#
Unveiling the Hidden Entropy in ZnFe_{2}O_{4}

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{8}

^{9}

^{*}

## Abstract

**:**

_{2}O

_{4}has been intensively investigated with results showing a lack of long-range order, spin frustrations, and a “hidden” entropy in the calorimetric properties for inversion degrees δ ≈ 0 or δ = 0. As δ drastically impacts the magnetic properties, it is logical to question how a δ value slightly different from zero can affect the magnetic properties. In this work, (Zn

_{1-δ}Fe

_{δ})[Zn

_{δ}Fe

_{2-δ}]O

_{4}with δ = 0.05 and δ = 0.27 have been investigated with calorimetry at different applied fields. It is shown that a δ value as small as 0.05 may affect 40% of the unit cells, which become locally ferrimagnetic (FiM) and coexists with AFM and spin disordered regions. The spin disorder disappears under an applied field of 1 T. Mossbauer spectroscopy confirms the presence of a volume fraction with a low hyperfine field that can be ascribed to these spin disordered regions. The volume fractions of the three magnetic phases estimated from entropy and hyperfine measurements are roughly coincident and correspond to approximately 1/3 for each of them. The “hidden” entropy is the zero point entropy different from 0. Consequently, the so-called “hidden” entropy can be ascribed to the frustrations of the spins at the interphase between the AFM-FiM phases due to having δ ≈ 0 instead of ideal δ = 0.

## 1. Introduction

_{1-δ}Fe

_{δ})

^{A}[Zn

_{δ}Fe

_{2-δ}]

^{B}O

_{4}, where A and B represent, respectively, the tetrahedral and the octahedral sites of the cubic structure and δ is the inversion degree parameter. There is a general agreement in the scientific community that δ plays the fundamental role in the magnetic properties of (Zn

_{1-δ}Fe

_{δ})

^{A}[Zn

_{δ}Fe

_{2-δ}]

^{B}O

_{4}with δ ≠ 0 [1,2,3,4]. For δ = 0, the ferrite is denominated normal, and it has been predicted to have an antiferromagnetic (AFM) transition around 10 K [5]. For δ ≠ 0, the zinc ferrite becomes ferrimagnetic (FiM) with magnetization that depends on δ as M = δ·(5.9 µ

_{B}) at 5 K [4] and can reach high magnetization values even at room temperature.

_{B}at the octahedral sites; a positive Curie–Weiss temperature and a magnetic entropy which is half that of the theoretical one. This reduced entropy has been observed in other spinel structures [15,19,20], and has been described as a “hidden” entropy, an entropy different from 0 at the zero point entropy [19]. Since in most of the experimental works on normal spinel, it is assumed δ ≈ 0 [21,22,23,24], the main question is: how critical is it to have δ ≈ 0 instead δ = 0.

## 2. Materials and Methods

^{3}capacity jar and 1 cm diameter balls stainless steel, at a rotating speed of 275 rpm. This milled sample was then subjected to an annealing temperature of 400 °C for 1 h in order to decrease the inversion degree. The sample name was ZFO-0.27.

_{exp}), the weighted summation of residual of the least-squares fit (R

_{wp}), and the goodness of fit (GoF or chi-square, whose limit tends to 1) [29].

^{57}Co(Rh) source, and a He closed-cycle cryorefrigerator.

## 3. Results and Discussion

_{wp}= 3.27, R

_{exp}= 2.65, and GoF = 1.26).

^{3+}magnetic ions are also occupying A sites. In addition, peak broadening can occur by reducing the crystallite size and/or increasing the lattice strain. Besides differences in the height and shape of the diffraction peaks associated with these two factors, it can be observed a broad peak at a smaller angle for the sample with δ = 0.05 (Figure 2), corresponding to the (1 0 ½) reflection, which indicates the presence of a short-range order (SRO) antiferromagnetic (AFM) order. The area of this peak decreases with increasing δ. The broadness of the peak at (1 0 ½) for δ = 0.05 describes a picture of lack of AFM long-range order that has also been reported by other authors [9].

_{T}) between 2 and 40 K is the sum of both the magnetic (C

_{m}) and vibrational (C

_{L}) heat capacities, while at 40 K, there are only vibrational contributions.

_{D}is the Debye temperature, and T the temperature. By subtracting the vibrational contribution from the total heat capacity, the magnetic contribution to the entropy can be calculated by integrating the C/T curve.

_{T}, magnetic ΔS

_{m,}and lattice ΔS

_{L}contributions to the entropy for δ = 0.05 at H = 0 T, and Table 2 collects the values at 40 K at different applied fields. The entropy increment for the sample with δ = 0.27 is field-independent.

^{3+}. The difference between the theoretical and the experimental value contains information about additional contributions to the entropy, and it is defined as the zero-point entropy at 0 K [19,20]. It is important to remark that for (Zn

_{1-δ}Fe

_{δ})

^{A}[Zn

_{δ}Fe

_{2-δ}]

^{B}O

_{4}, the parameter δ remains constant in the whole temperature range for each sample. Consequently, the entropy changes at low temperatures must be ascribed to an increment in the magnetic contribution and, to a lesser extent, to the lattice vibrational effect.

^{3+}in the B site. Consequently, the first neighboring Fe

^{3+}atoms in the B sites (with 4 Fe

^{3+}per site) surrounding the A site occupied by a Fe

^{3+}experience some kind of frustration due to the two competing AFM and FiM interactions. Figure 5 shows a scheme of this situation.

_{c}) is either 0 or 1/8 = 0.125, the last one corresponding to the exchange of a single Zn/Fe pair in the unit cell. Assuming that only a single Zn/Fe exchange can occur in a cell, around 40% of the unit cells suffer a single Zn/Fe cations exchange when the macroscopic δ is as small as 0.05. Those unit cells with δ

_{c}= 0.125 are FiM (40%) with a magnetic moment of 5.9 μ

_{B}, [4] whereas for δ

_{c}= 0 the cells are AFM (60%). This pictures the dramatic effect that δ ≈ 0 can have over the magnetic and calorimetric properties.

_{m}= 8.7 J/mol·K, which can be associated with AFM to PM transition. Assuming that for δ=0.05 a 60% of the sample is AFM, the expected ΔS

_{m}is close to $0.6\xb72RLn(2J+1$) ≈ 18 J/mol·K. Therefore, the small experimental value of 8.7 J/mol·K indicates that only 29% of the sample has evolved from AFM to the paramagnetic phase. In summary, instead of the expected 60%, only 29% of the sample seems to be AFM ordered.

_{m}could be ordered with transition temperature above 40 K or disordered in the whole 0–40 K range. It is normally associated with a hidden or missing entropy observed in other spinels [19,20].

_{m}rises up to 17.1 J/mol·K, indicating that a volume fraction of 57% has evolved from 0 entropy to 29.7 J/mol·K. This 57% is close to the calculated AFM fraction of 60%. At 40 K, only contributions to the entropy of the lattice or a magnetically ordered phase are expected; therefore, both experimental curves, with and without applied field, are matched at this temperature (Figure 6). As can be seen, the effect of the field is shown to promote a decrease of the low-temperature magnetic entropy and also to raise its increasing rate nearby 0 K. According to this result, at zero applied field, a fraction of the spins seems to be disordered and thereby contributing in a small amount to the entropy increment, so giving rise to the “hidden entropy.” However, at low temperatures, the field gradually orders these disordered spins, contributing to the magnetic entropy increment when the temperature rises and the PM phase is achieved. In conclusion, the effect of the field allows us to unravel the hidden entropy origin as the spin disordered volume fraction vanishes.

_{m}(0T, 40K) = 8.7 J/mol·K, which corresponds to a volume fraction of 29% of AFM state. (b) The contribution resulting from the spin disordered regions is field-dependent. The volume fraction of the spin disordered regions can be inferred by subtracting ΔS

_{m}(0T, 40K) = 8.7 J/mol·K to ΔS

_{m}(1T, 40K) = 17.1 J/mol·K, that leads to a spin disordered contribution of 8.4 J/mol·K, corresponding to a volume fraction of 28% of the spin disordered state. This volume fraction of spin disordered Fe

^{3+}is expected to be located at the FiM-AFM interphase. The remaining 43% fraction is expected to have a FiM ordering with Curie temperature above the 40 K, [30]; therefore, this fraction does not contribute to the entropy increment.

_{m}(µ

_{0}H,T) for H = 1 T and H = 9 T, according to the expression:

_{0}H is the applied magnetic field.

_{m}(T) at 9 T with respect to ΔS

_{m}(T) at 1T observed experimentally (see Figure 2), as illustrated by Figure 7.

^{3+}in octahedral oxygen coordination: δ

_{1}= 0.34 mms

^{−1}, Δ

_{1}= 0.26 mms

^{−1}and δ

_{2}= 0.34 mms

^{−1}, Δ

_{1}= 0.57 mms

^{−1}. The 77 K spectrum still shows a paramagnetic doublet. The 8.8 K spectrum, however, shows a broad unresolved magnetic pattern (Figure 8). This spectrum, characteristic of a system experiencing magnetic relaxation, indicates that the measurement temperature is close to the magnetic ordering temperature of this particular zinc ferrite sample. As mentioned previously, it is known that the critical temperature of well-crystallized, canonical zinc ferrite (which should be a direct spinel) is close to 10 K and that this temperature increases if it is partially inverse [31]. Thus, the present result is compatible with a zinc ferrite sample having a very small inversion degree, as is the case.

^{3+}(B)-Fe

^{3+}(B) AFM exchange interaction was present. Instead, the spectrum was best-fitted using three different contributions characterized by hyperfine magnetic fields of 46.6 T, 37.2 T, and 26.7 T accounting for 34%, 29%, and 37% of the spectral area, respectively. We want to mention that because of the small inversion degree of the sample and the complexity of the 8.8 K spectrum, the isomer shifts of the various components fitted do not appear to be sensitive to differences in the Fe

^{3+}cation coordination, being around δ = 0.50 mms

^{−1}(i.e., mainly characteristic of octahedral sites) in the three cases.

_{m}= 5.5 J/mol·K. In this condition, the volume fraction of the sample contributing to ΔS

_{m}is 18%, much smaller than the corresponding sample with δ = 0.05, which was 57%. The presence of long-range order is confirmed by the macroscopic FiM associated with the decrease of the area enclosed by the halo of the NPD experiment that has almost vanished.

## 4. Conclusions

^{3+}migrated in the A positions of the spinel produces a sufficient disturbance in the magnetic order.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Diffraction patterns recorded at room temperature for the sample with δ = 0.27 and δ = 0.05.

**Figure 3.**(

**A**) C/T for sample ZFO-0.05 (black line), ZFO-0.27 (red line), and lattice contribution C

_{L}/T (continuous line); (

**B**) Entropy increase at different applied magnetic fields for ZFO-0.05; (

**C**) for ZFO-0.27.

**Figure 4.**ΔS for sample with δ = 0.05, where the magnétic (red line), vibrational (blue line) and total entropy (black line) at H = 0 are shown separately.

**Figure 5.**(

**A**) Half-cells of ZnFe

_{2}O

_{4}with δ = 0, Fe

^{3+}(red circles), and Zn

^{2+}(green circles) in their corresponding octahedral (cells 1, 4) and tetrahedral (cells 2, 3) sites. Blue circles are oxygen. The black arrows indicate the magnetic moments. (

**B**) A pair of Zn-Fe cations interchanged their sites; the stronger AFM A-B super exchange interaction leads to an FM order in the B sites and promotes some kind of frustration in the first neighbor’s B sites (shadow circle).

**Figure 6.**Entropy increment of ZFO-0.05 without applied field lifts up at 10 K regarding the curve at 1 T at the same temperature.

**Figure 7.**Calculated entropy increase for the paramagnetic region under a magnetic field of 0 T (black line), 1 T (blue line), and 9 T (red line).

**Figure 9.**(

**A**) Illustration of magnetic arrangement of ZFO-0.05 sample, white circles representing ferrimagnetic particles, the blue area is AFM converted to PM at 40 K, and the green crown represents the disordered interphase between AFM and FM regions; (

**B**) Different orientation of interphase area depending on the applied field (0, 1 T), and temperature; (

**C**) Magnetic arrangement of ZFO-0.27 with percolating FM clusters and blue AFM regions.

**Table 1.**Microstructural parameters obtained from the Rietveld refinement of the diffraction patterns recorded using XRD and NPD.

Sample | Source | Lattice Parameter (Å) | Inversion Degree(δ) | O-Position (x = y = z) | Crystal Size (nm) | μ-Deformation (ε) |
---|---|---|---|---|---|---|

ZFO-0.05 | XRD | 8.4489(5) | 0.05(1) | 0.2416(9) | >150 | - |

NPD | 8.4498(5) | 0.05(1) | 0.2397(3) | >150 | - | |

ZFO-0.27 | XRD | 8.4322(5) | 0.28(2) | 0.2424(5) | 15(1) | 0.0020(2) |

NPD | 8.4373(5) | 0.20(2) | 0.2414(3) | 14(1) | 0.0019(2) |

**Table 2.**Total, magnetic, and lattice entropy increment (expressed in J/mol·K) for δ = 0.05 at 40 K.

H (T) | 40 K | ||
---|---|---|---|

ΔS_{T} | ΔS_{m} | ΔS_{L} | |

0 | 13.2 (1) | 8.7 (1) | 4.6 (1) |

1–5 | 21.7 (1) | 17.1 (1) | 4.6 (1) |

9 | 18.9 (1) | 14.2 (1) | 4.7 (1) |

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**MDPI and ACS Style**

Cobos, M.A.; Hernando, A.; Marco, J.F.; Puente-Orench, I.; Jiménez, J.A.; Llorente, I.; García-Escorial, A.; de la Presa, P.
Unveiling the Hidden Entropy in ZnFe_{2}O_{4}. *Materials* **2022**, *15*, 1198.
https://doi.org/10.3390/ma15031198

**AMA Style**

Cobos MA, Hernando A, Marco JF, Puente-Orench I, Jiménez JA, Llorente I, García-Escorial A, de la Presa P.
Unveiling the Hidden Entropy in ZnFe_{2}O_{4}. *Materials*. 2022; 15(3):1198.
https://doi.org/10.3390/ma15031198

**Chicago/Turabian Style**

Cobos, Miguel Angel, Antonio Hernando, José Francisco Marco, Inés Puente-Orench, José Antonio Jiménez, Irene Llorente, Asunción García-Escorial, and Patricia de la Presa.
2022. "Unveiling the Hidden Entropy in ZnFe_{2}O_{4}" *Materials* 15, no. 3: 1198.
https://doi.org/10.3390/ma15031198