Physical properties of RIr3 (R = Gd, Tb, Ho) compounds with coexisting polymorphic phases

TL;DR

This study investigates the magnetic and thermal properties of GdIr3, TbIr3, and HoIr3 compounds, revealing coexisting polymorphic phases with distinct magnetic behaviors, including ferromagnetism and spin-glass states, through comprehensive experimental analysis.

Contribution

It provides the first detailed analysis of the magnetic critical behavior and phase coexistence in RIr3 compounds with polymorphic phases, including critical exponents and magnetic class identification.

Findings

GdIr3 is ferromagnetic and lacks glassy features.

TbIr3 and HoIr3 exhibit spin-glass behavior.

Critical exponents classify the compounds into a specific magnetic universality class.

Abstract

The binary compounds GdIr3, TbIr3 and HoIr3 are synthesized successfully and found to form in macroscopic co-existence of two polymorphic phases: C15b and AuCu3-type. The dc magnetization and heat capacity studies confirm that C15b phase orders ferromagnetically, whereas the AuCu3 phase remains paramagnetic down to 2 K. The frequency dependent ac-susceptibility data, time dependent magnetic relaxation behavior and magnetic memory effect studies suggest that TbIr3 and HoIr3 are cannonical spin-glass system, but no glassy feature could be found in GdIr3. The critical behavior of all the three compounds has been investigated from the magnetization and heat capacity measurements around the transition temperature (TC). The critical exponents alpha, beta, gamma and delta have been estimated using different techniques such as Arrott-Noaks plot, Kouvel-Fisher plot, critical isotherm as well as analysis of specific heat data and study of magnetocaloric effect. The critical analysis study identifies the type of universal magnetic class in which the three compounds belong.

Figures

Tables

Table 1. Table 1: Crystallographic and fitting parameters from Rietveld refinement of XRD data of R Ir 3 ( R = Gd, Tb, Ho) compounds

Compound	Phase	a(Å)	RBragg	Rf
GdIr3	C15b (74%)	7.5495(1)	15.9	11.3
	AuCu3 (26%)	3.7917(3)	13.6	19.1
TbIr3	C15b (73%)	7.5267(1)	9.72	7.70
	AuCu3 (27%)	3.7769(2)	7.63	10.3
HoIr3	C15b (63%)	7.5076(1)	10.8	7.39
	AuCu3 (37%)	3.7937(2)	5.6	7.77

Table 2. Table 2: Value of critical exponents according to different ideal model. 42 ; 43 ; 43a ; 50

	$\alpha$	$\beta$	$\gamma$	$\delta$
Mean field	0	0.5	1.0	3.0
2D Ising	0	0.12	1.75	15
3D Ising	0.11	0.32	1.24	4.82
3D Heisenberg	-0.11	0.36	1.38	4.90
3D XY	-0.007	0.34	1.34	4.8
Tricritical mean field	0.5	0.25	1.0	5.0

Table 3. Table 3: Critical exponents obtained from magnetization, heat capacity and magnetocaloric data using different technique for R Ir 3 ( R = Gd, Tb, Ho) compounds

Compound	Critical exponents obtained from modified Arrot plot	Critical exponents obtained from MS(T) and ${\chi }_0^{-1}$(T) curves	Critical exponents obtained from Kouvel Fisher method	Critical exponents obtained from magnetic isotherms	Critical exponents obtained from Scaling	Critical exponent obtained from CP(T) curve	Critical exponents obtained from MCE data	Types of interaction
GdIr3	$\beta =0.45$	$\beta =0.44$	$\beta =0.45$	$\delta =3.33$	$\beta =0.45$	$\alpha =0.008$	$n=0.64$	Mean Field
	$\gamma =1.1$	$T_C=87.31$	$T_C=87.34$		$\gamma =1.1$		$\delta =3.38$
	$\delta =3.44$	$\gamma =1.08$	$\gamma =1.06$		$\delta =3.44$	$\alpha =0.001$	$\beta =0.45$
	$T_C=87.0$	$T_C=87.1$	$T_C=87.0$		$T_C=87.0$	$\left(T>T_C\right)$	$\gamma =1.07$
		$\delta =3.45$	$\delta =3.35$
TbIr3	$\beta =0.32$	$\beta =0.32$	$\beta =0.32$	$\delta =5.10$	$\beta =0.32$	$\alpha =-0.114$	$n=0.60$	3d Heisenberg
	$\gamma =1.32$	$T_C=39.70$	$T_C=39.83$		$\gamma =1.33$		$\delta =5.10$
	$\delta =5.12$	$\gamma =1.32$	$\gamma =1.32$		$\beta =5.15$	$\alpha =-0.12$	$\beta =0.32$
	$T_C=39.0$	$T_C=39.10$	$T_C=39.82$		$T_C=40$	$\left(T>T_C\right)$	$\gamma =1.34$
		$\delta =5.12$	$\delta =5.12$
HoIr3	$\beta =0.20$	$\beta =0.20$	$\beta =0.20$	$\delta =5.70$	$\beta =0.20$	$\alpha =0.6$	$n=0.40$	Tricritical Mean Feild
	$\gamma =1.0$	$T_C=12.70$	$T_C=12.30$		$\gamma =1.0$		$\delta =5.98$
	$\delta =6.0$	$\gamma =0.98$	$\gamma =1.0$		$\delta =6.0$	$\alpha =0.6$	$\beta =0.21$
	$T_C=12.0$	$T_C=11.8$	$T_C=11.5$		$T_C=12.0$	$\left(T>T_C\right)$	$\gamma =1.04$
		$\delta =5.9$	$\delta =6.0$

Table 4. Table 4: | Δ ​ S M | m ​ a ​ x superscript Δ subscript 𝑆 𝑀 𝑚 𝑎 𝑥 |\Delta S_{M}|^{max} , Δ ​ T a ​ d Δ subscript 𝑇 𝑎 𝑑 \Delta T_{ad} (for a fielf change of 0 → → \rightarrow 70 kOe) and RCP for compounds R Ir 3 ( R = Gd, Tb, Ho)

Compound	${\left|\Delta S_M\right|}^{max}$	RCP	$\Delta T_{ad}$
	J/kg-K	J/kg	K
GdIr3	4.7	232	3.5
TbIr3	8.3	287	5.6
HoIr3	10.5	248	10.5

Equations

$$\chi ={\chi }_0+\frac{C}{(T-{\theta }_p)}$$

$$C_p=\gamma T+C_{phonon}+C_{mag}$$

$$C{}_{\mathrm{E}}=\sum _{N}3N{}_{\mathrm{E}}\mathrm{R}\frac{x^2{\mathrm{e}}^x}{[{\mathrm{e}}^x-1{]}^2},$$

$$C_{\mathrm{D}}=9N_{\mathrm{D}}\mathrm{R}{\left(\frac{T}{{\theta }_{\mathrm{D}}}\right)}^3{\int }_0^{\frac{{\theta }_{\mathrm{D}}}{T}}\frac{x^2{\mathrm{e}}^{x}dx}{[{\mathrm{e}}^x-1{]}^2}$$

$$C_P=\gamma T+\sum _i{\eta }_i{C}_{Ei}(T)+\left(1-\sum _i{\eta }_i\right)C_D(T)$$

$$\delta T_f=\frac{\Delta T_f}{T_f\Delta (lo{g}_{10}\nu )}$$

$$\tau ={\tau }_0{\left(\frac{T_f-T_{SG}}{T_{SG}}\right)}^{-z{\nu }^{\prime }}$$

$$\nu ={\nu }_0\exp \left[-\frac{E_a}{k_B(T_f-T_0)}\right]$$

$$M(t)=M_0+M_g\exp \left[-{\left(\frac{t}{\tau }\right)}^{\beta }\right]$$

$$M_S(0,T)=M_0(-\epsilon {)}^{\beta }\epsilon <0,$$

$${\chi }_0^{-1}(0,T)=\left(\frac{h_0}{M_0}\right)(\epsilon {)}^{\gamma }\epsilon <0,$$

$$M(H,T_{\mathrm{C}})=A_0(H{)}^{\frac{1}{\delta }}\epsilon =0,$$

$$C(T)=C_0{\epsilon }^{-\alpha }$$

$$M(H,T)=F(T)\times G(T,H)$$

$$M(H,\epsilon )=(\epsilon {)}^{\beta }f{}_{\pm }[H/{\epsilon }^{\gamma +\beta }]$$

$$\delta =1+\left(\frac{\gamma }{\beta }\right)$$

$$m=f_{\pm }(h)$$

$$M=A_0{H}^{1/3}$$

$$(H/M{)}^{1/\gamma }=a\left(\frac{T-T_{\mathrm{C}}}{T_{\mathrm{C}}}\right)+b{M}^{1/\beta },$$

$$C_P^{\pm }=\left(\frac{A_{\pm }}{\alpha }\right)|\epsilon {|}^{-\alpha }+B+C\epsilon$$

$$\Delta S_M(T,H)={\int }_0^H\left[\frac{\partial M}{\partial T}\right]dH$$

$$\Delta T_{ad}=[T(S,H)-T(S,0){]}_S,$$

$$\Delta T_{ad}(T,H)=-{\int }_0^H\left[\frac{T}{C}\right]\left[\frac{\partial M}{\partial T}\right]dH$$

$$|\Delta S_M{|}^{max}\propto H^n$$

$$\mathrm{RCP}\propto {\mathrm{H}}^{1+1/\delta }$$

$$n=1+\frac{(\beta -1)}{(\beta +\gamma )}$$

$$\beta =\frac{1}{\delta (1-n)+1}$$

$$\gamma =\frac{(\delta -1)}{\delta (1-n)+1}$$

$$\Phi =\frac{(T-T_{\mathrm{C}})}{(T_r-T_{\mathrm{C}})}$$

$$\Phi =\{\begin{array}{ll}-\frac{(T-T_{\mathrm{C}})}{(T_{r1}-T_{\mathrm{C}})},& T\le T_{\mathrm{C}}\\ \frac{(T-T_{\mathrm{C}})}{(T_{r2}-T_{\mathrm{C}})},& T>T_{\mathrm{C}}\end{array}$$

Full text

††preprint: APS/123-QED

I Introduction

Physical properties of a material depend strongly on its states, viz. solid, liquid, gas, etc. It is the density of the constituent particles, or rather the relative arrangement of atoms that determines the state of the matter. One may find different arrangement of atoms in the same state of matter, e.g., in solid state itself, which is arguably the most common form for a large number of materials in normal temperature and pressure. One such example is solid elemental carbon that can exist with different atomic arrangements, viz., graphite, diamond, fullerene (C60), C70, nanotubes, etc. 1 ; 2 ; 2a ; 2b ; 2c ; 2d ; 2e ; 2f and all these allotropic forms of carbon are known to exhibit a wide range of diverse physical properties 1 ; 2 ; 2a ; 2b ; 2c ; 2d ; 2e ; 2f . It is not only the elemental solids, but many multi-element compounds are also known to exhibit different properties depending on their relative atomic arrangements. For example, while LaIr2Si2 forming in CaBe2Ge2-type crystal structure exhibit superconductivity, the same compound forming in ThCr2Si2-type crystal structure shows no traces of it down to lowest measureable temperature 3 . On the other hand, PrIr2Si2, forming in ThCr2Si2-type structure, exhibit antiferromagnetic ordering at low temperature, but the compound remain paramagnetic when forms in CaBe2Ge2-type structure 4 . The different atomic arrangements in the respective crystal structures have been argued to be the reason behind their different properties. Such a characteristic, where the chemical composition remains conserved yet having different crystal structure, is commonly known as polymorphism. Except a very few cases like RIr2Si2 (R = rare earth) mentioned above, where the structural change is achieved by annealing the material at high temperature, most often one realize polymorphic phases by tuning external parameters, viz., temperature, pressure, etc. However, an external parameter driven structural change does not allow us to compare physical properties under identical conditions, e.g., under the influence of same temperature, pressure, magnetic and electric field, etc. Therefore, to study the effect of polymorphism on physical properties, it is highly desirable to obtain different phases under similar environment, viz., temperature, pressure, etc. Similar to RIr2Si2 series of compounds, it has been recently shown that RPt3B-type (R = rare earth) of compounds that form in tetragonal crystal structure, changes to cubic perovskite like structure when annealed at high temperature 5 , although the structural change found to be associated with creation of partial vacancy in the body-center position. One may note here that the different members of binary RPt3 (R = rare earth) compounds are known to form in two different crystal structures: C15b type for R = La-Tb, and AuCu3-type for R = Dy-Lu 8 ; 9 ; 12 ; 13 . Additionally, it was found that C15b type of structure in TbPt3 appear to be metastable in nature, as it changes to AuCu3 type on annealing 8 . Another binary series, RRh3 (R = rare earth), are also reported to form in different crystal structures (CeNi3-, PuNi3-, AuCu3-types) for different rare earth analogues. Only LaRh3 have been reported to form in two different structures 10 ; 11 ; 6 . In such scenario, one may expect multiple crystal structures in RIr3 series too, since Ir has very similar outer electron configuration as that in Rh, and has only one electron less than that in Pt. In our work we found that although many members of RIr3 series of compounds could be synthesized with single chemical compositions, but with two different crystal structures coexisting together. We also report here various magnetic properties of a few members (R = Gd, Tb, Ho) of RIr3 series of intermetallic compounds.

II Experimental Details

The polycrystalline RIr3 (R = Gd, Tb, Ho) compounds were synthesized in an arc-furnace by melting the stoichiometric amount of constituent elements of high purity ($>$ 99.9%) on a water-cooled Cu hearth under flowing inert gas Ar atmosphere. The ingots were melted 5 - 6 times after flipping each time to get volume homogeneity. The weight loss after melting were less than 1%. The as-cast ingots were annealed subsequently under vacuum in a sealed quartz tube at 900*∘*C for 7 days. The structural characterization of the annealed compounds were characterised by powder x-ray diffraction (XRD) technique at room temperature using Cu-K$\alpha$ radiation on a TTRAX-III diffractometer (M/s Rigaku, Japan) having 9 kW power supply. The crystal structure and phase purity of the annealed compounds were checked by Rietveld analysis of XRD data using FullProf software package 7 . The scanning electron microscopy (SEM) Measurements were carried out in the instrument EVO 18 (M/s Carl Zeiss, Germany) and the elemental analysis were performed in energy dispersive x-ray spectroscopy (EDX)(Element EDS system, M/s EDAX Inc., USA). The dc magnetic measurements were carried out in SQUID VSM (M/s Quantum design Inc., USA) in the temperature range 2 - 300 K and magnetic fields up to 70 kOe. The ac magnetic susceptibility measurements were carried out in Ever Cool II VSM system (M/s Quantum design Inc, USA). The specific heat of the sample at zero field were measured on PPMS system (M/s Quantum design Inc, USA).

III Results and Discussions:

III.1 Structural analysis:

The powder XRD patterns of RIr3 (R = Gd, Tb, Ho) compounds at room temperature are shown in fig. 1. The presence of sharp peaks in the XRD patterns of all samples confirm well crystalline behavior. In literature, various members of the binary AB3 system are reported to form in a wide variety of crystal (C15b, AuCu3, CeNi3, PuNi3-type) structures depending on the rare earth and transition metals present in the system 6 ; 8 ; 9 ; 10 ; 11 ; 12 ; 13 . Since the crystal structure of these RIr3 compounds are not known 13b , we have generated XRD patterns for all possible crystal structures as mentioned above, using the PowderCell software 13c . Although the experimental XRD patterns of RIr3 compounds closely match with the XRD pattern of cubic C15b-type structure (space group: F$\bar{4}$3m, No.216) a few additional peaks nevertheless remain unindexed for all the RIr3 system (fig. 1(d)). These additional XRD peaks remain unchanged, even after the compounds were annealed at 900*∘C for 7 days. To check the homogeneity of the materials, EDX measurement have been carried out which confirms that the RIr3 compounds are chemically homogeneous with rare earth and transition metal ratio 1:3. It therefore appears to be quite likely that these compounds form in new crystal structure or the extra peaks may come from coexisting additional polymorphic phase similar to that reported earlier in RPt3B series of compounds 5 . It may also be pointed out here that binary TbPt3 compounds are also known to form in two different polymorphic phases TbPt3: C15b and AuCu3-type 8 . In our further analysis, we found that those additional XRD peaks of RIr3, which remain unaccounted in C15b structure could be indexed with the cubic AuCu3-type crystal structure (space group: Pm$\bar{3}$m). The detailed Rietveld analysis of the XRD patterns of all of the annealed RIr3 (R = Gd, Tb, Ho) compounds considering both the cubic C15b-type and cubic AuCu3-type phases are shown in fig. 1. In C15b-type unit cell, 4a* (0,0,0) site is occupied by R atoms, while 16e ($\frac{5}{8},\frac{5}{8},\frac{5}{8}$) site is occupied by Ir atoms. The remaining R and Ir atoms are randomly distributed among 4c site ($\frac{1}{4},\frac{1}{4},\frac{1}{4}$) in equal proportions (Fig. 2(a)). In AuCu3-type unit cell the R atoms sit in the cubic corner positions 1a (0,0,0) while Ir atoms occupy the face centre positions 3c (0, $\frac{1}{2},\frac{1}{2}$) (fig. 2(b)). The details of crystallographic parameters obtained from the Rietveld refinement are listed in table 1. The RIr3 system thus form as a macroscopic coexistence of two crystalline phases, cubic C15b and cubic AuCu3-type with different relative percentage for different rare earths (Table 1). Coexistence of similar polymorphic phases have earlier been reported in literature 5 ; 5a . For example it is recently reported that RPt3B compounds form in two polymorphic crystal structures, tetragonal CePt3B-type and and ideal cubic perovskite-type at room temperature 5 . However, after annealing at high temperature the percentage of the cubic phase increases beyond the tetragonal phase. On the other hand in RIr3 series the percentage of two phases remain conserved upon annealing at temperature upto 900*∘*C 13b .

III.2 dc magnetization

The temperature dependent dc magnetic susceptibilities ($\chi=M/H$) of RIr3 (R = Gd, Tb, Ho) compounds under both zero-field-cooled (ZFC) and field-cooled (FC) protocols at different applied external magnetic fields are shown in fig. 3. The magnetic transition temperatures have been determined from the first order temperature derivative $\left(\frac{dM}{dT}\right)$ of the magnetization measured at 1 kOe applied magnetic field under FC condition. In case of ferromagnetic (FM) transition, the minima in the $\frac{dM}{dT}$ curves are described as the transition temperature $T_{\rm{C}}$ from paramagnetic to ferromagnetically ordered state. On the other hand, the transition temperature $T_{\rm{N}}$ of an antiferromagnetically ordered compound is defined as the temperature, at which the $\frac{dM}{dT}$ curve changes its sign. The temperature dependent magnetic susceptibility curves for each of these three compounds exhibit a single ferromagnetic anomaly at low temperatures. As our system consist of macroscopic coexistence of two phases (C15b-type, AuCu3-type), at least one of these two phases must have undergone ferromagnetic transition at low temperatures. If we compare the magnetic properties of RIr3 compounds with those of RX3 (X = Pd, Pt) compounds, one may able to shed some light on which phase is responsible for ferromagnetic ordering. Since the AuCu3-type of structure in RX3 compounds is generally known to be conducive to antiferromagnetic interaction 13 ; 13a , the ferromagnetic interaction observed in RIr3 compounds is likely to arise from the other polymorphic phase C15b. Absence of any antiferromagnetic signal in experimental data suggests AuCu3 phase may remain paramagnetic down to lowest experimental temperature limit 2 K. This point will be discussed further later in this work, where quantitative analysis of magnetic entropy have been carried out.

The RIr3 series of compounds show ferromagnetic transition at low temperatures with $T_{\rm{C}}$ = 83 K, 43 K, 11 K, respectively for GdIr3, TbIr3 and HoIr3. Magnetic susceptibility measurements of these compounds reveal thermal hysteresis behavior below their respective ordering temperatures. As the strength of the applied magnetic field increases, the same thermal hysteresis tend to weaken gradually. The temperature below which the divergence appears also decreases with the increasing applied magnetic field. At high applied magnetic field e.g. at 70 kOe, we obtain a completely reversible nature of temperature dependence of magnetic susceptibility. Such a thermal irreversibility between the ZFC and FC magnetization are generally attributed to the magnetic anisotropy and/or /spin/cluster glass behavior present in this system. We will probe on further detail on this point while discussing the magnetic relaxation phenomenon.

The observed magnetic transition temperatures of these compounds found to get reduced with increasing atomic number of rare earth element in the RIr3 series of materials (R = Gd and heavier rare earth). Generally, such a reduction of magnetic ordering temperature can be explained using de-Gennes scaling behavior of isostructural series of compounds. However, we found that the measured magnetic ordering temperatures deviates strongly than those expected from the de-Gennes scaling with respect to the transition temperature of GdIr3 where the crystalline electric field effect is negligible. Conventionally, a good agreement between the experimental value of transition temperatures and that obtained from the de-Gennes scaling is an indication of the dominance of RKKY interaction over the crystalline electric field (CEF) effect. The discrepancy between the experimental and the scaled value may therefore indicates that the CEF level scheme might have a strong influence on the magnetic ordering temperature in these materials. Inelastic neutron scattering experiment may help us to identify the CEF level scheme of the rare earth ions in these materials. However, these measurements are beyond the scope of the present work.

In the paramagnetic region $T_{\rm{C}}$ $<$ T $\leq$ 300 K magnetic susceptibility curves follow the Curie-Weiss behavior:

[TABLE]

where C is the Curie constant, $\theta_{p}$ is the paramagnetic Curie temperature and $\chi$0 is the temperature independent contribution. The estimated values of effective magnetic moment ($\mu$eff) and paramagnetic Curie temperature ($\theta_{p}$) are mentioned in fig. 3. The $\mu_{eff}$ values for RIr3 compounds closely follow those of the theoretical free ion values $\sqrt{gJ(J+1)}$ of the respective R3+ ion, indicating that only the localized 4f shells of R3+ are contributing towards the magnetism. The estimated $\theta_{p}$ values appear to be very close to the experimental value of $T_{\rm{C}}$, as expected in most of the ferromagnetic materials.

III.3 Heat Capacity

As mentioned earlier, heat capacity measurement is generally considered to be an effective tool to establish the bulk nature of magnetic ordering. An estimation of magnetic entropy from the heat capacity measurements can also provide valuable information regarding the volume fraction involved in the magnetic ordering process. Fig. 4 shows the heat capacity data of RIr3 (R = Gd, Tb, Ho) compounds as a function of temperature at zero applied magnetic field. A $\lambda$-like transition is observed around 83 K, 43 K and 11 K respectively for GdIr3, TbIr3, HoIr3 that are in good agreement with the magnetization measurements. The single $\lambda$-like transition observed in the heat capacity measurement corresponds to the long range magnetic ordering of the respective RIr3 compounds.

The temperature dependent heat capacity can be described by the standard formula,

[TABLE]

Here the first, second and third terms correspond to the electronic, phononic and magnetic contributions, respectively. At high temperature the heat capacity approaches to the classical value 3NR ($\sim$100 J/mol-K for N = 4 in case of RIr3). The magnetic contribution to heat capacity is generally estimated by subtracting the heat capacity of isostructural non-magnetic analogue from the heat capacity of the magnetic member, as it is generally assumed that the lattice and electronic contribution of both the system remain essentially same. However since in our case the system effectively consists of two different polymorphic phases where effective volume fraction depends strongly on the rare earth members involved, the same standard procedure may turn out to be quite misleading. Insted, one may first estimate the electronic and phononic contribution to heat capacity by fitting the data in the paramagnetic region and then extrapolated the fit down to 0 K. By subtracting the fitted curve from the experimental data, the $C_{mag}$ can subsequently be determined.

The total heat capacity of a material in the paramagnetic region consists of two contributions: electronic ($\gamma$T) and Phononic ($C_{phonon}$). The phononic contribution was first explained by Einstein, who assumed that a solid composed of N atoms can be represented as 3N independent harmonic oscillators having same frequency 16 ; 16a . The Einstein contribution can be written as 16 ; 17 ,

[TABLE]

where $N{\rm{}_{E}}$ is the number of Einstein oscillators, x = $\theta{\rm{}_{E}}/T$, $\theta{\rm{}_{E}}$ is the Einstein temperature. However, it was found that the Einstein model appears to be quite inadequate to describe the experimentally observed specific heat behavior at low temperature region for most of the solids 16 ; 16a ; 16b . Following this, Debye had modified Einstein model by assuming that the solid consisting of a set of coupled oscillator instead of independent oscillators 16 ; 16a ; 16b , where the phononic contribution to heat capacity takes the following form 16 ; 17 ,

[TABLE]

where $N\rm_{D}$ is the number of Debye oscillators and x = $\theta{\rm{}_{D}}/T$, $\theta{\rm{}_{D}}$ being the Debye temperature. The modification proposed by Debye indeed able to explain the low temperature heat capacity data in much better way than Einstein model. The Debye model still cannot describe the experimental heat capacity behavior over the entire temperature region, as it works well below $\theta{\rm{}_{D}}/50$ and above $\theta{\rm{}_{D}}/10$ only 16 . The quantitative mismatch in the intermediate temperature region has its origin in the fact that the phonon dispersion phenomenon was not taken into account in the Debye model. Since neither a single Einstein model nor a single Debye model can describe the experimental outcome over the whole temperature range, a combination of both the contributions generally used to describe the overall heat capacity behavior 16c ; 16d ; 16e ; 16f ; 16g ; 16h ; 17 , that can be expressed as 16g ; 16h ; 17 ,

[TABLE]

The parameter $\eta$ determines the relative percentage of the two contributions. We have achieved a good fit for all these three compounds in the paramagnetic region by using eqn. (5) with Einstein and Debye contributions for GdIr3, TbIr3 and HoIr3 as 70% and 30%, 63% and 37%, 70%, 30% respectively. The corresponding fitting parameters are listed in fig.4. $C_{mag}$ has been evaluated by subtracting the experimental data from the fitted model after extrapolating to lowest temperature. After calculating $C_{mag}$ and integrating $\frac{C_{mag}}{T}$, over the entire temperature range, it is possible to estimate the magnetic entropy, $S_{mag}$ (= $\int_{0}^{T}\frac{C_{mag}}{T}dT$). For a bulk magnetic phenomenon, when all the R ions takes part in the magnetic ordering process, at high temperature, $S_{mag}$ saturates to the theoretical value R$ln(2J+1)$, where J is the total angular momentum and R (= 8.31 J/K) is the universal gas constant. Theoretically if all the R atoms would have ordered, one would expect $S_{mag}$ to be reached to Rln8, Rln13 and Rln17 for GdIr3 (J = 7/2), TbIr3 (J = 6), HoIr3 (J = 8) respectively. In our analysis, we however found a much reduced value of $S_{mag}$ as 1.3R for GdIr3, 1.7R for TbIr3 and 1.7R for HoIr3. Thus we have found from magnetic entropy calculation that only 70% of Gd atom, 70% of Tb atom and 62% of Ho atom would have contribute towards magnetism. Since in XRD it was found that RIr3 (R = Gd, Tb and Ho) consists of two different phases viz, C15b and AuCu3-type with relative percentage 74 and 26, 73 and 27, 63 and 37 respectively, the reduced value of estimated $S_{mag}$ indicates that only the C15b phase participate in magnetism.

III.4 ac succeptibility

The ac susceptibility measurements were carried out for GdIr3, TbIr3 and HoIr3 in an excited field of 0.124 Oe for different frequencies (f). Fig. 5 shows the variations of real and imaginary parts of magnetic susceptibility with temperature at various frequencies. For all the three samples, the peaks in $\chi^{\prime}$ as well as $\chi^{\prime\prime}$ could be found at the respective temperatures, which have been identified as magnetic ordering temperatures through dc magnetic susceptibility as well as heat capacity measurements. The non-zero values of $\chi^{\prime\prime}$ below the ordering temperatures suggest the magnetic ordering to be ferromagnetic type.

The peak temperature in both $\chi^{\prime}$ and $\chi^{\prime\prime}$ in GdIr3 remain invariant as a function of frequency indicating a long range nature of the ferromagnetic ordering in this compound. However, although the peak temperatures in $\chi^{\prime}$ for TbIr3 and HoIr3 do not exhibit any discernible shift as a function of frequency, a close examination of $\chi^{\prime\prime}$ for both the samples reveal a change in peak position. As the frequency increases, the peak in $\chi^{\prime\prime}$ tend to shift toward higher temperature. This feature is generally attributed to the presence of metastable spins, in the system that exhibit spin/cluster glass behavior. For TbIr3 the peak temperature shifts from 40.50 K to 41.92 K with increasing frequency from 37 Hz to 9997 Hz [fig. 5 (b)]. In case of HoIr3, the peak shifts from 9 K to 9.6 K with similar increase in frequency [fig. 5 (c)].

As mentioned above, such a shift in peak temperature manifests the presence of spin/cluster glass transition with $T_{f}$ (freezing temperature) to be 40 K for TbIr3 and 9 K for HoIr3. To find the catagory of spin/cluster glass system we have estimated the relative freezing temperature shift per decade of frequency which is defined as 18 .

[TABLE]

where $T_{f}$ is the freezing temperature, $\nu$ is the applied frequency. The value of $\delta T_{f}$ found to be 0.01 for TbIr3 and 0.02 for HoIr3, which suggest the systems belonging to canonical spin glass type 18 .

The spin glass nature can also be established through the power law behavior of $T_{f}$ that 18 ; 19

[TABLE]

where $\tau=\frac{1}{\nu}$ is the relaxation time corresponding to the applied frequency, $\tau_{0}$ is the relaxation time for single spin-flip, $T_{SG}$ is the temperature of spin/cluster glass with f = 0, $\nu^{\prime}$ is known as critical exponent for correlation length $\xi=\left(\frac{T_{f}}{T_{SG}}-1\right)^{-\nu{\prime}}$ and $\tau\sim\xi^{z}$. The term $z\nu^{\prime}$ is called as dynamical critical exponent. For canonical spin glass, the value of critical exponent $z\nu^{\prime}$ lies between 4 and 12 while $\tau_{0}$ lies between $10^{-12}-10^{-13}$ 20 . In our analysis, we have estimated the values of $z\nu^{\prime}$ & $\tau_{0}$ to be 4.46 & $10^{-10}$ sec for TbIr3 and 9, $10^{-11}$ sec for HoIr3 respectively. The derived values of $z\nu^{\prime}$ are in the range reported for spin-glass system but $\tau_{0}$ value is somewhat larger than canonical spin glass although it remains orders of magnitude larger than the values for cluster glass system ($\tau_{0}\sim 10^{-7}$) 20 ; 20a ; 20b ; 20c ; 20d .

Vogel-Fulcher relation 18 ; 21 is another dynamical scaling law for spin/cluster glass system where freezing temperature $T_{f}$ depends on frequency as

[TABLE]

Here $E_{a}$ denotes the activation energy, $\nu_{0}$ the characteristic attempt frequency and $T_{0}$ the Vogel-Fulcher temperature. For typical canonical spin-glass, $\frac{E_{a}}{k_{B}}/T_{0}$ value should be close to 1. We found the values of $\frac{E_{a}}{k_{B}}$, $T_{0}$ and $\frac{E_{a}}{k_{B}}/T_{0}$ as 21.10, 37.61 K and 0.6 for TbIr3 and 10.54, 7.45 K and 1.41 for HoIr3. These values again establish these two compounds to be canonical spin glass type.

III.5 Nonequilibrium dynamics

The presence of magnetically frustrated spins in the system can also be established by studying the magnetic relaxation behaviors. The relaxation process have been studied under ZFC protocol, where the sample is cooled in the absence of any external applied field from the paramagnetic region to the desired temperature which is below $T_{\rm{C}}$. After reaching the desired temperature, the sample is kept at zero field for a certain time at that temperature. Subsequently a small amount of field is applied and time evolution of magnetization (M(t)) is monitored. The ZFC relaxation of RIr3 (R = Gd, Tb, Ho) compounds at temperature 2.5 K are displayed in the fig.6. As expected earlier, GdIr3 which does not show glassy behavior, exhibit no magnetic relaxation [5(a)].

The time dependent magnetization follow the exponential behavior as 22 ; 23 ; 25

[TABLE]

where $M_{0}$ is intrinsic magnetization, $M_{g}$ is the glassy component of magnetization, $\tau$ is the relaxation time, $\beta$ is the stretching exponent. The value of $\beta$ depends on the nature of energy barriers involves in the relaxation process. $\beta$ = 0 implies no relaxation and $\beta$ = 1 is for single time constant relaxation process. Since typical spin glass systems are characterized with a distribution of energy barriers, value of $\beta$ lies between 0 and 1 18 ; 24 .

The time evolution of magnetization of TbIr3 and HoIr3 are fitted with eq. (9). For compounds TbIr3 and HoIr3 the relaxation times obtained are 1244 sec, 600 sec and $\beta$ values have been estimated to be 0.29, 0.27 respectively. The $\tau$ value of TbIr3 lies within the range of earlier reported different glassy systems 18 ; 24 but $\tau$ value in case of HoIr3 indicates a weaker nature of glassy behavior.

III.6 Magnetic memory effect

In the foregoing part it has been observed that TbIr3 and HoIr3 show time dependent magnetic relaxation behavior which is absent in GdIr3. Besides the magnetic relaxation behavior, magnetic memory effect is another tool for distinguishing various class of glassy systems.

The memory effect in RIr3 (R = Tb, Ho) samples have been investigated in both FC and ZFC protocols 26 . In the FC protocol, the samples were cooled with low applied field of 100 Oe from the paramagnetic region (T = 125 K for TbIr3 and T = 60 K for HoIr3) to the lowest measurable temperature 2.5 K with single intermediate stop at Tstop = 15 K (for TbIr3) and at Tstop = 4 K (for HoIr3) for a duration of $t_{W}$ = 1.5 h. The magnetizations measured during this process is represented as $M^{STOP}_{FCC}$ as shown in fig. 7.

At the respective stopping temperatures of the two samples, the magnetic field was switched off and after the lapse of time $t_{W}$ = 1.5 h, the same field was reapplied with resumed cooling. After reaching lowest temperature 2.5 K, the samples are heated up to the paramagnetic region with the same applied field, as well as same rate and the measured magnetization curve is depicted as $M^{mem}_{FCW}$. The $M^{mem}_{FCW}$ curve thus obtained show a tendency to follow $M^{STOP}_{FCC}$ curve yielding a signature to remember the past history [fig. 7]. The standard FC magnetization curve $M^{ref}_{FCW}$ for both compounds are also displayed in the fig. 7 (a), (b). From the figures it is clear that in case of TbIr3, the observed memory effect below $T_{\rm{C}}$ is relatively stronger than that observed in case of HoIr3. Such type of memory effects are quite well known behavior observed in various glassy systems 23 ; 27 . In this context it should be mentioned that the estimated relaxation time constant estimated for HoIr3 is smaller than that of TbIr3.

The memory effect under ZFC protocol is also carried out in both these compounds. In the ZFC protocol the samples were first cooled down in zero field from the paramagnetic region to some stopping temperatures (Tstop = 15 K for TbIr3 and Tstop = 4 K for HoIr3) where the temperatures were kept on hold for tW = 1.5 h. The cooling was then recommenced down to the lowest temperature 2.5 K. The magnetization M(T) was then recorded during heating from 2.5K to paramagnetic region under application of 100 Oe magnetic field. The M(T) curve obtained in this process is leveled as $M^{mem}_{ZFCW}$. The reference ZFC magnetization for 100 Oe field is also measured. This is indicated as $M^{ref}_{ZFCW}$. The ZFC memory effect of these two samples are shown in the fig. 7. Fig 7 also show the difference in magnetization of the two measurement processes of RIr3 samples. The difference, $\Delta M$ (= $M^{mem}_{ZFCW}$- $M^{ref}_{ZFCW}$) shows memory dip around the stopping temperatures for both the samples.

It may be pointed out here that the memory effect is also observed in phase-separated or superparamagnetic systems in FC process 28 . Only the ZFC memory effect can differentiate spin glass class from superparamagnetic system because superparamagnetic compound does not show memory effect in ZFC protocol 28 . Thus the observed memory effect in ZFC mode confirm the presence of spin glass state in the two compounds.

III.7 Critical behavior of the magnetization and susceptibility

In the earlier analysis, we have seen that while GdIr3 exhibit a long range magnetic order, an addition glassy feature is observed in both TbIr3 and HoIr3. It thus raises a question whether the magnetic ordering observed in these three compounds are indeed have a long range character and if so one would be interested to know more about the universality class of these magnetic systems. These information could be extracted by studying the critical behavior of different physical properties, viz. M(H,T), C(T), etc, around their respective magnetic transition temperatures. The critical analysis of these physical properties helps us to understand and classify a system according to the nature and strength of their respective magnetic interactions. Critical analysis study utilizes the fact that any phenomena that takes place in the vicinity of the phase transition temperature can be associated with a power law behavior of the reduced temperature ($\varepsilon$ = $\frac{T-T_{\rm C}}{T_{\rm C}}$). For example, magnetic correlation length $\xi$ can be expressed as $\xi$ = $\xi_{0}|\varepsilon|^{-\nu}$, where $\nu$ is known as critical exponent.

Following the same argument, one can express several other physical quantities viz., $M_{S}(T)$, $\chi_{0}(T)$, $M(H,T=T_{\rm{C}})$, C(T), etc. with similar power law expression as 29 ; 40 ; 32 :

[TABLE]

where $M_{0}$, $\frac{h_{0}}{M_{0}}$ and $A_{0}$, $C_{0}$ are the critical amplitudes, $M_{S}$ is the spontaneous magnetization, $\chi_{0}$ is the initial susceptibility. Depending on the characteristic of various universality classes, viz. 2D Ising model, 3D Ising model, mean field, 3D Heisenberg model, tricritical mean field, XY model etc the critical exponents $\alpha$, $\beta$, $\gamma$ and $\delta$ can assume different set of values (see table 2). Conversely by carrying out critical analysis and obtaining the values of $\alpha$, $\beta$, $\gamma$ and $\delta$, one may associate the compound with the universality class it belongs to. The values of critical exponents associated with different universal class is given in table 2.

It must be pointed out here that although, the power law behavior expressed in eqs. (10), (11), (12), (13) are independent to each other, but the critical exponents are not so. The critical exponents can be linked using different scaling relations. For example, magnetization $M(H,T)$ can be expressed using two independent functions of H and T as,

[TABLE]

where F(T) is a function of T alone, while G(T,H) is a function of both T and H. Solving analytically one can rewrite equation 14 as,

[TABLE]

where $f_{+}$ and $f_{-}$ are the functions of temperatures above and below $T_{\rm{C}}$, respectively 29 ; 40 . Using different boundary conditions, one can obtain a scaling relationship,

[TABLE]

which is widely known as Widom scaling relation 32 .

If the scaled or renormalized magnetization and magnetic field are defined as, $m=|\varepsilon|^{-\beta}M(H,\varepsilon)$ and $h=|\varepsilon|^{-(\gamma+\beta)}H$, eq. (15) reduces to a simple form

[TABLE]

This equation is quite significant as it shows that with appropriate choice of a particular set of $\beta$, $\gamma$ and $\delta$ the scaled magnetization (m) as a function of scaled field (h) taken at different temperatures can essentially be converged to two different universal curves: $f_{+}(h)$ for temperatures above $T_{\rm{C}}$ and $f_{-}(h)$ for temperatures below $T_{\rm{C}}$.

As shown in eqs. (10), (11), (12), (13) different measurements can be employed to estimate different critical exponents. For example, by studying the isothermal magnetization close to critical temperature, eq. (12) indicates that one can obtain information on $\delta$ (and subsequently on $\alpha$ and $\beta$). From table 2 we see that for mean fielf like variation $\delta$ is close to 3, that is eq. (12) reduces to

[TABLE]

where $A_{0}$ is a constant. This is generally known as Arrott equation 30 . Using this equation a set of magnetic isotherms obtained experimentally near $T_{\rm C}$ can be turned into another set of parallel straight lines in the $M^{2}$ vs $H/M$ representation. This reconstructed magnetic isotherms are called Arrott plot 30 . The magnetic isotherm of Arrott plot that passes through origin defines the $T_{\rm C}$. However, the material that does not obey mean field approximation cannot produce such set of parallel straight lines. A more generalized equation has been provided by Arrott and Noaks as 34 ,

[TABLE]

(where a and b are constants) which is used to obtain a set of parallel straight lines in the $M^{1/\beta}$ vs $\left(H/M\right)^{1/\gamma}$ representation. This plot obeying Arrott-Noak equation of state is often referred as modified Arrott plot 34 . Thus a self consistent values of $\beta$, $\gamma$ and $\delta$ can be obtained by same set of data (isothermal magnetization) using different analytical approach as presented in eqs. (10), (11), (12), (15) and (19).

Magnetic isotherms of GdIr3 in the temperature range 60-130 K in an interval of 2 K near $T_{\rm{C}}$ are shown in fig. 8(a). To test the applicability of Arrott equation in this system, magnetic isotherms are plotted in $M^{2}$ vs $H/M$ depiction in fig. 8(b). The nonlinear nature of the same plot suggests that the material does not belong to the ideal mean field family. A set of parallel straight lines however could be obtained in the temperature range 70-100 K by redrawing the Arrott plot using eq. (19) by considering $\beta$ = 0.45 and $\gamma$ = 1.1 (fig. 8(c)). We are able to obtain such set of parallel straight lines in the modified Arrott-plot over quite a large region -0.19 $<$ $\varepsilon$ $<$ 0.14 around the critical temperature. Incidentally, not many system exist where the critical region spans over such a wide temperature zone 43a . The modified Arrott plot helps us to estimate the magnetic ordering temperature far more accurately ($T_{\rm{C}}\sim$ 87 K). To test the validity of $\beta$ and $\gamma$, we have estimated the same set of parameters using different methods as described in eqs. (10) & (11) near the critical temperature region. Fig. 8(d) shows the extracted values of $M_{S}(T)$ and $\chi_{0}^{-1}(T)$ and fitted using eqs. (10) & (11). The nonlinear fitting of $M_{S}$ vs. T and $\chi_{0}^{-1}$ vs. T suggest $\beta=0.44$, $T_{\rm{C}}=87.31$ K and $\gamma=1.08$, $T_{\rm{C}}=87.1$ K, respectively. A linear dependency with temperature can be obtained using a method suggested by Kouvel and Fisher 33 where $M_{S}\left(dM_{S}/dT\right)^{-1}$ and $\chi_{0}^{-1}\left(d\chi_{0}^{-1}/dT\right)^{-1}$ are plotted (Fig. 8(e)) against temperature having slopes $1/\beta$ and $1/\gamma$ respectively. Fig. 8(e) suggests a value of $\beta=0.45$ and $\gamma=1.06$ for GdIr3 using the Kouvel-Fisher technique.

The above mentioned set of analysis helps us to determine the Curie temperature of GdIr3 with resonable confidence level. To estimate the other parameter $\delta$, we have chosen the magnetic isotherm measured experimentally at a temperature close to $T_{\rm{C}}$. Fig. 8(f) shows the magnetic isotherm at $T\simeq T_{\rm{C}}$ for GdIr3. Inset of this figure shows logarithmic behavior of same isotherm. A linear fit of the inset data using eq. (12) suggests the value of $\delta=3.33$. Widom relation 32 , presented earlier in eq. (16) also suggest an alternative method to estimate the value of $\delta$ when the exponents $\beta$ and $\gamma$ are known. The value of $\delta$ thus around to be 3.35, (taking the values of $\beta$ and $\gamma$ from Kouvel-Fisher technique as shown in fig. 8(e)), which is very close to that obtained earlier from different methods.

The values of $\beta$ and $\gamma$ can also be independently estimated by using two sets of scaling relations (eq. 17) to different magnetic isotherms, above and below $T_{\rm{C}}$, respectively. Tuning the values of $\beta$ and $\gamma$ we have been successfully able to merge all the rescaled relations into two different universal curves (Fig. 8(g)) for $T<T_{\rm{C}}$ and $T>T_{\rm{C}}$. The rescaled curves in logarithmic scale converges near $T=T_{\rm{C}}$ as shown in the inset of fig. 8(g). The parameters thus obtained also found to match with the same set of parameters estimated earlier using different methods (table 3).

Apart from magnetization, specific heat at constant pressure and in absence of external magnetic field also follow a power law at temperatures close to $T_{\rm{C}}$ (eq. (13)). We have fitted the measured heat capacity in the critical region as a function of reduced temperature using the following eq., 42

[TABLE]

where $\alpha$ is the critical exponent, while $A_{\pm}$, B and C are constants. The subscript ‘+’ is for $\varepsilon>0$ i.e. for $T>T_{\rm{C}}$ and ‘-’ stands for $\varepsilon<0$ i.e. for $T<T_{\rm{C}}$. Fig. 8(h) shows the $C_{P}(T)$ behavior for GdIr3, near its transition temperature. This data is fitted with eq. (20) below and above $T_{\rm{C}}$ and the critical exponent $\alpha$ are obtained for both the scaled curves (Table 3).

The same set of analysis described above in this section have also been carried out for other two compounds TbIr3 and HoIr3, and the critical exponents are obtained by all these methods (Table 3). The same type of figures for TbIr3 and HoIr3 compounds are shown in figs. from 9(a) to 9(h) and from 10(a) to 10(h) respectively. The critical exponents estimated for GdIr3, TbIr3 and HoIr3 by different methods closely match with those reported in mean field theory, 3-D Heisenberg magnetic class and tricritical mean field theory, respectively 36 ; 37 ; 38 ; 39 ; 41 ; 43 . Thus the above analysis suggest that while GdIr3 obeys mean field theory, TbIr3 belongs to 3-D Heisenberg class and HoIr3 follows tricritical mean field theory.

Studying the universality class of the magnetic phase transition also helps us in understanding the range of exchange interaction J(r) 35 . The renormalization group theory analysis for such systems by Fisher et al. 33 suggests that the exchange interaction, J(r) varies as, 1/rd+σ, 35 where d is the dimension of the system and $\sigma$ is the range of the exchange interaction. For a 3D system the exchange interaction is J(r) = 1/r3+σ with $\dfrac{3}{2}$ $\leq$ $\sigma$ $\leq$ 2. For 3-D Heisenberg system $\sigma$ is equal to 2, thus J(r) varies with r as r-5 and the interaction strength decays fastest among all classes. The mean field exponents hold if J(r) varies with r as r-4.5 (for $\sigma$ = 3/2). In the intermediate range, the exponents belong to a different universality class which depends upon the value of $\sigma$. Thus for GdIr3 sample, the interaction strength is of long range, but in case of TbIr3, the interaction is of short range 3D Heisenberg-type. For intermediate values of $\sigma$, the critical exponents follow different kind of universality class such as triclinic mean field class for HoIr3.

III.8 Magnetocaloric Effect

In the preceeding section, we have discussed about the nature of magnetic interaction present in these three RIr3 (R = Gd, Tb, Ho) compounds through critical analysis study. It was concluded that GdIr3 follows the mean field theory, TbIr3 belongs to the 3D Heisenberg magnetic class while HoIr3 obeys tricritical mean field theory. The validity of such conclusion can also be checked through the study of magnetocaloric effect (MCE).

MCE is an environment friendly and alternative technique over the conventional gas compression/expansion methods to achive cooling process. It is defined as a change in the temperature (heating or cooling) of materials due to the application of a magnetic field and can be estimated either through heat capacity or isothermal magnetization measurements.

In the later procedure, the magnetic entropy change, $\Delta S_{M}$ using Maxwell relation 46 is defined as

[TABLE]

$\Delta$S${}_{M}(T,H)$ have been obtained from a set of isothermal magnetization data by solving eq. (21) using numerical approximation method. Figs. 11 (a), (c), (e) show the magnetic entropy changes as a function of temperature for different field changes of RIr3 (R = Gd, Tb, Ho) compounds, respectively. The maximum values of |$$\Delta S_{M}$$| are 4.7 J/kg-K, 8.3 J/kg-K, 10.5 J/kg-K for GdIr3, TbIr3, HoIr3 at temperatures 83 K, 43 K, 11 K, respectively for a field change of 0 $\rightarrow$ 70 kOe respectively [fig. 11 (a), (c), (e)]. Similar values of $\Delta S_{M}$ are also reported for different intermetallic compounds in this temperature range subject to similar field change 57 ; 58 .

Along with magnetic entropy change, the amount of heat transfer between the hot and cold reserviors in an ideal refrigerant cycle of the material is quantified by relative cooling power (RCP). RCP for a particular field H is defined as the product of maximum entropy change ($\Delta S_{M}$) and full width at half-maximum ($\delta T_{FWHM}$) of $|\Delta S_{M}|$-T curve 47 . A large value of RCP can be achieved either by getting large $\Delta S_{M}$ or widespread of $\Delta S_{M}$ over a large temperature range, or both. RCP values for compounds GdIr3, TbIr3, HoIr3 have been estimated to be 232 J/kg, 287 J/kg and 248 J/kg respectively at field of 70 kOe indicating their appropriate usage in cooling technology.

However, to judge the applicability of a good MCE material, another important parameter is adiabatic temperature change ($\Delta T_{ad}$), that is defined as

[TABLE]

where T(S,H) and T(S,0) are the temperatures at applied field H and no applied field respectively, for a particular entropy S. Using thermodynamic relation, it can be written as

[TABLE]

where C is the specific heat of the system at zero applied field. We have estimated $\Delta T_{ad}$ using $\Delta S_{M}$ and the zero field heat capacity data. The maximum values of $\Delta T_{ad}$ are 3.5 K, 5.6 K, 10.5 K for GdIr3, TbIr3, HoIr3 at temperatures 83 K, 43 K, 11 K, respectively, at a field change of 0 $\rightarrow$ 70 kOe [fig. 11 (b), (d), (f)]. One may notice an appreciably large value of $\Delta T_{ad}$ in HoIr3 in this temperature region and similar field sweep 25 . The values of $|\Delta S_{M}|^{max}$, RCP, $\Delta T_{ad}$ for fielf change 0 $\rightarrow$ 70 kOe are represented in table 4 for RIr3 (R = Gd, Tb, Ho) compounds.

As mentioned earlier in this section, MCE can also be used independently to determine the critical exponents by studying the variation of $\Delta S_{M}$ and RCP as a function of applied magnetic field 59 .

It is generally found, the field dependence of the magnetic entropy change at the critical temperature associated with a second order magnetic phase transition follows the relation 49

[TABLE]

and the RCP varies as 50

[TABLE]

The insets in fig. 11 (a), (c), (e) shows the plot of $|\Delta S|^{max}$ as a function of H and plot of* RCP* as a funtion of H. The power law fit of eqs. (24) and (25) give the values of n and $\delta$ for the three compounds which are shown in table 3.

The parameter n in eq. (24) has been expressed in terms of $\beta$ and $\gamma$ as, 51

[TABLE]

Using both eqs. (26) and (16), $\beta$ and $\gamma$ can be expressed as a function of n and $\delta$ as 59

[TABLE]

Using eqs. (27), (28) and (16) the values of $\beta$, $\gamma$ and $\delta$ are estimated for RIr3 (R = Gd, Tb, Ho) compounds. The obtained values are displayed in table 3 and closely matches with the values acquired by other methods discussed in section G. The closeness of the critical exponents ($\beta$, $\gamma$, $\delta$) suggest the self consistency of the analysis.

Another interesting feature of MCE is that if the magnetic entropy change, $\Delta S_{M}$ can be described in terms of appropriately chosen reduced temperature scale for any particular magnetic field, the shape and value of $\Delta S_{M}(T,H)$ curves for any arbitrary magnetic field can be generated, even without knowing the critical exponents, subject to an assumption that no major change in magnetic interaction takes place abruptly in the system 51 ; 53 ; 54 ; 55 ; 56 . Additionally, the same universal curve can also be used to predict the $\Delta S_{M}(T,H)$ curves for other members of same series of compounds, that may be having similar nature of magnetic interaction and magnetic ordering temperature is known 53 . Thus, the utility of such universal or master curve is to use to extrapolate data in the temperature ranges where the sample was not even measured 53 ; 56 .

In general, such a master curve is obtained by normalizing the parameter $\Delta S^{\prime}=|\Delta S_{M}|/|\Delta S_{M}|^{max}$, as a function of rescaled temperature 47 ; 51

[TABLE]

where, $T_{r}$ is the temperature at which $|\Delta S_{M}|=a\times|\Delta S_{M}|^{max}$ (‘a’ is an adjustable parameter which can take value between 0 and 1). However, if the sample is magnetically inhomogeneous or the measuring field is too low, then one need to use two scaling parameters $T_{r1}$ and $T_{r2}$ instead of a single one ($T_{r}$) 47 ; 51 . Therefore temperature, $\Phi$ is defined as 43 ; 47 ; 51

[TABLE]

For TbIr3, GdIr3 and HoIr3 compounds, after rescaling the temperature using eq. (30) and choosing a = $\frac{1}{2}$, we find that $\Delta S^{\prime}(\Phi$) for all the applied fields collapse in single curves for each individual samples [fig. 12 (a), (b), (c)].

Furthermore, if these three universal curves corresponding to the three different compounds, GdIr3, TbIr3, HoIr3 are plotted together then it is found that they overlap for positive $\Phi$ $>$ 0 that is in the paramagnetic region (T $>$ TC), while they slightly differ from each other for $\Phi$ $<$ 0 that is within the ordered region (T $<$ TC) [fig. 12]. This property allows the prediction of |$$\DeltaSM$|$(T) curves even in other compounds having related compositions 53 .

III.9 Summary

In summary, we report the successful synthesis of three compounds GdIr3, TbIr3, HoIr3 which found to form in two polymorphic phases (C15b, AuCu3). The dc magnetization measurements show that these compounds orders ferromagnetically, while magnetic entropy calculation from heat capacity measurement indicates that C15b phase is responsible for ferromagnetic ordering and AuCu3 phase remain paramagnetic down to 2 K. The ac susceptibility measurement and time dependent relaxation measurement indicates the presence of glassy nature in TbIr3 and HoIr3 but is absent in GdIr3. The detailed study of dynamical scaling of ac susceptibility, magnetic relaxation and memory effect measurements established both TbIr3 and HoIr3 to be canonical spin glass in nature. The modified Arrott plot, MS(T), $\chi_{0}^{-1}(T)$ curves, Kouvel-Fisher method and specific heat analysis confirm that GdIr3 obeys mean field theory, TbIr3 lies in 3-D Heisenberg universality class and HoIr3 follows tricritical mean field theory. The aforesaid critical analysis complies with the MCE studies. The $\Delta$Tad value of HoIr3 found to be quite appreciable.

Acknowledgements

RNB thanks to CIF, Pondicherry University for ac susceptibility measurements. The work has been carried out by the CMPID project at SINP and funded by Department of Atomic Energy, Govt. of India. We thank Dr. Santanu Pakhira for his help during data analysis.