#### 3.3. Mössbauer Spectra

Figure 4 depicts the Mössbauer spectra of synthesized NiDy

_{x}Fe

_{2−x}O

_{4} (0.0 ≤

x ≤ 0.1) NPs at room temperature.

Table 2 enlists various Mössbauer parameters calculated by spectral fitting using three sextets (A for the tetrahedral sites and B and B

_{1} for the octahedral sites). Fe

^{3+} ions in the tetrahedral A site are characterized by a large hyperfine field with an insignificant isomer shift. Conversely, the other two sextets with a comparatively smaller hyperfine field signify the occupation of Fe

^{3+} at two dissimilar environments in the B-site [

17]. Besides the ferromagnetic sextets, a minute paramagnetic doublet with quadrupole-splitting was evidenced for NiDy

_{0.01}Fe

_{1.99}O

_{4}, NiDy

_{0.07}Fe

_{1.93}O

_{4}, and NiDy

_{0.1}Fe

_{1.9}O

_{4} NPs. The occurrence of such a paramagnetic doublet was attributed to the fractions of Fe

^{3+} with fewer nearest neighbors that possessed magnetically ordered spins. Interestingly, in the spinel ferrite structure, Fe

^{3+} did not contribute to the super exchange interaction [

18].

The achieved relative area for the A and B sites clearly indicated the occupation of Ni

^{3+} in the A and B sites. Kumar et al. acknowledged the preferential occupation of Dy

^{3+} in the octahedral B sites of Co-ferrites [

19,

20]. Thus, the cation distribution in the proposed Dy

^{3+} substituted nanoferrites was obtained following the formula unit of (Ni

_{y}Fe

_{1−y})

_{A} (Ni

_{1−y}Dy

_{x}Fe

_{1+y−x})

_{B}. The distribution of Fe

^{3+} over the A and B sites was observed to be relative to the proportional area of A and B in the Mossbauer sub-spectra.

Table 2 summarizes the approximate cation distribution obtained from the Mössbauer spectra. The results in

Table 2 revealed that Fe

^{3+} cations emigrated from the B site to A site due to Dy

^{3+} substitution. The line width of the A site was randomly altered, whereas for B and B

_{1} sites, it was enhanced with the substitution of Fe

^{3+} (0.64 Å) in the B sites by Dy

^{3+} (0.91Å) having larger ionic radii than the former one. This observation authenticated the increase in the degree of disorder due to the substitution on B sites.

The values of the hyperfine magnetic fields for the A and B sites (

Table 2) in the studied nanosized spinel ferrites were first reduced with an increase in Dy

^{3+} contents up to 0.05 and then enhanced at 0.07. Eventually, the hyperfine field for A site was continuously enhanced, but for the B site it was diminished. This alteration in the hyperfine field for the A and B sites was attributed to the addition of the diamagnetic Dy

^{3+} that replaced the ferromagnetic Fe

^{3+} with a higher magnetic moment (5 µ

_{B}) and lowered the average number of magnetic linkages (

${\mathrm{Fe}}_{\mathrm{A}}^{3+}-\mathrm{O}-{\mathrm{Fe}}_{\mathrm{B}}^{3+}$). Thus, Fe

^{3+} nuclei experienced a reduction in the magnetic field at both the sublattices up to the Dy3+ content of 0.05. Beyond 0.05, the number of Fe

^{3+} at the A site was augmented, thereby increasing the hyperfine magnetic field and magnetic moment of Fe

^{3+} at the A site.

#### 3.4. ZFC-FC Magnetizations

Figure 5 shows the curves of zero-field-cooled (ZFC) and field-cooled (FC) temperature dependencies of the magnetization, M

_{ZFC} (

T) and M

_{FC} (

T), of NiFe

_{2−x}Dy

_{x}O

_{4} (where

x = 0.00, 0.03 and 0.09) NPs. These measurements were performed in a temperature interval ranging between 2 and 400 K under a DC field of 100 Oe. For M

_{ZFC} (

T) measurements, the sample was cooled, first of all, from room temperature (RT) to a very low temperature in the absence of an applied field and subsequently the magnetization was recorded by increasing the temperature in the presence of the field. However, in the M

_{FC} (

T) measurements, the magnetization was recorded by cooling the product in the presence of applied field. A splitting and a large irreversibility between M

_{ZF} (

T) and M

_{FC} (

T) curves for different synthesized products can be clearly seen in

Figure 5. The M

_{FC} (

T) increased gradually and remained constant below temperature

${T}_{\mathrm{s}}$, while the M

_{ZFC} (

T) decreased with a lowering of the temperature down to about 4 K. The dispersion of M

_{ZFC}-M

_{FC} versus

T curves is congruent with the poly-disperse character of magnetic NPs, with a correlated distribution in particle size and individual anisotropy axes [

21]. The enlargement could also owe to dipolar interactions among particles [

21].

It is reported in the literature that the manifestation of a peak in the M

_{ZFC} (T) plots is associated to the blocking temperature (

T_{B}) [

22]. The curves of M

_{ZFC} (T) of the prepared products showed an incomplete maximum or broad maximum at around the temperature noted

T_{B}. This is typical for superparamagnet (SPM) materials, which show the properties of classical paramagnet materials (PM) above

T_{B}, where the total spin is equal to the spin of a whole NPs but behave as ferromagnetic (FM) materials below their blocking temperature (

T_{B}). Below

T_{B}, the M

_{FC} (

T) and M

_{ZFC} (

T) curves considerably diverge, and the various ferrite NPs are in the FM state (blocked state). Above

T_{B}, the M

_{FC} (

T) and M

_{ZFC} (

T) curves coincide, which is because all NPs are in the same SPM state. At

T =

T_{B}, the thermal activation overcomes the magnetic anisotropy barrier, which leads to fluctuations in magnetization [

23]. Therefore, the wide peak at

T_{B} in the M

_{ZFC} (

T) curves is an indication of a broadened energy barrier distribution. Further, it can be seen that the blocking temperature varies by increasing Dy substitution content. The non-substituted product NiFe

_{2}O

_{4} shows a blocking temperature around

T_{B} ≈ 390 K. The

x = 0.03 product exhibits a well-defined

T_{B} at around 300 K. By further increasing the amount of Dy, the product synthesized with

x = 0.09 was not able to reach the

T_{B} within 400 K, so the

T_{B} value is superior than 400 K for

x = 0.09. It can be seen clearly that the blocking temperature decreases for a lower Dy content (

x = 0.03) and then increases for higher content (

x = 0.09). The dependence of the

T_{B} on particles size has been reported in previous studies [

24]. The lower

T_{B} is attributed to smaller particle size or narrow size distribution. However, the higher

T_{B} represents a larger particle size. Nevertheless, the different

x = 0.00, 0.03 and 0.09 products show approximately same particles size. Therefore, the variations in blocking temperature with substitution effects are not predominantly influenced by the grain size. Thus, in addition to the particle size effect, the

T_{B} could also be affected by numerous other extrinsic factors, mostly related to interactions among particles and intrinsic factors that principally include a magneto–crystalline, surface and shape anisotropy [

22,

25]. We noticed in the synthesized product with

x = 0.03 that the M

_{ZFC} (

T) exhibits a breaking at temperatures indicated by the dashed circle in

Figure 5.

On the other hand, the M

_{FC} (

T) curves increase smoothly for different samples with a decrease of temperature, while a kind of saturation in the magnetization is noticed below the temperature noted by

${T}_{\mathrm{s}}$ for all samples. It is reported in the literature that for SPM nanoparticles, the curve of M

_{FC} (

T) increases continuously [

26,

27]. Nevertheless, in the case of super-spin glass (SSG) systems in which the interactions among particles are strong, a flat type or a slow increase is observed [

26,

27]. Therefore, the detected flat nature below

${T}_{\mathrm{s}}$ in the M

_{FC} (

T) curves establishes the occurrence of an SSG-like state. The origin of the observed magnetic features in ZFC-FC magnetization part will be discussed in detail in the analyses of the AC susceptibility measurements. The latter are a useful way to identify the freezing dynamics of the spin-glass (SG) materials.

#### 3.5. AC Susceptibility

For measurements of AC susceptibility (

${\chi}_{\mathrm{ac}}$), an AC magnetic field (

${H}_{\mathrm{ac}}$) is applied to the sample and, as a consequence, a resultant magnetic moment is measured. The

${\chi}_{\mathrm{ac}}$ is represented as follows:

where

χ’ real and

${\chi}^{\u2033}$ imaginary parts are, respectively, the in-phase and out-phase components of

${\chi}_{\mathrm{ac}}$. It should be noted that the relaxation time (

$\tau $) of the AC susceptibility measurement is not based upon the energy barrier (

${E}_{\mathrm{a}}={K}_{\mathrm{eff}}V$ where

${K}_{\mathrm{eff}}$ is the effective anisotropy constant and

$V$ is the volume of particles). However, it is influenced by the external excitation frequency. The AC susceptibility measurements give important details about the dynamics of the systems and the strength of exchange interactions between the magnetic nano-particles (MNPs) and between the different cations.

Firstly, we will discuss the

χ’ real part measurements of the two NPs samples with

x = 0.00 and 0.03.

Figure 6 presents the curves of

χ’versus

T, ranging from 350 to 2 K, for

x = 0.00 and 0.03 products, performed in the presence of an

${H}_{\mathrm{ac}}$ = 10 Oe and in a frequency range of 50–10

^{4} Hz. The magnitude of

χ’ for

x = 0.03 increased slightly compared to the non-substituted product (

x = 0.00), which is in accordance with M

_{ZFC} (T) and M

_{FC} (T) measurements. The in-phase AC susceptibility data of the different samples showed dispersion and a decrease in magnitude while increasing the applied frequency from 50 Hz to 10 kHz. The

χ’(

T) curve of the non-substituted NiFe

_{2}O

_{4} NPs exhibited a peak at around 300 K. However, the

x = 0.03 product did not show any peak up to 350 K.

Figure 7 shows the

${\chi}^{\u2033}\left(T\right)$ curves for NiFe

_{2-x}Dy

_{x}O

_{4} (where

x = 0.00 and 0.03) performed in a

${H}_{\mathrm{ac}}$ = 10 Oe and in the frequency range of 50–10

^{4} Hz. It can be seen that both samples display two peaks—the first at the higher temperature indicated by

T_{B} in the figure, which is associated with magnetic blocking of huge core spins, and the second indicated by

${T}_{\mathrm{s}}$, which can be associated with the spin-glass freezing on the surface of a single NP [

28]. The

χ’(

T) curves do not offer any information about these two peaks, hence from now on we will focus only on analyses of

${\chi}^{\u2033}\left(T\right)$ curves.

At the same applied frequency, the

T_{B} and

${T}_{\mathrm{s}}$ shifted to lower temperatures with Dy substitution compared to the non-substituted one. This is consistent with the M

_{ZFC} analyses. Both the blocking temperature

T_{B} and spin-glass freezing temperature

${T}_{\mathrm{s}}$ are affected by frequency. Both show a shift to higher temperatures upon increasing the value of the applied frequency (

$f$). Similar behavior is observed in the spin-frustrated system of CoFe

_{2}O

_{4} NPs dispersed in an SiO

_{2} matrix [

28]. The shifting with

f is helpful for evaluating dynamic magnetic behaviors, deducing the anisotropic energy, the magnetic anisotropy, and the interaction strength between MNPs.

Various physical laws can be used to investigate the f-dependence shift of

T_{B} and

${T}_{\mathrm{s}}$ temperatures. The Neel–Arrhenius (N–A) law was first tested to fit the experimental data (

Figure 8). This theory is valid for thermal excitations of non-interacting single-barrier NPs and is expressed as follows [

29,

30]:

where

$\tau =1/f$ is the measured time,

${\tau}_{0}$ is the jump attempt time (in the range of 10

^{−9}–10

^{−}^{13} s),

${k}_{\mathrm{B}}$ is the Boltzmann constant, and

${E}_{\mathrm{a}}={K}_{\mathrm{eff}}V$ is the activation energy barrier. The estimated values of

${\tau}_{0}=1/{f}_{0}$,

${E}_{\mathrm{a}}/{k}_{\mathrm{B}}$ and

${K}_{\mathrm{eff}}$ for different samples are given in

Table 3. The best N–A fit offers very unreasonable values for

${\tau}_{0}$ and

${E}_{\mathrm{a}}/{k}_{\mathrm{B}}$. This indicates that the synthesized products do not obey the thermally activated N–A law and, as a consequence, they are non-interacting.

The Vogel–Fulcher (V–F) law is a useful model for investigating the interactions between NPs. This law uses an additional parameter,

${T}_{0}$, that represents the strength of inter-particle interactions. Based on this model, the relaxation is described as follows [

29,

30]:

The fitting data using the V–F law of the plots of

$f$ vs.

T_{B} and

$f$ vs.

${T}_{\mathrm{s}}$ for the prepared products are illustrated in

Figure 9a,b, respectively. The different estimated parameters are summarized in

Table 3. The analysis of f-dependent

T_{B} now gives reasonable

${\tau}_{0}$ and

${E}_{\mathrm{a}}/{k}_{\mathrm{B}}$ values. Obviously, the

T_{0} values are not negligible compared to

T_{B}. The occurrence of

T_{0} confirms the presence of moderate inter-particle interactions between the NPs [

28,

29,

30]. It is found, moreover, that

${\tau}_{0}$ increased more for the

x = 0.03 product than for the x = 0.00 one. The increase in the

${\tau}_{0}$ for

x = 0.03 product suggests the strengthening of interactions between NPs [

29,

30]. Compared to the

x = 0.00 product, the values of

${E}_{\mathrm{a}}/{k}_{\mathrm{B}}$ and

${K}_{\mathrm{eff}}$ improved with Dy substitution for

x = 0.03. This improvement iresulted from the strengthening of magnetic interactions among different NPs and the increase of magnetic anisotropy sources [

29,

30].

In other hand, the investigation of

f vs.

${T}_{\mathrm{s}}$ provides unphysical values for

${\tau}_{0}$. Therefore, the critical slowing down (CSD) law is used to study the presence of SG behavior in the synthesized NPs. Based on this model, the relaxation is expressed as [

30]:

where

${\tau}_{0}^{*}$ is associated to the coherence time of coupled individual “atomic” spins in the NP (in the range 10

^{−6}–10

^{−13} s) [

31],

${T}_{\mathrm{g}}$ is the SG freezing temperature, and

${T}_{\mathrm{s}}$ is the

f-dependent freezing temperature. The

$\u201cz\upsilon \u201d$ is the critical exponent that offers information about the SG, and it varies from 4 to 12 for various SG systems [

28]. We fit the same

f vs.

${T}_{\mathrm{s}}$ data using the CSD law, in order to examine the possibility of the SG nature (

Figure 10). The various deduced parameters are listed in

Table 3. The obtained reasonable values of

${\tau}_{0}$,

${T}_{\mathrm{g}}$ and

$\u201cz\upsilon \u201d$ proved the existence of SG behavior in the prepared samples. Similar comportment has been reported in numerous products, such as CoFe

_{2}O

_{4}/(SiO

_{2})

_{x} systems [

28], Fe

_{3}O

_{4} MNPs (

$z\upsilon =8.2$ and

$\tau ~{10}^{-9}s$) [

32,

33], soft ferrite Ni

_{0.3}Zn

_{0.7}Fe

_{2}O

_{4} NPs (

$z\upsilon =8.01$ and

$\tau ~{10}^{-12}s$) [

34], and La

_{0.9}Sr

_{0.1}MnO

_{3} NPs [

35]. It is reported that the strength of magnetic interactions increases based on the decreasing

$\u201cz\upsilon \u201d$ exponent. The non-substituted NiFe

_{2}O

_{4} product exhibits a

$\u201cz\upsilon \u201d$ value equal to 5.11, and it decreases to 3.95 with Dy substitution for

x = 0.03. This result indicates the improvement of the magnetic interactions among NPs for the

x = 0.03 product.