Composition dependent elastic and thermal properties of LiZn ferrites

Journal of Alloys and Compounds 728 (2017) 1091e1100

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Composition dependent elastic and thermal properties of LieZn ferrites A.V. Anupama a, 1, V. Rathod b, c, 1, V.M. Jali b, B. Sahoo a, * a

Materials Research Centre, Indian Institute of Science, Bangalore, 560012, India Department of Physics, Gulbarga University, Gulbarga, 585106, India c Department of Physics, Government College, Gulbarga, 585105, India b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 3 July 2017 Received in revised form 8 September 2017 Accepted 9 September 2017 Available online 9 September 2017

We report the composition dependent elastic and thermal properties of LieZn ferrite (Li0.5-x/2ZnxFe2.5-x/ 2O4, x ¼ 0.0, 0.2, 0.4, 0.6, 0.8 and 1.0) ceramics. The IR absorption frequencies (obtained from the Fourier transform infrared spectra), the lattice parameters and the mass-densities were used to evaluate the elastic and thermal parameters such as force constant, elastic moduli and Debye temperature of the studied LieZn ferrite samples as a function of Zn-content x. The observed force constants were low in value for the octahedral site complexes than that of the tetrahedral site complexes, but both the force constants were increasing with increasing Zn content. The longitudinal elastic wave velocities in the sample were higher than the transverse wave velocities. The elastic moduli such as Young's modulus (E), bulk modulus (B) and rigidity modulus (G) showed increasing trends with Zn concentration in the samples. The calculated Debye temperature (qD) is near to the literature values obtained experimentally. Our results provide a better understanding of the variation of mechanical, elastic and thermal properties of LieZn ferrites. © 2017 Elsevier B.V. All rights reserved.

Keywords: FTIR spectroscopy LieZn ferrites Sound velocity Elastic modulus Thermal properties

1. Introduction The mechanical properties of materials can be understood from the nature of the chemical bonds and their vibrational properties. Most of the properties such as magnetic, mechanical, vibrational, elastic and thermal etc., depend on the nature of binding forces of materials. As the binding forces depend on the type of atoms/ions and the inter-atomic/inter-ionic forces involved [1], substitution of host elements by dopants invariably result in varied physical properties [2]. Hence, lithium ferrite is often doped with other cations to optimize its electrical, magnetic, and elastic properties [3]. Lithium ferrites are spinel ferrites having general formula AB2O4, which is represented as (A)[B2]O4. Here, the parenthesis represents the tetrahedral site and the square bracket represents the octahedral site. Depending on the position of the divalent cations, i.e., tetrahedral, octahedral or both sites, the spinel ferrites are divided into three categories: (i) direct (or normal) spinel ferrites (where Zn2þ, Mg2þ, Mn2þ etc. occupy the tetrahedral (A) site [4,5]),

* Corresponding author. E-mail address: [email protected] (B. Sahoo). 1 These authors contributed equally. http://dx.doi.org/10.1016/j.jallcom.2017.09.099 0925-8388/© 2017 Elsevier B.V. All rights reserved.

(ii) inverse spinel ferrites (where Ni2þ, Fe2þ etc. occupy the octahedral (B) site [6,7]) and (iii) mixed spinel ferrites (where the occupation of divalent cations is distributed in A- and B-sites [8]). Note that LieZn ferrites (ZnxLi(1-x)/2Fe(5-x)/2O4) are typical examples of mixed spinel ferrites, where the Zn2þ occupy the tetrahedral sites and the rest of the required divalent-metal cations (other than Zn2þ) are formed by the combination of cations, [Liþ0.5Fe3þ0.5], which occupy the octahedral sites of two adjacent sub-unit cells [9]. However, in certain spinel compounds (e.g., LiCrGeO4) Liþ ion is observed to occupy tetrahedral position [10]. Ferrites are widely used for microwave and magnetic memory applications [11e17]. The data reading and writing on magnetic storage devices or operation of microwave based devices produces heat. In microwave devices, such as circulators and filters, a part of the microwave is often absorbed in the material producing heat energy. Accumulation of such heat is detrimental for the magnetic behavior of the materials and also for the adherence of the material to the circulator (due to thermal expansion and contraction). Dissipation of this heat is important for proper functioning and durability of all such devices. Hence, the mechanical (elastic), thermal and vibrational properties of such microwave ferrites is very important. In this work, we have used Fourier transform

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infrared (FTIR) spectroscopy to understand these properties. The infrared (IR) spectroscopy deals with the interaction between a molecule and IR radiation, which causes the excitation of the (stretching and bending) vibrations of bonded atoms within the molecule. The vibrations of cations which are bonded to the oxygen ions in ferrites produce different vibrational frequencies that appear as IR absorption bands in the FTIR spectrum. Hence, the position of the spectral bands depends on the type and strength of bonding, coordination between the ions involved, crystal/local symmetry and type (mass) of the ions involved. The principle of FTIR spectroscopy can be well understood using spring-model of vibration of atoms, wherein the wave number n is defined as:

n¼

1 2pc

2.2. Cation occupation and lattice parameter The lattice parameters and bond lengths are important parameters for the understanding of the elastic and thermal properties of materials. To compare the lattice parameters obtained from the Rietveld refinement of the XRD data of our samples from theoretically estimated values, we have calculated the theoretical lattice parameters (ath) for all the analyzed samples based on the ionic radii of cations (Liþ(Oh): 0.76 nm, Fe3þ(Oh): 0.645 nm, Fe3þ(Th): 0.49 nm and Zn2þ(Th): 0.6 nm) and anion (O2: 0.132 nm) in cubic close packed geometry. We have used the following expression [19]:

ath ¼

qffiffiffiffiffiffiffiffiffiffi k=mr

(1)

where k is the spring (force) constant, mr is the reduced mass of ions involved and c is the velocity of light. The force constant is a measure of bond strength and an important quantity controlling elastic properties of materials. Hence, it is quite interesting to obtain the force constants (k) and other elastic moduli along with the thermal properties of LieZn ferrites for better understanding of the materials and development of high performance devices. In this work, we report the detailed study of elastic and thermal properties of Li0.5-x/2Znx Fe2.5-x/2O4 ceramics through analysis of their FTIR spectra. The Li(0.5-x/2)ZnxFe(2.5-x/2)O4 ceramic powder samples with x ¼ 0.0, 0.2, 0.4, 0.6, 0.8 and 1.0, were labeled as LZF0, LZF2, LZF4, LZF6, LZF8 and LZF10, respectively. Using the reduced mass (m) and observed absorption frequencies in the FTIR spectrum, the force constant (k) was calculated via the spring model of atomic vibrations (eq. (1)). Other mechanical parameters such as elastic wave velocity, elastic moduli and Debye temperature (qD) were also calculated based on these results. The obtained results were compared with the available literature values on similar ferrite materials. 2. Materials and methods 2.1. Synthesis of LieZn ferrite samples The combustion synthesis method [9,18] is a simple, single-step and cost effective method for the synthesis of ceramic powders. We have used this method for the synthesis of all the LieZn ferrite samples studied here. The detailed synthesis-procedure is described elsewhere [9]. The samples were previously character€ssbauer spectroscopy ized in detail by X-ray diffraction (XRD), Mo and pulse field magnetometry [9]. Note that the samples are phase pure for lower concentrations of Zn (x < 0.6), however, for higher Zn-content (x  0.6), presence of very small amounts (less than a few %) of impurity phases such as ZnO, Fe3O4 and a-Fe2O3, were observed [9].

i pffiffiffi 8 h ðrA þ R0 Þ þ 3ðrB þ R0 Þ 3√3

(2)

where, rA and rB are the average cationic radii at A- and B-sites, respectively. RO is the oxygen ion radius (0.132 nm) [20]. In all our samples, as different number and type of cations are present in the tetrahedral and in the octahedral sites, we have used the average cationic radius in eq. (2). For the calculation of the average cationic radii of the tetrahedral (r A ) and octahedral (r B ) site-ions the cation distribution obtained from the Rietveld refinement of the XRD data [9] were considered. The ionic radii of all the elements were taken from Shannon [20] and David van Horn [21] data base. As per the Rietveld refinement, all the Zn2þ ions preferentially occupy the Asites, all the Liþ ions occupy the B-sites, and the rest of the A and Bsites are occupied by Fe3þ cations. Hence, the average cationic radii are given by:

rA ¼ ½xrZn2þ þ ð1  xÞrFe3þ  rB ¼

(3)

1 ½ð0:5 þ 0:5xÞrLiþ þ ð1:5  0:5xÞ rFe3þ Þ  2

(4)

where, rZn2þ , rFe3þ and rLiþ are the ionic radii of the Zn2þ, Fe3þ and Liþ ions, respectively. The calculated lattice parameters (ath) are listed in Table 1, along with the lattice parameter (ax-ray) values (of the intended LieZn ferrite phases) obtained from the Rietveld refinement of the XRD data [9]. The plots of ath and ax-ray versus Zn concentration (x) is shown in Fig. 1. 2.3. FTIR spectra The structural, elastic and thermal properties of the synthesized LieZn ferrite samples were studied via FTIR spectroscopy. Fig. 2 shows the measured FTIR spectra of all the studied samples, where splitting of the FTIR absorption bands are seen. The FTIR spectra were recorded at RT in the wave number range of 250e1000 cm1 using KBr disc technique using Bruker Tensor 27, FTIR spectrometer. The sub-band positions are listed in Table 2 and named in Fig. 2. The vibrational modes corresponding to the sub-

Table 1 The r, a, u and M-O represent the average cation radii, lattice parameters, oxygen position parameters and cation to oxygen-ion bond lengths, respectively, for the Li(0.50.5x)ZnxFe(2.5-0.5x)O4 samples.. A and B correspond to the tetrahedral and octahedral sites, respectively. rA and rB are the average cation radii at tetrahedral and octahedral sites derived using equations (3) and (4), respectively. ‘ath’ and ‘ax-ray’ are the theoretical and experimental values of lattice constant, respectively. ‘uth’ and ‘ux-ray’ are the theoretical and experimental values of oxygen position parameters, respectively. (AeO)th and (AeO)x-ray are the A-site ion to oxygen bond length and (BeO)th and (BeO)x-ray are B-site ion to oxygen bond length, where the subscripts ‘th’ and ‘x-ray’ represent theoretical and experimental values, respectively. Sample

rA (nm)

rB (nm)

ath (nm)

uth (nm)

ax-ray (nm)

ux-ray (nm)

(AeO)th (nm)

(BeO)th (nm)

(AeO)x-ray (nm)

(BeO)x-ray (nm)

LZF0 LZF2 LZF4 LZF6 LZF8 LZF10

0.069 0.073 0.077 0.080 0.084 0.088

0.074 0.073 0.072 0.071 0.070 0.069

0.8342 0.8373 0.8403 0.8434 0.8464 0.8495

0.3849 0.3871 0.3892 0.3913 0.3934 0.3954

0.8328 0.8348 0.8377 0.8401 0.8427 0.8437

0.3852 0.3875 0.3896 0.3918 0.3940 0.3964

0.1950 0.1988 0.2026 0.2064 0.2102 0.2140

0.2003 0.1992 0.1982 0.1971 0.1961 0.1950

0.1950 0.1990 0.2030 0.2060 0.2100 0.2140

0.1997 0.1983 0.1972 0.1959 0.1947 0.1928

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tetrahedral (A) site (kt) and octahedral (B) site (ko) were calculated following the method by Waldron [23], through the use of equations (5) and (6) given below [24].

kt ¼ 7:62  MA  n21  107 N=m ko ¼ 10:62 

Fig. 1. Dependence of the calculated (ath) and experimentally obtained (ax-ray) lattice parameters on Zn concentration (x) in Li(0.5-0.5x)Znx Fe(2.5-0.5x) O4 samples.

(5)

MB  n22  107 N=m 2

(6)

where, MA and MB are the average molecular weights of cations at the A-sites and B-sites, respectively. For calculating molecular weights of A- and B- sites we have considered the ideal site occupancies, i.e., the whole amount of Liþ goes to the octahedral sites and Zn2þ goes to the tetrahedral sites; the rest of the available Aand B- sites are occupied by Fe3þ ions to complete the stoichiometric AB2O4 spinel structure. The calculated molecular weights are listed in Table S1 (Supplementary information (SI)). In eqns. (5) and (6), n1 and n2 are the wave numbers corresponding to the vibrational frequencies of the cations at the tetrahedral and octahedral sites, respectively. The averaged wave numbers are given in Table 2. As all the different atoms at a particular site are not vibrating at a single frequency and the probability of IR absorption is different for different frequencies, it is not accurate to consider only a single frequency, as given in Table 2, to calculate the average force constants. Hence, to calculate the average wave number corresponding to vibrations of the atoms, we have adopted three different ways

Fig. 2. The FTIR spectra of Li(0.5-0.5x)Znx Fe(2.5-0.5x)O4 samples with x ¼ 0.0, 0.2,0.4, 0.6, 0.8 and 1.0. For clarity, the spectra of samples with x ¼ 0.0, 0.2, 0.4, 0.6 and 0.8 are shifted by 155, 120, 85, 65 and 55 units, respectively.

bands were same as assigned earlier [22]. 2.4. Force constants

Fig. 3. Dependence of force constants (kt, ko and kav) on Zn concentration “x” in Li(0.5Fe(2.5-0.5x)O4 samples. The superscripts wt, no and pb correspond to weighted-, number- and primary-band average values of wave numbers, respectively.

0.5x)Znx

The force constant per atom (Fig. 3) corresponding to the

Table 2 FTIR absorption band positions corresponding to A and B sites of Li(0.5-0.5x)ZnxFe(2.5-0.5x)O4 samples. The sub-bands n1(1) and n2(1) are taken as primary bands n1* and n2*, av respectively. nav 1 and n2 represent number averaged values of sub-band positions corresponding to A- and B- sites, respectively. Sample

A-site (cm1) þ

3þ

Li eO/Fe

LZF0 LZF2 LZF4 LZF6 LZF8 LZF10

eO

nav 1

3þ

Li eO/Fe

n1(2)

n1(3)

n1(1)

n1(4)

710 707 706 707 702

670 665 667 667 666 667

583 578 574 576 578 602

545 538 547 545 540 543

627 622 623 624 621 603

nav 2

B-site (cm-1) þ

2-

eO

2-

2þ

Fe

n2(2)

n2(1)

n2(3)

n2(4)

467 466 470 466 462 459

395 392 396 398 394 395

374 374 374 372 366 371

327 324 328 326 324 326

393 389 392 390 386 387

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through the use of the absorption bands of the FTIR spectrum as explained below. (i) Considering the weighted average of the vibrational wave numbers (frequencies) within vibrational bands of A- or Bsites. For calculating weighted average of the wave number (nwt av ) we have used the formula:

P

f ti i fi

i i nwt av ¼ P

(7)

where fi and ti are the wave numbers (frequencies) and the corresponding transmission probabilities in the measured FTIR spectrum. The transmission probabilities are recalculated after normalizing the transmission intensity in the wave number range of 900e1000 cm1 in the FTIR spectra to 100%. Using the values of nwt av we calculated the corresponding force constants and the values are listed in Table S1 (SI). Fig. 3 shows the variation of the force constants (kwt, wt is given as superscript) with Zn content (x). (ii) Averaging over the wave number values of only the minima of all the absorption sub-bands as seen in Fig. 2. The subband positions taken into account for this averaging are given in Table 2 and the obtained number averaged wave numbers (nno av ) are given in Table S2 (SI). Fig. 3 shows the variation of the force constants (kno, no is given as superscript) with Zn content. (iii) Considering only the primary band positions (n*) (neglecting the split-band frequencies), as given in Table S3 (SI). This is the often reported method in the literature, which may not lead to accurate results. The variation of this force constant kpb (pb given as superscript) with Zn content is shown in Fig. 3. Note that among these three methods the weighted average of vibrational frequencies can be the most reliable averaging, because it considers all the vibrational frequencies with their absorption probabilities. Hence, throughout the paper, the weighted average of vibrational frequencies was assumed as more accurate and all the elastic and thermal parameters obtained using this averaging method, were compared with the literature values. The calculated average force constant per atom as a function of x is shown in Fig. 3. The average force constant per atom (kav) was calculated by averaging over the force constants of two octahedral atoms and one tetrahedral atom. The force constants, kt and ko, calculated using all the above three different ways of averaging vibrational wave numbers, are given in Fig. 3. 2.5. Elastic moduli and sound velocity The elastic constants such as Bulk modulus (B), Rigidity modulus (G or Shear modulus (m)), Young's modulus (E) and Poisson's ratio (P) for all the samples were calculated by using the following formulae. The force constants used for the calculation were obtained by all three different ways of averaging vibrational frequencies. The bulk modulus is [25]:

For isotropic materials including spinel ferrites and garnets (which have cubic symmetry), C11 is nearly equal to 3C44 [26]. Furthermore, for cubic crystals the stiffness constants are related as: C12 z C13 z C44 and C11 z 3C44 [26]. Note that Mazen et al. (in LieNi ferrites) [27] and Ravinder et al. (in NieCd ferrites) [28] assumed that C11 z C12, but this may lead to high errors, hence, we have not used their approximation. The stiffness constant is defined as the force constant per lattice constant. Hence, the average stiffness constant can be written as,

kav a

Cav ¼

where kav is the average force constant per atom and ‘a’ is the lattice parameter. kav per atom is calculated as, kav ¼ 2k03þkt . The values of kt and ko were obtained from the measured FTIR spectra using eqs. (5) and (6) as described above. Using the relations, C12 z C13 z C44, C11 z 3C44 and eq. (8), C11 and C12 can be written in terms of Cav as,

C11 ¼

9Cav 5

(9)

C12 ¼

3Cav 5

(10)

The longitudinal sound wave velocity (Vl) and the transverse sound wave velocity (Vt) are closely related to the stiffness constants [26] and for cubic materials (using eqs. (9) and (10)) they can be expressed as,

Vl ¼

sffiffiffiffiffiffiffiffi C11

sffiffiffiffiffiffiffiffiffiffi 9Cav ¼ r 5r

1 ðC þ 2C12 Þ 3 11

Vt ¼

sffiffiffiffiffiffiffiffi C12

B ¼ Cav

(8a)

r

¼

sffiffiffiffiffiffiffiffiffiffi 3Cav 5r

(12)

The elastic wave velocities Vl and Vt calculated using all the three methods of assigning average wave numbers are listed in Table S4, S5 and S6 (SI). The r values taken for calculation include both X-ray and experimental (bulk) densities (rX-ray and rexp, respectively) as listed in Table 3. The X-ray densities were obtained from Rietveld refinement of the XRD data and the experimental densities were measured using Archimedes principle (liquid: Benzene, pellet diameter: 1 cm, force applied during pelletization: 5 ton). The corresponding plots of Vl and Vt calculated considering X-ray density (for intended phase only) and experimental density in eqs. (11) and (12), are given against Zn-content (x) in Fig. 4 (a) and (b),

Table 3 Mass densities (r) of all samples, where, rth, rx-ray, rav and rref refer to the X-ray densities corresponding to theoretically calculated lattice parameters, refinement results for intended phases, average X-ray density considering all constituent phases and from literature values, respectively. rexp is the experimental (bulk) density of the samples. For the references of the rref are the Refs. [26,27] of Ref. [9].

(8)

where, C11 and C12 are the stiffness constants. The bulk modulus is also expressed in terms of the average stiffness constant, hence,

(11)

and, therefore,

Sample

B¼

(8b)

LZF0 LZF2 LZF4 LZF6 LZF8 LZF10

density (g/cc)

Porosity (%)

rth

rx-ray

rav

rexp

rref

4.83 4.88 5.00 5.09 5.19 5.33

4.76 4.93 5.09 5.16 5.29 5.50

4.76 4.93 5.09 5.19 5.32 5.32

4.47 4.69 4.80 4.46 4.46 4.66

4.76 4.88 4.98 5.09 5.20 5.33

6.0 4.8 5.7 14.0 16.2 12.0

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Fig. 4. Dependence of elastic wave velocities: longitudinal (Vl), transverse (Vt) and mean (Vm) versus Zn concentration “x” of Li(0.5-0.5x)Znx Fe(2.5-0.5x)O4 samples. The superscripts wt, no and pb correspond to weighted-, number- and primary-band average values of wave numbers, respectively. Figures (a) and (b) correspond to elastic wave velocities calculated using X-ray (for intended phases only) and experimental densities, respectively.

respectively. The other elastic moduli were calculated using the formulae given below [29]:

Rigidity modulus G ¼ r  Vt2 ; or; m ¼

E 2ð1 þ PÞ

(13) Vm ¼

εt ð3B  2GÞ Poisson0 s ratio P ¼ ¼ ð6B þ 2GÞ εl

(14)

Young0 s modulus E ¼ ð1 þ PÞ  2G

(15)

The above elastic constants G, E and P were calculated by using both X-ray (rx-ray) and bulk densities (rexp) and are tabulated in Table S7, S8 and S9 (SI). Fig. 5 shows the Zn-content dependence of the elastic moduli (B, G (or m) and E) calculated using eqs. (8), (13) and (15). Fig. 6 shows the variation of rigidity modulus with Young's modulus for different ways of taking band frequencies as described earlier. 2.6. Debye temperature (qD )

hcnav k

(16)

where, nav is the average wave number of the absorption band in the FTIR spectrum, h is Planck's constant, k is Boltzmann constant and c is velocity of light. We have calculated the values of qDW using the average wave numbers obtained via all the three ways of averaging as discussed previously. The calculated qDW values are shown in Fig. 7 and listed in Table S1, S2 and S3 (SI). In the second method, Debye temperature (qDA ) was calculated using Anderson formula [30]:

qDA ¼

1  h 3Nqr 3   Vm kB 4pm

1  3

2 1 þ Vt3 Vl3

!1 3

(18)

Using this formula (eq. (18)), we have calculated the mean elastic wave velocity Vm and the values are plotted in Fig. 4 and tabulated in Tables S4, S5 and S6 (SI), corresponding to the three methods of averaging vibrational wave numbers as described above. Fig. 7 shows the plot of Debye temperatures versus Zn content (x), calculated using both the above methods (eqs. (16) and (17)), along with experimentally obtained Debye temperatures reported in the literature [31].

3. Results and discussion 3.1. Elastic properties (force constant, sound velocity and elastic moduli)

To calculate the Debye temperature we have adopted two methods: In the first method, the Debye temperature (qDW ) was calculated using the formula suggested by Waldron [23],

qDW ¼

Avogadro number, m is molecular weight of the sample (given as MA and MB in Table S1), q is number of atoms in a molecule (in case of our LieZn ferrites q is 7), r is the density of the sample and Vm is the mean elastic wave velocity as defined below:

(17)

where, h is the Planck's constant, kB the Boltzmann constant, N is

The force constant involved in the vibration of atoms in a certain material depends on the distance between the vibrating atoms (MeO bond length), the size of the ions, the valence (charge) and mass of the ions forming the bond and their interaction strength (frequency of vibration). The sound velocity and the elastic moduli (B, G and E) depend on the force constant. All these elastic parameters of the studied LieZn ferrite samples were obtained from the analysis of their FTIR spectra. The lattice parameters (aX-ray) as a function of Zn-content (x) for all the studied LieZn ferrite samples obtained from the Rietveld refinement of the XRD patterns [9] are shown in Fig. 1. We observed that aX-ray increase with increasing Zn-content (x), because the bigger sized Zn2þ cations (ionic radius: 0.6 nm) replace the smaller Fe3þ cations (ionic radius: 0.49 nm) at the A-site. The effect of decrease in lattice parameter due to replacement of bigger Liþ ions (ionic radius: 0.76 nm) by smaller Fe3þ ions (ionic radius: 0.645 nm) at the octahedral site, is less (because, size of the octahedral void is bigger than that of the tetrahedral void). As a result the lattice parameter increases with Zn content. In Fig. 1, although the ath increases linearly with ‘x’, aX-ray showed slight deviation at higher

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Fig. 6. Variation of Rigidity modulus (G) with Young's modulus (E) for Li(0.5samples (calculated using rx-ray). The abbreviations wt, no and pb correspond to G and E calculated using weighted-, number- and primary-band average values of wave numbers, respectively.

0.5x)ZnxFe(2.5-0.5x)O4

Fig. 7. Variation of Debye temperature (qD) with x for Li(0.5-0.5x)ZnxFe(2.5-0.5x)O4 samples. The superscripts wt, no and pb correspond to weighted-, number- and primaryband average values of wave numbers, respectively. The subscripts W and A represent Debye temperatures calculated by Waldron and Anderson methods, respectively, while X-ray and exp correspond to the values calculated using X-ray and experimental (bulk) densities in Anderson method, respectively. The experimental Debye temperature obtained from the ultrasonic pulse transmission technique is qUPT [38]. D

Fig. 5. Variation of (a) Bulk modulus (B), (b) Rigidity modulus (G) and (c) Young's modulus (E) with x for Li(0.5-0.5x)ZnxFe(2.5-0.5x)O4 samples. The superscripts wt, no and pb correspond to weighted-, number- and primary-band average values of wave numbers, respectively.

‘x’ (¼ 1.0). This can be attributed to the decrease in lattice strain due to the phase separation of Zn, forming ZnO, Fe3O4 and a-Fe2O3

impurity phases [9]. Overall, both ath and ax-ray were found to be linearly increasing with Zn concentration validating Vegard's law. Vegard's law is an empirical rule which states that, at constant temperature, a linear relation exists between the crystal lattice parameter of an alloy/ substitutional solid solution and the concentrations of the constituent elements. However, small differences in absolute values of the lattice parameters (between ath and ax-ray, at any particular composition) are observed in Fig. 1. This discrepancy could be attributed to the nano-size effects of samples where bond length tends to contract at free surfaces [32] of nanoparticles, in comparison to the free cationic radii used in our calculations. Furthermore, deviation in effective cationic radii in crystal field of oxygen

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anions has its contribution in lattice contraction of unit cell. In LieZn ferrite, as the Zn concentration increases, the bigger divalent Zn2þ cations are gradually replacing the smaller trivalent Fe3þ cations at the A-site. Simultaneously, there is a decrease in number of bigger monovalent Liþ ions at the B-site. The AeO and BeO bond lengths obtained theoretically [33] and from the Rietveld refinement of the XRD data are given in Table 1. The theoretically calculated bond lengths are in very good agreement with the bond lengths obtained from the Rietveld refinement of the XRD data [9]. As expected, the AeO bond lengths and the rA values increase with Zn content in the sample, but the BeO bond lengths and the rB values decrease. Note that the increase in AeO bond length is ~ 10%, when Zn content varies from x ¼ 0 to 1, whereas the decrease in BeO bond length is only ~ 3% (although ionic size of Fe3þ is much smaller than Liþ at the B-site). The reason behind the latter is that the size of the octahedral void is much bigger than that of the tetrahedral void, as mentioned earlier, and the occupations of bigger Liþ ions do not produce much strain. In Fig. 2, the FTIR spectrum of each of our analyzed LieZn ferrite samples shows typically two main absorption bands [9]: the one at ~ 510 - 800 cm1 (n1) assigned to the stretching vibrations of the tetrahedral (M-O) complexes and the other low frequency band at ~ 250 - 510 cm1 (n2) assigned to the stretching vibrations of octahedral (M-O) complexes [34,35]. Furthermore, the splitting of the IR absorption bands (both tetrahedral and octahedral) into several sub-bands is seen. For the LZF0 sample very prominent splittings of both the bands are observed. This splitting of the bands arises due to the low atomic mass of Liþ ions occupying the octahedral site in comparison to all other cations (Zn2þ and Fe3þ). Furthermore, splitting of both the bands (tetrahedral and octahedral) clearly demonstrates that the vibrations of the tetrahedral complexes are correlated with that of the octahedral complexes. This also suggests that the divalent [Liþ0.5Fe3þ0.5] species in this sample are very well ordered in the LZF0 sample. However, as concentration of Zn increases the sub-bands start to smear out and shift to other frequencies. For the pure Zn-ferrite (LZF10) sample, only two broad bands are observed (along with a band at ~ 300320 cm1 corresponding to the Fe2þ ions in the sample) [9,22]. In Fig. 3, irrespective of the sample and the method used to obtain the average wave numbers corresponding to the vibrations of the cations, the force constant of A-site (kt ) is found to be higher than that of the B-site (ko ). This can be due to the stronger interaction between tetrahedral (Fe3þ or Zn2þ) cations and the surrounding four oxygen (O2) ions than the octahedral cations and the six O2 ions coordinating the octahedral sites. This reason is further substantiated by the shorter AeO bond length than the BeO bond length (Table 1) and the presence of mono-valent Liþ ions at the octahedral sites (except for LZF10 sample). Note here that, higher the valence of the cations, stronger is its interaction with oxygen ions. Furthermore, as seen in Fig. 3, the method of weighted averaging of the vibrational modes leads to higher force constant values for the A-site than the number averaging method, while the force constant obtained using the method of primary band averaging is the lowest. The opposite is observed for the B-site. It is clear that both the force constants, kt and ko , increase linearly with Zn concentration. However, this increase over the whole range of Zn contents (x) is only slightly, because the Zn2þ ions are not much heavier than Fe3þ ions which they replace at the tetrahedral site. The elastic wave velocities Vl, Vt and Vm, for all the studied samples were calculated using the X-ray densities (rx-ray) of the intended phases and the experimental densities (rexp) of the samples through eqs. (11) and (12). Fig. 4 (a) and (b) show the variation of elastic wave velocities versus Zn concentration. The

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elastic wave velocities, i.e., VX-ray and Vexp (Fig. 4, Tables S4, S5 and S6 (SI)) show an increasing trend with Zn-content ‘x’, though small for VX-ray (Fig. 4(a)). This increase in elastic wave velocities can be due to the larger force constant/stronger bond between Zn- and Oions. In Fig. 4 (b), the value of Vexp suddenly becomes higher with Zn- concentration beyond x ¼ 0.4. According to eqs. (11) and (12), this increase can be understood as due to the decrease in experimental density of the sample. Hence, Fig. 4 (b) (in comparison with Fig. 4 (a)) provides the experimental estimation in fluctuation of the wave velocities during practical use of the materials. However, for comparison with the literature data we have used only the parameters obtained using X-ray densities. In Fig. 5 (and Tables S7, S8 and S9 (SI)), the variations of all the elastic moduli with Zn content (x) are shown. The values of E, B and G have increasing trends with increasing Zn-content. Furthermore, the elastic moduli obtained using the weighted average wave numbers of the vibrations are believed to be near to the ideal/expected values than the values obtained by the number-averaging and primary band position averaging methods. The increasing value of elastic constants with increase in Zn content was interpreted earlier via the increased strength of the interatomic binding of Zn2þ ions with its neighbors [36]. We observed that all the elastic moduli vary similarly (linearly) with ‘x’. This is due to the linear relation between these moduli (according to eq. (13)e(15)). A plot of the variation of G with respect to E, for each of the samples is given in Fig. 6, which shows a linear relation. This linear dependency suggests the isotropic elastic nature of these ferrites. It can be observed in Tables S7, S8 and S9 (SI), that the Poisson's ratio (P) remains almost a constant (~ 0.25) for all the compositions, although, it can vary between 1 and þ0.5, for isotropic elastic materials [37]. The elastic and thermal parameters available in the literature for different ceramic systems are compared with our results. Those systems include LieZn ferrites, LieCo ferrites, LieNi ferrites, LieMn ferrites, LieMgeZn ferrites and CueZn ferrites etc., as given below. 3.1.1. LieZn ferrites Previously, Ravinder and Raju [38] studied the elastic properties of LieZn ferrites, Li0.5-x/2ZnxFe2.5-x/2O4 (x ¼ 0.2, 0.4 … 1.0) samples prepared by conventional double sintering method. They used ultrasonic pulse transmission (UPT) technique at RT. According to their results, Vl decreases from ~ 6243 to 4934 m/s and Vt decreases from ~3806 to 3028 m/s with increasing value of ‘x’. Their measured elastic constants, E, G and B, show a random variation with ‘x’. They explained that, this observation was due to the random variation of the atomic binding strength with Zn, although not satisfactory. P was reported to be nearly a constant (~ 0.20e0.21), except for x ¼ 0.6 sample (P ¼ 0.17). The sound velocities (Vl and Vt), the elastic constants and P, showed an abrupt decrease at x ¼ 0.6. However, no proper explanation was given for this observation. Also the end member, pure Zn ferrite, had low values of Vl, Vt and elastic moduli. In the present work, our results (Fig. 4) show that the Vl values increase from ~ 6751 to 6980 m/s and Vt values increases from ~ 3897 to 4030 m/s with increasing value of ‘x’ from 0.0 to 1.0. The mean elastic wave velocity (Vm) increases from 4327 to 4474 m/s. The elastic constants, B increases from 121 to 149 GPa, G increases from 72 to 89 GPa and E increases from 182 to 221 GPa (Fig. 5). However, unlike the earlier reported values [38], our results are more systematical. Accuracy of our results can be rationalized considering that, increasing the amount of Zn in the sample gradually increases the number of ZneO bonds and as the ZneO bonds are stronger, it increases the strength of the materials. Hence, the value of all elastic parameters increases systematically in our

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samples. However, our observed values of P are nearly constant (~ 0.25 ± 0.01) for all our samples (Table S7, S8 and S9). 3.1.2. LieMgeZn ferrites Shaikh et al. [39] reported the increase of force constants with Zn- substitution in single phase LixMg0.4Zn0.6-2xFe2þxO4 (x ¼ 0.0, 0.05, 0.1 … 0.3) samples synthesized by standard double sintering ceramic method. According to their report, the kt increases from ~ 182.8 to 189.4 N/m and ko increases from ~ 110 to 116.8 N/m, with increasing x (from 0 to 0.3). The trends observed for force constants here are similar to the present study on LieZn ferrites, except that the absolute values of the force constants are a little higher. 3.1.3. LieCo ferrites Watawe et al. [35] studied the variation of bond lengths and force constants with Zn- content (x) in Li0.5-x/2CoxFe2.5-x/2O4 (x ¼ 0.1, 0.2 … 0.7) samples prepared by standard double sintering route. According to their results, the force constants kt increases from ~ 160 to 190 N/m and ko from ~ 95 to 100 N/m as x increases from 0.1 to 0.7, although both the bond lengths, at A- and B- sites, were increasing with x. The reason for this, as stated, was the increase in interaction of Zn2þ ions with oxygen anions. According to our results on the LieZn ferrite samples (Table 1), the AeO bond length increases but BeO bond length decreases. This is understood as the replacement of smaller Fe3þ ions by bigger Zn2þ ions at the A-site, and replacement of bigger Liþ ions by smaller Fe3þ ions at the B-site. The increase in both bond lengths observed in LieCo ferrite system could be related to the occupancy distribution of Co in both sites (mixed spinel nature of Co-ferrites). We have observed an increase in the force constants kt from ~ 166 to 177 N/m and ko from ~ 67 to 100 N/m as x increases from 0.0 to 1.0. These results are not very different from the force constants observed in LieCo system, because the Co2þ and Zn2þ cations are not very different in terms of charge and mass. 3.1.4. LieNi ferrites Mazen and Elmosalami [27] studied the behavior of force constants and elastic moduli of Li0.5-x/2NixFe2.5-x/2O4 (x ¼ 0.0e1.0) through the analysis of the FTIR spectra. According to their results the ko increases from ~ 65 to 97 N/m and kt increased from ~ 135 to 150 N/m with increasing ‘x’ (from 0 to 1.0). B increases from ~ 125 to 150 GPa, E increases from ~ 125 to 150 GPa and G increases from ~ 42 to 50 GPa. The Poisson's ratio P was a constant (0.35) for all values of ‘x’. All these results are in the same trend as those of our results of the present study. Comparatively higher absolute values of all the elastic parameters for the LieZn ferrite samples confirms that the LieZn samples are much stiffer. This clearly suggests that the inter-atomic bonding between Zn- and O-ions is much stronger than other cations. A similar observation was made in MneZn ferrites [1,40] as discussed below. 3.1.5. LieMn ferrites The variation of elastic properties in single phase LieMn ferrite (Li0.5-0.5xMnxFe2.5-0.5xO4 (0  x  1.0)) samples synthesized by standard ceramic method were previously studied by Mazen et al. [24]. They reported that, the ko increases from ~ 66.5 to 83.9 N/m, while kt decreased from ~ 145.4 to 124.9 N/m with increasing ‘x’ (from 0 to 1.0). Both the sound velocities decrease with increasing ‘x’: Vl decreases from ~ 5171 to 4961 m/s and Vt decreases from ~ 2985 to 2864 m/s. Hence, Vm shows a decreasing trend, from ~ 3314 to 3179 m/s. The elastic constants also show a decreasing trend with increase in ‘x’: B decreases from ~ 127 to 123 GPa, E decreases from ~ 114.5 to 110.5 GPa and G decreases from ~ 42.4 to 40.9 GPa. The Poisson's ratio P was a constant (0.35) for all values of ‘x’.

These results on LieMn ferrites are contrastingly opposite to the results of the present study on LieZn ferrite samples. This opposite trend in elastic parameters can be understood if Mn-ions at the tetrahedral and/or octahedral sites do not interact strongly with oxygen ions (weak MneO bond strength). This can lead to decrease of force constant and other parameters as Mn-contents in the sample increase. To understand this aspect we have taken help of the gamma irradiation study [1,4,40] and annealing temperature study [5] on Mn and Zn containing spinel ferrites. Interestingly, in these studies, it was clearly demonstrated that Mn-ions in the spinel structure are quite less stable, whereas Zn-atoms are more stable. Hence, the observation of decrease in values of the elastic parameters in LieMn ferrites with increasing Mn content, is reasonable. On the other hand, all these results suggest that MneO bond is much weaker than ZneO bond, mutually supporting/ proving the earlier results [1,4,5,40] observed in Mn1-xZnxFe2O4 samples by gamma-irradiation or annealing studies. 3.1.6. CueZn ferrites Ravinder et al. [31] directly measured the elastic wave velocities and the elastic moduli of CueZn ferrite (Cu1-xZnxFe2O4; x ¼ 0.0e1.0) samples synthesized by double sintering ceramic route using ultrasonic pulse transmission (UPT) technique. According to their results, the Vl values increased from ~ 5826 to 6848 m/s and Vt values increased from ~ 3112 to 3848 m/s with increase in ‘x’ from 0.0 to 1.0. The elastic constants: B increased from ~ 101 to 145 GPa, G increased from ~ 46 to 79 GPa and E increased from ~ 120 to 200 GPa, with ‘x’ increasing from 0 to 1, while P showed a decreasing trend (from 0.33 to 0.27). Although we have observed similar trends in the elastic parameters (Figs. 4 and 5) with Zn content, the absolute value of these elastic parameters for LieZn ferrite samples are higher. This suggests that LieZn ferrites are mechanically strong. 3.1.7. NieCd ferrites The variation of elastic properties in NieCd ferrite (Ni1xCdxFe2O4 (0.2  x  0.8)) samples were previously studied by Ravinder and Manga [28]. According to their report, both the sound velocities decrease with increasing ‘x’: Vl decreases from ~ 8104 to 6812 m/s and Vt decreases from ~ 3610 to 2564 m/s. The elastic moduli also show decreasing trends with increase in ‘x’: B decreases from ~ 245 to 202 GPa, E decreases from ~ 185 to 111 GPa and G decreases from ~ 67 to 39 GPa. The Poisson's ratio P increased from 0.38 to 0.43 with x increasing from 0.2 to 0.8. The trends of variations of the elastic parameters are similar to those of LieMn ferrites and opposite to the present study on LieZn ferrites. However, these values are comparable to those presented in this study on LieZn ferrites. Taken together, in all our LieZn ferrite samples the elastic moduli derived from the FTIR absorption spectra are higher. The elastic wave travels faster in LieZn ferrites than other spinel ferrite compounds reported earlier. 3.2. Thermal properties (Debye temperature) Fig. 7 shows the dependence of Debye temperature (qD ) calculated using Waldron method (qDW ) versus Zn content (x). The values of the Debye temperature (qDA ) calculated via Anderson's equation (eq. (17)), using rx-ray and rexp, are also shown in Fig. 7. Both these sets of qDA values show good agreement with each other (see also Table S10 (SI)). Among the different methods of obtaining average wave number of the vibration of the atoms, the number averaging leads to higher qD, whereas primary band averaging leads to lowest qD . On an average, the calculated values of qD (both qDW and qDA ) show similar behavior for all values of ‘x’, i.e., qD is almost

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independent of Zn content. However, the Debye temperature calculated using Waldron method (qDW z 720 K) is higher than that calculated using Anderson's equation (qDA z 590 K). The lower value of experimental qDA can be due to the low value of the density obtained by the experimental determination. This is expected because, the X-ray density of intended phase (rx-ray) and the experimental density (rexp) are similar but having low values, maybe due to the porosity present in the samples (Table 3) and the impurities as well as the possible defects present. The only reported experimental values of qD in LieZn ferrite system [38] are close to the values that we obtained by Anderson approach, although the data points are scattered, as seen in Fig. 7. There are no other reports available to determine the accuracy of the models. In literature, Mazen et al. [24] calculated the Debye temperature of LieMn ferrite samples using Waldron method and reported that qDW decreases from ~ 693 to 663 K with increasing ‘x’ (from 0 to 1). This decreasing behavior of qDW was also seen in Ni-Cd ferrites [28], where qDW decreases from 542 K to 394 K, as x increases from 0.2 to 0.8. For LieNi ferrites [27] a small increase in qDW was observed, where qDW increases from 445 K to 462 K as x increases from 0 to 1. In CueZn ferrites, Ravinder et al. [31] reported an increasing trend of qDA , from ~ 470 to 590 K, with increasing ‘x’ (calculated by Anderson method). In LieZn ferrites (Li0.5-x/2ZnxFe2.5-x/2O4 (x ¼ 0.2, 0.4, …1.0)), Ravinder et al. [38] reported that the direct experimental values of qD shows a fluctuation (between 400 K and 600 K) with ‘x’, similar to the variation of the related entities such as sound velocities (Vl and Vt), elastic constants and Poisson's ratio as discussed above. In the present work, the obtained qDW z 720 K is higher than that of the LieMn system [24], but the qDA z 590 K is closer to the experimental value obtained earlier [38], at least for a few compositions of LieZn ferrites. It is known that qD is the temperature needed to activate all phonon modes in a crystal, i.e., larger qD implies stiffer crystal. Since optical phonons have a higher frequency, it requires greater energy to activate the optical phonons. The observed Debye temperatures of our samples are higher than that of some other ferrites, as discussed above. This illustrates high stiffness (high bond strength) of our LieZn ferrite crystallites. It is worth mentioning that the high stiffness (elastic moduli) and Debye temperature of our samples could be related to the well grown rod-shaped crystallites [9]. 4. Conclusions The elastic and thermal properties of Zn- substituted lithium ferrite (Li0.5-x/2ZnxFe2.5-x/2O4, where x ¼ 0.0, 0.2, 0.4, 0.6, 0.8 and 1.0) samples were studied through FTIR spectroscopy. The tetrahedral and octahedral IR absorption bands of the studied LieZn ferrites split due to the presence of light weight Liþ ions at the octahedral site along with heavy Fe3þ ions. The band splitting observed in case of pure lithium ferrite gradually disappears with increasing Zn content due to increased disorder in cation distribution at octahedral sites. The weighted average of the wave numbers of vibrations could be better suited for the determination of elastic and thermal parameters of the materials from their FTIR spectra. The measured force constants for the tetrahedral sitecomplexes are higher than that of the octahedral site-complexes. The elastic sound wave travels faster longitudinally than that in the transverse direction. According to our results, the elastic (Young's, Bulk and Rigidity) moduli calculated using the method by Waldron show a progressive increase in their values with Znconcentration in the samples. A comparison between Debye temperatures (qD) estimated by Waldron method and Anderson methods showed that Anderson method resulted in lower value of qD, but it is more close to the reported experimental value. The high

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elastic moduli and Debye temperature of our samples, due to stronger interaction between cation and anions, proves the stiff nature of the LieZn ferrite particles. Acknowledgment VR would like to thank UGC, New Delhi for financial support in the form of minor research project. The financial support through the sponsored research program of ISRO-IISc Space Technology Cell (Code number: ISTC/CMR/BS/355) is gratefully acknowledged by BS. Appendix A. Supplementary data Supplementary data related to this article can be found at http:// dx.doi.org/10.1016/j.jallcom.2017.09.099. References [1] J.V. Angadi, A.V. Anupama, H.K. Choudhary, R. Kumar, H.M. Somashekarappa, M. Mallappa, B. Rudraswamy, B. Sahoo, Mechanism of g-irradiation induced phase transformations in nanocrystalline Mn0.5Zn0.5Fe2O4 ceramics, J. Solid State Chem. 246 (2017) 119e124. [2] G.P. Rodrigue, A generation of microwave ferrite devices, Proc. IEEE 76 (1988) 121. [3] P.D. Baba, G.M. Argentina, W.E. Courtney, G.F. Dionne, D.H. Temme, Fabrication and properties of microwave lithium ferrites, IEEE Trans. Magn. 8 (1972) 83e94. [4] J.V. Angadi, A.V. Anupama, R. Kumar, H.K. Choudhary, S. Matteppanavar, H.M. Somashekarappa, B. Rudraswamy, B. Sahoo, Composition dependent structural and morphological modifications in nanocrystalline Mn-Zn ferrites induced by high energy gamma-irradiation, Mater. Chem. Phys. 199 (2017) 313e321. [5] S.V. Bhandare, R. Kumar, A.V. Anupama, H.K. Choudhary, V.M. Jali, B. Sahoo, Annealing temperature dependent structural and magnetic properties of MnFe2O4 nanoparticles grown by sol-gel auto-combustion method, J. Magn. Magn. Mater. 433 (2017) 29e34. [6] A.V. Anupama, W. Keune, B. Sahoo, Thermally induced phase transformation in multi-phase iron oxide nanoparticles on vacuum annealing, J. Magn. Magn. Mater. 439 (2017) 156e166. [7] Harshada Nagar, Naveen V. Kulkarni, Soumen Karmakar, B. Sahoo, I. Banerjee, P.S. Chaudhari, R. Pasricha, A.K. Das, S.V. Bhoraskar, S.K. Date, W. Keune, €ssbauer spectroscopic investigations of nanophase iron oxides synthesized Mo by thermal plasma route, Mater. Charact. 59 (2008) 1215e1220. [8] S.K. Date, P.A. Joy, P.S. Anil Kumar, B. Sahoo, W. Keune, Structural, magnetic €ssbauer studies on nickel-zinc ferrites synthesized via a precipitation and Mo route, Phys. Status Solidi (c) 1 (2004) 3495e3498. [9] V. Rathod, A.V. Anupama, V.M. Jali, V. Hiremath, B. Sahoo, Combustion synthesis, Structure and magnetic properties of Li-Zn ferrite ceramic powders, Ceram. Int. 43 (2017) 14431e14440. [10] P. Tarte, Infra-red spectrum and tetrahedral co-ordination of Li in the spinel LiCrGeO4, Acta Cryst. 16 (1963) 228. [11] R. Valenzuela, Novel applications of ferrites, Phys. Res. Int. 2012 (2012) 1e9. [12] E. Schloemann, Advances in ferrite microwave materials and devices, J. Magn. Magn. Mater. 209 (2000) 15e20. [13] V.G. Harris, A. Geiler, Y. Chen, S.D. Yoon, M. Wu, A. Yang, Z. Chen, P. He, P.V. Parimi, X. Zuo, C.E. Patton, M. Abe, O. Ache, C. Vittoria, Recent advances in processing and applications of microwave ferrites, J. Magn. Magn. Mater. 321 (2009) 2035e2047. [14] M.P. Horvath, Microwave applications of soft ferrites, J. Magn. Magn. Mater. 215e216 (2000) 171e183. [15] L. Thourel, The Use of Ferrites at Microwave Frequencies, Pergamon Press, Paris, 1964. € [16] Ümit Ozgür, Yahya Alivov, Hadis Morkoç, Microwave ferrites, part 2: passive components and electrical tuning, J. Mater. Sci. Mater. Electron. 20 (2009) 911e952. [17] V. Kuncser, W. Keune, M. Vopsaroiu, P.R. Bissell, B. Sahoo, G. Filoti, Easy axis distribution in modern nanoparticle storage media: a new methodological approach, J. Optoelectron. Adv. Mater. 5 (2003) 217e226. [18] R.M. Sangshetti, V.A. Hiremath, V.M. Jali, Combustion synthesis and structural characterization of Li-Ti mixed nanoferrites, Bull. Mater. Sci. 34 (2011) 1027e1031. [19] S.A. Mazen, M.H. Abdallah, B.A. Sabrah, H.A.M. Hashem, The effect of Ti on some physical properties of CuFe2O4, Phys. Stat. Sol. (a) 134 (1992) 263e271. [20] R.D. Shannon, Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides, Acta Cryst. A 32 (1976) 751e767. [21] J. David Van Horn, Electronic table of Shannon ionic radii. http://v.web.umkc. edu/vanhornj/shannonradii.htm, 2001.

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