=<sup>β</sup>* gives credible values of inverse susceptibility *χ*�<sup>1</sup> <sup>0</sup> ð Þ *T*, 0 and spontaneous magnetization *Ms*ð Þ *T*, 0 , respectively. These temperature dependent values, *χ*�<sup>1</sup> <sup>0</sup> ð Þ *T*, 0 as a function of T and *Ms*ð Þ *T*, 0 versus *T*, are displayed in **Figure 4**. As already mentioned by Eqs (5) and (6), based on the experimental results (open line) may be adjusted to two solid graphs (continuous line). It allows to give two novel cases in *β* = 0.320 with *T*<sup>c</sup> ¼ 318*:*300 *K* and *γ* = 1.296 with *T*<sup>c</sup> ¼ 317*:*819 *K*. Therefore, these findings are very similar to the 3D-Ising model.

Susceptible, these critical exponents and *T*<sup>c</sup> may be more accurately determined by the Kouvel-Fisher (KF) approach [20]:

$$\frac{M\_s(T)}{\text{d}M\_s(T)/\text{dT}} = \frac{T - T\_c}{\beta} \tag{8}$$

**Figure 2.** *Modified Arrott graphs: (*M*1/*<sup>β</sup> *versus (μ*0H*/*M*)1/*<sup>γ</sup> *) over compound La0.67Ca0.18Sr0.15Mn0.98Ni0.02O3.*

$$\frac{\chi\_0^{-1}(T)}{\mathrm{d}\chi\_0^{-1}(T)/\mathrm{d}T} = \frac{T - T\_\mathrm{c}}{\gamma} \tag{9}$$

By conforming to this method, *Ms*ð Þ <sup>d</sup>*Ms=*d*<sup>T</sup>* �<sup>1</sup> as a function of T with *χ*�<sup>1</sup> <sup>0</sup> d*χ*�<sup>1</sup> <sup>0</sup> *<sup>=</sup>*d*<sup>T</sup>* �<sup>1</sup> as a function of *<sup>T</sup>* is expected to give straight lines of slopes 1/*<sup>β</sup>* and *Study of the Critical Behavior in La0.67Ca0.18Sr0.15Mn0.98Ni0.02O3 Manganite Oxide DOI: http://dx.doi.org/10.5772/intechopen.105053*

**Figure 3.** *Variation of RS with temperature of specimen La0.67Ca0.18Sr0.15Mn0.98Ni0.02O3.*

#### **Figure 4.**

M*<sup>S</sup> (left) with χ*�<sup>1</sup> <sup>0</sup> *(right) versus temperature for La0.67Ca0.18Sr0.15Mn0.98Ni0.02O3 specimen (solid lines are model fits).*

1/*γ*, respectively. *T*-axis intercepts fit correctly to *T*<sup>c</sup> if those lines are extrapolated to the zero ordinate. As shown by **Figure 5**, adjustment results using KF method yield the exponents as well as *T*<sup>c</sup> to the deviation: *β* = 0.324 with *T*<sup>c</sup> ¼ 319*:*900 *K* and *γ* = 1.238 with *T*<sup>c</sup> ¼ 317*:*120 *K*.

Certainly, using the KF method, the resulting critical exponent values and *T*<sup>c</sup> and the values obtained by using the modified Arrott of tricritical mean-field model are in agreement. For a better verification on the dependability as regards the above critical exponents, we may consider how the three critical exponents *β*, *γ*, and *δ* relate to each other.

**Figure 5.** *K–F graphs of La0.67Ca0.18Sr0.15Mn0.98Ni0.02O3 specimen (solid lines represent model fits).*

In this place, first we should find out *δ* value. By the terms of Eq. (7), we can directly get *δ* value by tracing the critical isotherm at *T*c. According to **Figure 6**, we distinguished *M* versus *H* plot at 319 K as being critical isothermal from the above discussion. Moreover, it is illustrated through the slide of **Figure 6**, which displays the graph on *M* versus *H* in the form of a log–log scale. Using Eq. (7), there is an appropriate result of the full straight edge and a slope of 1/*δ* is obtained. Based on a straight-line fit, a third critical exponent *δ* = 4.863 was derived. Following statistical theory, all three of these critical exponents should complete Widom's scale relationship:

$$\delta = \mathbf{1} + \frac{\mathbf{y}}{\beta} \tag{10}$$

Using the above obtained data *β* and *γ* As a result, Eq. (10) provides values of *δ* = 4.965 *β* and *γ* as obtained from **Figure 4**, as well as *δ* = 4.820 *β* and *γ* as evaluated using **Figure 5**. Respectfully, the above values are very similar to the ones calculated using the critical isotherm. As a result, the above relationship was proved with a plot of *M T*ð Þ <sup>¼</sup> *<sup>T</sup>*<sup>0</sup> as a function of *<sup>μ</sup>*0*Hβ β*ð Þ <sup>þ</sup>*<sup>γ</sup>* <sup>¼</sup> *<sup>μ</sup>*0*H*<sup>1</sup>*=<sup>δ</sup>* and proving that the curve is linear as illustrated in **Figure 6**.

Given that two critical exponents *δ* are similar to *δ* estimated based on critical isotherms near *T*c. As a result, the critical exponents obtained in this study absolutely obey the Widom scaling relationship, which entails the implication that both *β* and *γ* resultant data agree. In the critical region, the magnetic equation is expressed as follows:

$$\mathcal{M}(\mu\_0 \mathcal{H}, \varepsilon) = \varepsilon^{\beta} f \pm \left(\mu\_0 \mathcal{H} / \varepsilon^{\beta + \gamma}\right) \tag{11}$$

Where *<sup>ʄ</sup>* <sup>þ</sup> for *<sup>T</sup>* <sup>&</sup>gt;*T*<sup>c</sup> and *<sup>ʄ</sup>* � for *<sup>T</sup>* <sup>&</sup>lt;*T*<sup>c</sup> are regular functions [30]. As indicated by the Eq. (11), *Mε*�*<sup>β</sup>* as a function of *Hε*�ð Þ *<sup>β</sup>*þ*<sup>γ</sup>* leads to two universal plots: The first plot *Study of the Critical Behavior in La0.67Ca0.18Sr0.15Mn0.98Ni0.02O3 Manganite Oxide DOI: http://dx.doi.org/10.5772/intechopen.105053*

**Figure 6.**

*M versus μ*0*H graph for La0.67Ca0.18Sr0.15Mn0.98Ni0.02O3 specimen that occurs on* T*<sup>C</sup> = 319 K. the inset the enclosed shows the similar graph scaled to ln–ln.*

for temperature *T* >*T*<sup>c</sup> and the second plot for temperature *T* < *T*c. Therefore, there is a comparison between the obtained results and the scaling theory prediction by Eq. (11). As already seen in **Figure 7**, both experimental data fall on two curves, one over *T*<sup>c</sup> and one under *T*c, by correspondence with the scaling theory. This same graph is shown inside the inset in **Figure 7** in the form of a log–log scale. Same, all

**Figure 7.** *Renormalized magnetization M=*j j *<sup>ξ</sup> <sup>β</sup> versus <sup>μ</sup>*0*H=εβ*þ*<sup>γ</sup> for the La0.67Ca0.18Sr0.15Mn0.98Ni0.02O3 sample.*

points fall in two different curves. We can conclude from these results that the obtained values of the critical exponents and those of *T*<sup>c</sup> are reliable. On the other hand, there is accurate characterization of critical properties in our current system with the 3D-Ising model.
