Effect of hard magnetic ferrite (Ba 0.5 Sr 0.5 Fe 12 O 19 ) nanoparticles on the mechanical properties of the (Bi, Pb)‑2223 phase

To manifest the effect of hard magnetic Ba 0.5 Sr 0.5 Fe 12 O 19 nanoparticles on the mechanical performance of the (Bi,Pb)-2223 superconducting phase, nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 ) x /Bi 1.8 Pb 0.4 Sr 2 Ca 2 Cu 3.2 O 10+δ , with x = 0.00, 0.01, 0.02, 0.03, 0.04, 0.05, 0.10, and 0.20 wt%, were synthesized using a conventional solid-state reaction method. The X-ray diffraction (XRD) data revealed that adding nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 ) to the host (Bi,Pb)-2223 phase preserved the orthorhombic structure. The porosity ( P %) calculations revealed a decrease until x = 0.04 wt%, which suggests that the addition of nano-(Ba 0.5 Sr 0. 5 Fe 12 O 19 ) x reduces the number of voids and improves inter-grain connections, as confirmed by SEM micrographs. The superconducting transition temperature ( T c ) increased to 112 K with the inclusion of nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 ) up to x = 0.04 wt%. Vickers microhardness ( H V ) measurements were conducted at various applied loads (0.245–9.800 N) and a duration time of 45 s. The H V number increased with the addition of x up to x = 0.04 wt% but then decreased with further addition. Various models were employed for analysis and modelling of Vickers hardness ( H V ) versus test load (F), including Meyer’s law, Hays–Kendall (H–K) model, the elastic/plastic deformation (EPD) model, the proportional sample resistance (PSR) model, the modified proportional sample resistance (MPSR), and indentation-induced cracking (IIC) model. It was found that the PSR model was the most appropriate theoretical model for describing the microhardness of nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 ) x /(Bi,Pb)-2223 composites. Moreover, the elastic modulus ( E ), yield strength ( Y ), fracture toughness ( K ), brittleness index ( B ), and elastic stiffness coefficient ( C 11) were estimated as a function of the inclusion of nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 ) x . Furthermore, the indentation creep test (time-dependent Vickers microhardness) revealed that the dislocation creep mechanism exists in composite samples with low concentrations ( x < 0.05 wt%), whereas the dislocation climbs creep mechanism was observed for x ≥ 0.05 wt%.


Introduction
From the application point of view, the Bi-Sr-Ca-Cu-O (BSCCO) high-temperature superconductors (HTSCs) have attracted considerable interest from scientists all over the world because of their extraordinary qualities and numerous applications in various fields of technology [1]. Bi 2 Sr 2 Ca n-1 Cu n O 2n+4+δ , is the general formula of BSCCO superconductors, where n = 1, 2, and 3 denote the number of CuO 2 layers along the c-axis, which ultimately determines the number of unit cells in the superconductor [2]. The superconducting transition temperature T c at which these materials transition to superconductivity rises as the number of CuO 2 layers increases. There are three main phases that define the family of BSCCO superconductors, including the very low Bi-2201 phase (n = 1, T c ~ 20 K), the low Bi-2212 phase (n = 2, T c ~ 85 K), and the high Bi-2223 phase (n = 3, T c ~ 110 K), respectively. The high Bi-2223 phase has many characteristics, that make it the most desirable of these phases, including high T c (~ 110 K), critical current density, J c (~ 106 A/cm 2 ), the extremely high value of upper critical magnetic field at liquid helium temperatures (H c2 (4.2 K) > 100 T), atmospheric stability, and especially an excellent mechanical performance [3][4][5][6][7][8][9]. As a result, it was used in numerous industrial setups, including power cables [10], magnetic sensors [11], high-field magnets [12,13], and magnetic resonance imaging systems [14,15]. However, it still has many manufacturing issues, such as very low ductility and high brittleness due to their ceramic nature, resulting in poor mechanical properties for practical applications [16][17][18]. Therefore, to improve these restrictions, the microstructure and mechanical parameters of ceramic superconducting materials should be improved. Therefore, many researchers directed their efforts towards overcoming this limitation in diverse ways. The most efficient way is by adding nanoparticles into the pores between the grains in the structure of superconductor composites [19,20]. The Vickers microhardness (H V ) technique is a favorite for calculating the mechanical parameters of superconducting materials, because it is easy, efficient, simple, accurate, and nondestructive. Additionally, crucial structural factors like strength and deformation characteristics can be identified [19,21]. With the addition of various nanoparticles with different percentages of the (Bi,Pb)-2223 superconducting phase, many researchers have been working to examine and enhance the phase formation, microstructure, electrical transport, and mechanical properties. Rahal et al. [22] investigated the superconducting and mechanical properties of the composite (SnO 2 ) x (Bi,Pb)-2223 with different sintering temperatures, and found that the addition of nano-SnO 2 particles enhanced the mechanical properties of superconductors. Habanjar et al. [23] reported the impact of BaFe 12 O 19 nanoparticles on the (Bi,Pb)-2223 phase. Their results showed a rise in the values of microhardness, H v , in the (Bi, Pb)-2223 phase with the addition of nanoparticles. Awad et al. [24] discovered that adding SnO 2 nanoparticles improved the mechanical properties of the (Bi,Pb)-2223 phase until x = 0.4 wt%. Also, El Ghouch et al. [19] found that the addition of two transition metals (FeSiW 11 and MnSiW 11 ) nanoparticles into the (Bi,Pb)-2223 phase increased the mechanical parameters up to 0.04 wt% and 0.08 wt%, respectively.
The main purpose of this work is to investigate the effect of hard magnetic ferrites' Ba 0.5 Sr 0.5 Fe 12 O 19 nanoparticles on the mechanical characteristics of the (Bi 1.8 Pb 0.4 ) Sr 2 Ca 2.1 Cu 3.2 O 10+δ superconductor phase, besides phase formation, microstructure, and electrical resistivity measurements. In addition, the microhardness values (H V ) have been examined according to several theoretical models in the load-dependent and independent (plateau region) regions.
Moreover, the variation of microhardness values (H V ) with time in the range of duration (15-60 s) using the indentation creep mechanism was investigated.

Experimental procedure
Conventional solid-state reaction technique was used to prepare the superconducting phase nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 ) x / (Bi 1.8 Pb 0.4 )Sr 2 Ca 2.1 Cu 3.2 O 10+δ with x = 0.00, 0.01, 0.02, 0.03, 0.04, 0.05, 0.10, and 0.20 wt%. Appropriate stoichiometric amounts of Bi 2 O 3 , PbO, SrCO 3 , CaO, and CuO (Sigma-Aldrich chemicals, > 99% purity) were mixed, ground in an agate mortar, and sifted using a 125 μm sieve. The resulting powder of the homogeneous mixture was calcinated twice in air at 1093 K for 24 h, with grinding and sifting between the two stages of the calcination process. The resulting powder was ground, sifted, and then pressed into a disc (1.3 cm in diameter and 0.3 cm in thickness) under a pressure of 0.98 GPa, using a Pye Unicam hydraulic press. The pellets were then sintered in air for 96 h at 1118 K at heating and cooling rates of 4 K/min and 2 K/min, respectively. At the end of the process, the final pellets were ground and sieved to become a homogenous powder, and the Ba 0.5 Sr 0.5 Fe 12 O 19 nanoparticles of average grain size (48.987 nm) synthesized via the wet chemical coprecipitation technique, as previously reported [25], were added with different weight percentages (x = 0.00, 0.01, 0.02, 0.03, 0.04, 0.05, 0.10, and 0.20 wt%) to the resultant powders. The resulting mixture was pressed again at 0.98 GPa and sintered in air at 1118 K with a heating rate of 2 K/ min and a cooling rate of 1 K/min to afford the desired nanoparticle-superconductor composite.
X-ray diffraction (XRD) was used to examine the crystal structure of the prepared nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 ) x /(Bi,Pb)-2223 composite samples using the X'Pert Pro (PANalytical) device with a Cu target (λ = 1.5406 Å) (at 40 kV and 40 mA) in the range 3° ≤ 2θ ≤ 65°. The surface morphologies of the samples were studied via scanning electron microscopy (SEM) using a JSM-IT200 device. The electrical resistivity measurements were carried out using the four-probe technique from room temperature down to zero-resistivity temperature (T 0 ) via a closed cryogenic refrigeration system (Displex) employing helium gas as a working medium. The Vickers microhardness device, model IN-412A, was used to measure the microhardness of composite samples at the various indentation test loads from 0.245 to 9.8 N at a dwell time of 45 s, at room temperature. The same process was performed five times at the various locations on the sample surface to find a more accurate average value. In addition, an indentation creep test was performed on the prepared samples at various dwell times (t = 15, 30, 45, and 60 s) and for fixed loads of 2.94, 4.90, and 9.80 N.

Theoretical approach
The Vickers microhardness tester is the most commonly used for hardness measurement, which involves applying a fixed load to a diamond indenter and measuring the dimension of the resultant indentation on the surface of the specimen [26]. The Vickers microhardness (H V ) values are calculated according to the following equation: where F and d are the applied test load and diagonal length of the indentation in newton (N) and micrometer (μm), respectively. To studying the mechanical properties of ceramic materials, several models have been proposed to interpret the Vickers microhardness data (H V ) behavior and estimate the true microhardness value.

Meyer's law
Meyer's law is used to explain the indentation size effect (ISE) behavior of materials. The relation between the applied load F(N) and the indentation diagonal length d (μm) is given by: where A is the standard hardness constant, and n is Meyer's number. The Meyer number is a leading indicator for defining a compound's mechanical behavior as defined by a stress applied to the surface. In this context, if the value of n > 2, the material shows normal indentation size effect (ISE) behavior, whereas if n < 2, the material will show reverse indentation size effect (RISE) behavior, and if n = 2, the microhardness is independent of the applied load [27,28].

Hays-Kendall approach (H-K)
H-K suggested that there is a threshold load (W) to begin the noticeable indentation in the sample surface [29]. As a result, they suggested an effective indentation load F ef f = F − W HK , which produces the measured indentation as follows: where A 1 is an independent constant from the applied load.
The true hardness of the plateau region according to the Hays-Kendall model, H HK , could be calculated according to Eq. (4)

Elastic/plastic deformation model (EPD)
The elastic/plastic deformation (EPD) model focuses on the elastic recovery that is created by the closeness of the indentation impression. Bull et al. [30] suppose that the indentation includes a portion of elastic distortion, which relaxed when the applied load was removed. Therefore, the measured semidiagonal plastic indentation (d) must be combined with an elastic component (d 0 ) to determine the microhardness [31]. Hence, the load force must be corrected according to the following relationship: where A 2 is constant.
Thus, the true hardness according to the EPD model, H EPD , could be calculated using Eq. (6):

Proportional specimen resistance model (PSR)
According to Li and Bradt [32], the Hays-Kandel approach was modified by replacing the constant threshold load W by a load-dependent term 1 d, where α 1 refers to the test sample's elastic resistance as well as the fractional resistance of the surface cracks [33]. The ISE behavior is governed by Eq. (7), as follows: where 2 is constant related to the true microhardness.
Accordingly, the true hardness, H PSR , can be calculated using the following relation:

Modified proportional specimen resistance (MPSR) model
Gong et al. [34] suggested a modified proportional specimen resistance (PSR) model according to the following equation: Here, the parameter 0 is related to the residual surface stresses associated with the surface grinding and polishing processes during sample preparation. The parameters 1 and 2 are constants as defined in the PSR model and have the same physical meaning.
Using the following equation, one can find the values of load independence with respect to the MPSR model:

Indentation-induced cracking (IIC) model
Finally, the indentation-induced cracking (IIC) model is a useful approach to deciding what behavior (ISE or RISE) of the material is present and calculating the microhardness parameters [35], according to the following equation: In this relation, λ 1 is constant, K 1 is constant depending on the indenter's geometry, and K 2 is constant depending on the applied load. For brittle materials (ceramic), the constant of λ 1 ≅ 0, so Eq. (11) becomes Herein, the parameters of K and m are ascribed to the load-independent constants.
Furthermore, the value of m is like the n value in Mayer's law, where both can be used to identify the indentation size effect (ISE) behavior of materials. The m value has a crucial value of 0.6; if m > 0.6, the material shows normal ISE behavior, whereas m < 0.6 exhibits RISE. Additionally, the values of load-independent parameters in the plateau region are calculated such as elastic modulus (E), yield strength (Y), fracture toughness (K f ), brittleness index (B i ), and elastic stiffness coefficient (C 11 ), according to the following formulizations [36]:

Indentation creep
Indentation creep is an important indentation in many applications of ceramic materials. It is defined as the (11) relative change in the indentation depth, while the applied load is kept constant [37]. Using the Sargent-Ashby model [38], you can investigate indentation creep behavior in terms of the following relationship: H V (t) denotes the time-dependent Vickers microhardness. ε 0 and c are constants, and is the stress exponent. The values of stress exponent μ illustrated the details of the deformation mechanism. It was reported that: if the value of μ is close to 1, the sample has diffusion creep [39], and if it is close to 2, the grain boundary will slide [40]. Dislocations climbs exist for values of μ ranging from between 4 and 6 [41]. When the values of μ are between 3 and 10, dislocation creep dominates [42,43].

Ba 0.5 Sr 0.5 Fe 12 O 19 characterization
The characteristics and analyses of Ba 0.5 Sr 0.5 Fe 12 O 19 nanoparticles were previously reported by Yassine et al. [25], and the findings are summarized in Table 1.

X-ray power diffraction (XRD)
The XRD patterns of the nano-(Ba 0.5 Sr 0. 5 [45,46]. The superconducting phase of (Bi,Pb)-2212 frequently forms as a minority phase in the (Bi,Pb)-2223 high phase and cannot be categorized as an impurity phase [47,48]. While the partially melted liquid Ca 2 PbO 4 phase formed during the heat treatment of Pbdoped Bi-based superconductors (Pb-rich compound) is essential for the development of the (Bi,Pb)-2223 phase [49]. Pb lead reacts with CaCO 3 to produce the highly reactive intermediate phase Ca 2 PbO 4 , which greatly aids in the phase formation of Bi-2223. This intermediate phase readily combines with other substances to form a liquid phase, acting as a reservoir and a fast cation diffusion pathway, and assisting in the formation of the Bi-2223 phase [50,51]. The mixture of the Bi-2212 phase, which formed during the sintering process at temperatures between 750 and 800 °C, and the partially melted liquid Ca 2 PbO 4 phase, which formed above at a temperature of 820 °C, provides the best approximation of the formation of the Bi-2223 phase among several different methods for formation of the Bi-(2223) phase [52][53][54][55].
To stop the growth of secondary phases, which become less reactive as they grow larger, it is crucial to quickly synthesize the Bi-2223 phase during the sintering step. Therefore, this reaction must proceed quickly to prevent segregation and the growth of unnecessary secondary phases, to maximize the conversion of the calcined material to the Bi-2223 phase. A suitable phase composition (Bi-2212, Ca 2 PbO 4 , Ca 2 CuO 3 , CuO) must be obtained during calcination to enable rapid Bi-2223 phase formation [52,[56][57][58]. However, the presence of the minor phases (Bi-2212 and Ca 2 PbO 4 ) after the sintering process may be due to their not having completely transformed into the Bi-2223 phase. Although the secondary phases (Bi, Pb)-2212 and Ca 2 PbO 4 have increased as x increases up to 0.20 wt%, the major phase (Bi, Pb)-2223 remained. This majority could be due to the substitution   [22,65], and ZnO [66] nanoparticles into the (Bi,Pb)-2223 phase. The X-ray diffraction patterns for all samples were refined using the Fullprof program [67]. The details of the refinement procedure were done as follows: In the first step, the intensities of the experimental data were fitted using the pseudo-Voight function to refine the peak shape parameters, the linear interpolation function was used to describe the background between 38 points, and the Berar-Baldinozzi method was applied for the asymmetry correction.
The parameters, such as scale factors, lattice parameters, instrumental profile, peak shape parameters, and structure parameters, were then refined. At the end of the refinement, the agreement factors (R P , R wp ) of (7.92, 10.20), (8.20, 10.50), and (8.29, 10.50), and the goodness-of-fit parameters (χ 2 ) of 1.58, 1.67, and 1.84, respectively, were obtained for samples with x = 0.00, 0.04, and 0.20 wt%, respectively, as shown in the graphs of Fig. 1b. Furthermore, Fig. 1c visualizes the crystal structures of a pure (Bi,Pb)-2223 sample, drawn with VESTA software [68] through Fullprof output data. As shown, the unit-cell structure consists of two blocks of BiO-SrO-CuO 2 -Ca-CuO 2 -Ca-CuO 2 -SrO-BiO sequences along the c-axis. Moreover, the estimated average bond length of Cu-O in the CuO 2 plane is 1.89 Å, which is within the expected equilibrium distance range (1.80-2.00 Å), whereas the average Bi-O-bond length is 2.72 Å, which is within the expected equilibrium distance range (2.30-2.70 Å) [69][70][71].
The lattice parameters of nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 ) x / (Bi,Pb)-2223 composite samples were obtained from the Rietveld refinement and are listed in Table 2. It is clear that the lattice parameters (a and b) are almost unchanged, while there is a slight fluctuation without systematic trend in c-lattice parameter values that may be caused by a change in the distance between CuO 2 layers, which could affect how charges are transferred to these layers after the addition of some nanoparticles and the occurrence of a disordered rearrangement of oxygen vacancies. Similar behavior was mentioned early on for (Bi,Pb)-2223 phase inclusion with different nanoparticles, such as (K 5 [72] and TiO 2 [73]. Moreover, the orthorhombic distortion OD (b-a/a) is calculated and presented in Table 2. It is obvious that all values of OD are less than 1% for all composite samples, approving the stability of the (Bi, Pb)-2223 phase, and there is no transition to a tetragonal structure [74]. This suggests that the nano-(Ba 0. 5 Sr 0.5 Fe 12 O 19 ) x addition does not enter the crystal structure of the (Bi, Pb)-2223 phase but rather settles between the grains at low concentrations while sticking above the grains at higher concentrations [23,62,75,76], as discussed in detail in the SEM section.
The relative intensity method was used to compute the values of the relative volume fraction of the main phase and the impurity phases [77], and their values are presented in Table 2. The volume fraction of the (Bi,Pb)-2223 phase increased to 87.87% for x = 0.04 wt% by enhancement percentage of 7.588% and then decreased for further addition. Conversely, the volume fraction of (Bi,Pb)-2212 and Ca 2 PbO 4 impurity phases show opposite trends. According to these findings, a small addition of nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 ) up to 0.04 wt% improved the phase formation of (Bi, Pb)-2223, whereas a higher concentration degraded the main phase. This improvement could be explained by the addition of a low concentration of nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 ), which improved the viscosity, homogeneity, and rate of formation of the (Bi, Pb)-2223 phase at the reaction temperature [64]. While the  [64], MgO [11], and Eu 2 O 3 [78] nanoparticles were added to the (Bi,Pb)-2223 superconducting phase. Moreover, the ferrimagnetic nature of Ba 0.5 Sr 0.5 Fe 12 O 19 nanoparticles, with the saturation magnetization (M s = 56.619 emu/g) and the coercivity field (H C = 5813.9 G), which encourages Cooper pair breaking and retards the formation of the (Bi,Pb)-2223 phase at high concentrations, may also be responsible for this delay [79]. The porosity percentage (P%) of nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 ) x /(Bi, Pb)-2223 composites was calculated using Eq. (19) and is plotted in Fig. 2 where ρ exp and ρ th are the experimental and theoretical densities, respectively.
It was discovered that the value of (P %) for our samples ranged from 20 to 40%, which is acceptable for ceramic superconductor materials [80]. Furthermore, as the nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 ) concentration increased up to x = 0.04 wt%, the porosity decreased by 5.688%, indicating that the addition of a small amount of nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 ) improved the microstructure of these samples. Then, for x = 0.05 wt%, a slight increase and a noticeable increase for x = 0.10 and 0.02 wt% resulted in a significantly increased porosity, increasing the chance of the formation of voids and pores, which weakens the links between the grains for these samples, as will be discussed in detail in the SEM section.

SEM
SEM micrographs of selected nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 ) x / Bi 1.8 Pb 0.4 Sr 2 Ca 2 Cu 3.2 O 10+δ composites, with x = 0.00, 0.01, 0.04, 0.05, and 0.20 wt%, are shown in Fig. 3. All samples shared flaky layers of random alignment and large platelet-like structures, which are hallmarks of the main phase (Bi,Pb)-2223 [64]. Besides, the appearances of some grains with irregular shapes were designated for the impurity phase Ca 2 PbO 4 with different percentages in all micrographs of composite samples, as in the composite sample with x = 0.05, and 0.10 wt%, respectively. Moreover, there are still tiny spherical grains that might be signs of Ba 0.5 Sr 0.5 Fe 12 O 19 nanoparticles in the (Bi,Pb)-2223 phase, which agglomerate and stack on the grain's surface as the concentration of nanoparticles increases, as seen in x = 0.20 wt%. The average grain size was calculated using the ImageJ program [81]. The distribution of grain sizes can be clearly seen from the histogram in Fig. 3. It was noted that the average grain size increased as the concentration of nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 ) increased up to x = 0.04 wt% and then decreased for further concentrations, as shown in Table 2. Meaning that the addition of a small amount of nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 ) improved the grain's growth, and this indicates an enhancement in the grain's conductivity. The nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 ) added to fill voids and cracks reduced the porosity of these samples [82][83][84]. While the voids in the composite samples with x = 0.05 wt% increased slightly and the grain size began to degrade, they were still better than the pure sample. The voids increased significantly for samples as the nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 ) concentration increased, as seen in the micrograph for x = 0.20 wt%, and their grain size decreased. This is due to the agglomeration of the Ba 0.5 Sr 0.5 Fe 12 O 19 nanoparticles at high concentration, which produces the nano-(Ba 0.5 Sr 0. 5

Electrical resistivity measurements
The electrical resistivity dependence on temperature of nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 ) x /(Bi,Pb)-2223 samples with 0.0 ≤ x ≤ 0.2 wt% is represented in Fig. 4. All samples show metallic-like behavior from room temperature down to the onset temperature ( T onset c ) , followed by a superconducting transition at T c . A small curvature in the electrical resistivity data is observed above T c , indicating superconducting thermodynamic fluctuations [33]. These fluctuations are due to the appearance of Cooper pairs in a finite range above T c [88]. The values of T c are determined from dρ/dT versus T plot as the temperature corresponds to the maximum dρ/dT as shown in the inset of Fig. 4, and their values are displayed in Table 3. The zero-resistivity temperature, T 0 , also exhibits the same patterns as T c , rising from 93.250 to 103.500 K when x increased from 0.00 to 0.04 wt%, and then falling to 80.00 K when x = 0.20 wt%. As seen from Table 3, as x increases from 0.00 to 0.04 wt%, the T c increases from 107.831 to 112.440 K, and then, it is reduced for further addition. The increases in T c can be attributed to the increase in the (Bi, Pb)-2223 volume fraction phase, the improvement in grain connectivity, and the increasing grain size in the low concentrations, as confirmed by the SEM result. The decrease in T c values, on the other hand, is attributed to an increase in the impurity phases, and a weak link in grains conductivity due to nanoparticles' agglomeration, which results in scattering and the breaking of the Cooper pairs due to the ferrimagnetic nature of the nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 ), with saturation magnetization (M s = 56.619 emu/g) and the coercivity field (H C = 5813.9 G) [89]. The values of the superconducting transition width, ∆T = T c -T 0 , which determine the quality of the superconducting transition and the purity of the sample, are shown in Table 3. It is noted that the transition to T 0 for the samples with x = 0.00 to 0.05 wt% seems to be nearly sharp, which indicates that the phase formations of these samples are nearly identical to single-phase formations [90]. On the other hand, the samples with x = 1.0 and 0.2 wt% showed a broadening in the transition to T 0 ; this broadening being attributable to the dominance of impurities on the phase formation of these samples, which resulted in a decrease in the T c value. As a result, for x > 0.05 wt%, high The normal state resistivity (ρ room ) at 300 K, residual resistivity ( 0 ) at 0 K, and the resistivity temperature coefficient ( ) were obtained by the fitting of the resistivity data in range of temperature 150 ≤ T ≤ 300 K according to Matthiessen's rule [91], and their values are presented in Table 3 As seen in Table 3, the lower values of the resistivity parameters ρ room , ρ 0 , and of nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 ) x / (Bi,Pb)-2223 samples at x = 0.04 wt%, indicate a high purity of this sample, as obtained from XRD data, and good links between superconducting grains, as well as (20) ρ(T) = ρ 0 + αT.  . Each sample clearly exhibits non-linear microhardness behavior (loaddependent region) and then progresses to sustain saturation as the applied load exceeds 3 N (load-independent region). This behavior can be explained based on the penetration depth of the indenter [92]. The indentation size effect (ISE), which describes non-linear behavior in a load-independent region, has been widely reported in the literature  ) has a noticeable influence on increasing microhardness, indicating an improvement in the mechanical properties of these samples. This improvement is attributed to the SEM results, which revealed an increase in grain connectivity due to porosity reduction and an increase in specimen resistance to crack propagation among grains due to the addition of nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 ). While it is noted that there was a slight decrease in H v values for x = 0.05 wt%, it is still higher than the composite sample with x = 0.03 wt%. The values of H v then decreased significantly with an additional increase up to x = 0.2 wt%. This decrease is attributed to the unsatisfactory results for the porosity and SEM micrograph analysis of these samples.

Analyses and modelling of microhardness
The (ISE) behavior for nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 ) x /(Bi,Pb)-2223 composite samples was investigated, using Meyer's law according to Eq. (2) and the graph of lnF versus lnd , as shown in Fig. 6 for some selected samples. The values of the constants A and Meyer's number n were determined from the fitted curves of the experimental data, as shown in Fig. 6. It is clear that from  [95]. Moreover, the values of Meyer's constant A are not appreciated for high-temperature superconductors due to their small values. These results are similar to those found by Abdeen et al. [33] in their study to investigate the electrical and mechanical properties of (Bi,Pb)-2223 by substitution of Ca 2+ by Ho 3+ ions.
The  Table 4. In addition to that, the loadindependent microhardness H H-Kin values were calculated by Eq. (4) and are listed in Table 5.
According to Table 4, the nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 ) x /(Bi,Pb)-2223 composite samples positive W HK values show that the applied load is sufficient to cause both elastic and plastic deformation [96]. Furthermore, the microhardness H V values of nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 ) x / Bi 1.8 Pb 0.4 Sr 2 Ca 2 Cu 3.2 O 10+δ composite obtained from Eq. (3) are lower than our experimental H V values as seen in Fig. 8 for samples with x = 0.00, 0.03, 0.04, and 0.20 wt%. Additionally, the true microhardness H H-Kin values, which were calculated by Eq. (4) as seen in Table 5, are lower compared with the values of H Vexp that were obtained in the plateau region. Therefore, the H-K approach is not suitable to describe our experimental data for all samples.
The  d as given in Fig. 9 and are listed in Table 4. Moreover, the load-independent microhardness H EPDin values were calculated using Eq. (6) and are tabulated in Table 5.
From Table 4, the values of d 0 are positive, which confirms that all composite samples exhibit the ISE behavior. On the other hand, the smallest value of d 0 (9.584 μm) and the best value for the parameter A 2 for x = 0.04 wt%.
It means that adding nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 ) 0.04 improved the mechanical properties of this sample. Furthermore, as shown in Fig. 8 true microhardness values H EPDin calculated using Eq. (6) were lower than the H Vexp values obtained in the plateau region. Thus, the (EPD) model is not suitable to explain our H V experimental data.
The PSR model was used to analyze the microhardness of nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 ) x /(Bi,Pb)-2223 composite samples. According to Eq. (7), the H V values were also calculated through a plot of F/d against d that yielded a straight line as seen in Fig. 10, where the best-fit parameters of 1 and 2 can easily be obtained by regression analysis and are depicted in Table 4. Moreover, the H PSRin values are estimated from Eq. (8) and summarized in Table 5.
It is noted that in Table 4, the parameter 1 values are positive for all samples, confirming that all samples present normal (ISE) behavior as well as the production of both elastic and plastic deformations at the same time.  Fig. 8c. In addition, it is apparent from Table 5 that the true microhardness values of H PSRin were calculated from Eq. (8). It was found that their values were the closest to the H Vexp values that were obtained in the plateau region from the other models. As well, it can be seen in Table 5 that the PSR model has the smallest deviation percentage (D%) in comparison with the other models. It implies that the theoretical PSR model is appropriate for explaining the general mechanical behavior of (Bi, Pb)-2223 superconducting ceramic systems.
The microhardness data for the prepared nano-(Ba 0.5 Sr 0.

Fe 12 O 19 )
x /(Bi, Pb)-2223 composite samples were also analyzed using the MPSR model. The H V values were determined using Eq. (9). The values of 0 , 1 , and 2 parameters are obtained from the quadratic regressions of a plot of F against d, as seen in Fig. 11, and they are tabulated in   Table 4. Furthermore, the H MPSRin values are obtained from Eq. (10) and summarized in Table 5. Table 4 shows that the nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 ) x /(Bi, Pb)-2223 composite samples exhibit typical ISE behavior due to positive MPSR parameter α 2 values. The values of parameter α 2 increased with the increment in the nanoaddition concentration up to the value of x = 0.04 wt% and then decreased to x = 0.20 wt%. This means that the sample with the highest nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 ) concentration has the best mechanical durability, strength, and fracture toughness against the applied test load [97]. Furthermore, the H V values calculated by Eq. (9) for nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 )x/(Bi, Pb)-2223 composite samples are nearly identical to the H V experimental values, as shown in Fig. 8. However, it is noted that the values of H MPSR conducted in the low load range are slightly increased, which is attributed to the effect of the existence of the indentation load term on hardness testing conducted in the low load range [34,98]. Based on this, we can conclude that the MPSR model can provide a satisfactory description of our experimental data within a specific range. Moreover, the true microhardness values H MPSRin which are obtained from Eq. (10), as seen in Table 5, are slightly close to the H Vexp values that were obtained in the plateau region. In particular, the samples with x = 0.03 and 0.04 wt% are much closer than other samples, as shown in the inset figure in Fig. 8b. However, the MPSR model is considered unsatisfactory, because it does not interpret all our experimental results compared with the PSR model. Finally, IIC model was used to examine the crucial changes in the general mechanical characteristic features in addition to the load-independent microhardness parameters for the nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 ) x /(Bi,Pb)-2223 composite samples. The H V values are calculated from Eq. (12). The parameter values of m and k are extracted from the plotting curves in Fig. 12 and listed in Table 4.    (14), (15), (16), and (17), respectively. All parameters follow the same trend, increasing systematically with increasing the nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 ) concentration up to x = 0.04 wt%, and then decreasing to x = 0.20 wt%, as shown in Table 6. The toughness (K f ) of high-temperature superconductors is the most important parameter, because it affects the material's mechanical behavior, which is the material's capacity to withstand cracks [99]. The toughness increased at the optimum of 0.04 wt% by 6.529% of the pure sample due to an increase in the surface energy α 1 . That is, a small amount of nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 ) enhanced the mechanical properties of the superconducting ceramic systems of the (Bi, Pb)-2223 phase. On the contrary, they decreased for high doping x > 0.04 wt%. which was attributed to the poor results that were obtained from the microstructure analysis for these samples. These observations were reported in the literature [22,23,79,100].   x /(Bi,Pb)-2223 composite samples. As seen in these figures, the values of Vickers microhardness H V decrease slightly and linearly with increasing dwell time for all samples. This behavior supports the indenter's basis for penetration depth and appears to represent the typical relationship between microhardness and time [101]. This fall in the hardness values of the (Bi,Pb)-2223 phase with dwell time indicates material-bearing creep deformation [19].  Table 7. It is clear that the values of the stress exponent ranged between 10 and 12.2 for samples with low nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 ) concentrations up to 0.04 wt%, which indicates that the operative creep mechanism of these samples is dislocation creep [22]. The stress exponent for high-doped x > 0.05 wt%, on the other hand, ranges between 4.2 and 7.6. Thus, the mechanism of these samples is dislocation climbs creep.

Conclusion
Polycrystalline nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 ) x /(Bi, Pb)-2223 superconducting composites with x = 0.0, 0.01, 0.02, 0.03, 0.04, 0.05, 0.1, and 0.2 wt% have been successfully prepared. The XRD results showed the hard magnetic ferrite nano-(Ba 0.5 Sr 0.5 Fe 12 O 19 ) does not enter the crystal structure of the prepared phase and instead is distributed and settled between the superconducting grains. The SEM analysis showed that the average grain sizes increased up to x = 0.04