Effect of nickel substitution on structural, infrared and elastic ... - NOPR

Indian Journal of Pure & Applied Physics Vol. 45, July 2007, pp. 596-608

Effect of nickel substitution on structural, infrared and elastic properties of lithium ferrite S S Bhatu, V K Lakhani, A R Tanna, N H Vasoya, J U Buch, P U Sharma, U N Trivedi, H H Joshi & K B Modi* Department of Physics, Saurashtra University, Rajkot 360 005, India *[email protected] Received 8 June 2006; revised 12 December 2006; accepted 9 April 2007 The structural and elastic properties of Li0.5(1-x)NixFe2.5-0.5xO4 (x = 0.0-1.0) spinel ferrite system have been studied by means of X-ray diffraction and infrared spectroscopic measurements at 300K. The X-ray diffraction data has been used to determine the lattice constant, X-ray density, distributions of cations among the tetrahedral and octahedral sites of spinel lattice, anion parameters, site radii, ionic radii, bond angle and bond length. The nature and change in the position of IR bands have been explained on the basis of cations involved in the system. The force constants have been used to calculate elastic moduli like bulk modulus, rigidity modulus, Young’s modulus, Poisson’s ratio, Debye temperature and corrected to zero porosity. The observed variation of elastic constants with nickel substitution has been explained on the basis of strength of interatomic bonding. The applicability of heterogeneous metal mixture rule for estimating elastic constants has been tested. The Debye temperature obtained from elastic constant data is higher than that of the X-ray diffraction analysis, mainly due to existence of peaks in the vibrational spectra at lower frequencies. Keywords: Ferrites, Infrared spectroscopy, Elastic properties, Lithium ferrite, X-ray diffraction IPC Code: G01J3/28

1 Introduction The magnetic, electric and dielectric behaviour of spinel ferrites decisively depends upon the structural properties. Therefore, the non-destructive methods of characterization such as X-ray diffraction and infrared spectroscopy especially suited for such investigations. The wavelengths of X-ray, electrons and neutrons are comparable to the interplaner distances in solids. Because of high penetrating power, X-ray can provide important information regarding structural properties of matter. The angle of diffraction and intensity of diffracted beam together are characteristics of a particular crystal structure since no two atoms have exactly the same size and X-ray scattering ability; the intensities of diffracted beam will be unique for every material. This uniqueness helps to identify the structure and determine the structural parameters of the material. Infrared spectroscopy is one of the most powerful analytical techniques, which offers the possibility of chemical identification. One of the prime advantages of infrared spectroscopy over the other methods of structural analysis is that it provides useful information about the structure of molecule rapidly and also without cumbersome evaluation methods.

The technique is based upon the simple fact that a chemical substance shows marked selective absorption in the infrared region. After the absorption of infrared radiations, the molecules of a chemical substance vibrate at many rates of vibrations, giving rise to closed packed absorption bands, called IR absorption spectrum, which may be extended over a wide wavelength range. Various bands present in IR spectrum are corresponding to the characteristic function groups and bonds present in chemical substance. Hence, an IR spectrum of a chemical substance is the fingerprint for its identification. When the symmetry of the spinel structure is lower than cubic or/and supplementary ordering of cation exists, more IR bands may appear. A splitting of IR band is observed in some spinels containing octahedrally coordinated Jahn-Teller ions. The bands may split also due to the presence of two different kinds of ions in the same sub lattice. The compositional dependence of structural and elastic properties of Ni2+ substituted lithium ferrite with general formula Li0.5(1-x)NixFe2.5-0.5xO4 (x = 0.0, 0.2, 0.4, 0.6, 0.8 and 1.0) by means of X-ray diffraction and infrared spectroscopic measurements has been studied. This work is in continuation of our

BHATU et al.: EFFECT OF NICKEL SUBSTITUTION ON STRUCTURAL, INFRARED AND ELASTIC PROPERTIES

work on bulk magnetic properties of Li-Ni ferrites1 . Earlier, the compositional and temperature dependent thermoelectric power measurement on Li-Ni mixed ferrites has been performed by Reddy and Rao et al 2. However, no information is available in the literature regarding the structural and elastic properties of Li-Ni ferrite system. 2 Experimental Details The powdered samples of Li0.5(1-x)NixFe2.5-0.5xO4 spinel ferrite system have been prepared by usual double sintering ceramic technique, with compositions x = 0.0, 0.2, 0.4, 0.6, 0.8 and 1.0. The starting materials were of analytical reagent (AR) grade (99.3% pure) oxides Li2CO3, NiO and Fe2O3 supplied by E Merck. These oxides were pressed in the form of cylindrical disc and pre-sintered at 1100°C for 12 h in air. The pre-sintered mixtures were again well-ground and resulting powders were then compressed in the form of pellets using polyvinyl alcohol as a binder. The pellets were finally sintered in air at 1200°C for 12 hr and then slowly furnace cooled to room temperature at the rate of 2°C/min. The X-ray diffraction pattern of all the samples was recorded at 300K with a Philips (PM 9220) diffractometer using FeKα radiation. The infrared spectra for all the compositions were recorded in the wave number range 400-1000 cm-1. For the present study, BRUKER IFS 66V FT-IR spectrometer was used to carry out the infrared spectroscopic studies in KBr medium at 300K.

597

phase easy. The values of lattice constant for different nickel concentrations, are given in Table 1. The lattice constant increases with increasing x, thus obeying the Vegard’s law3. Usually in a solid solution of spinels within the miscibility range, a linear change in lattice constant with concentration of components is observed 3 . A slow linear increase in lattice constant is due to the replacement of smaller Li1+ ions (0.68 Å) and Fe3+ ions (0.64 Å) by the larger Ni2+ ions (0.69 Å) in the Li0.5(1-x)NixFe2.5-0.5xO4 system. The variation of X-ray density (ρ) as well as the bulk density (d) with chemical composition (x) is shown in Table 1. The Xray density and bulk density increase with increase in x, i.e. the bulk density follows the same behaviour of the theoretical density (Table 1). The X-ray density increases with Ni2+ substitution because of the fact

3 Results and Discussion 3.1 X-ray diffraction and structural properties

The room temperature (300K) X-ray diffractograms for the samples with x = 0.2, 0.4 and 0.6 are shown in Fig. 1. The diffractograms showed the presence of cubic spinel phase with no extra lines corresponding to any other phase. The X-ray lines were found to be sharp, which makes the detection of any impurity

Fig. 1Typical X-ray diffraction patterns for x = 0.2, 0.4 and 0.6 composition at 300 K

Table 1Lattice constant (a), X-ray density (ρ), bulk density (d), porosity (P) and shrinkage (S) for Li-Ni-Fe-O system Content (x)

a(nm) ±0.0002 nm

ρ(kg/m3) ×103

d(kg/m3) ×103

P(%)

S(%)

0.0 0.2 0.4 0.6 0.8 1.0

0.8316 0.8323 0.8338 0.8342 0.8349 0.8358

4.7829 4.8993 4.9981 5.0847 5.2017 5.3353

3.947 4.042 3.995 4.107 4.193 4.312

17.48 17.50 20.06 19.22 19.39 19.18

7.6 6.1 5.2 4.9 5.0 5.5

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INDIAN J PURE & APPL PHYS, VOL. 45, JULY 2007

that the rate of increase of molecular weight is faster than that of the volume of the unit cell per 20% of nickel concentration. The percentage porosity (P) was calculated using the relation4: P = (1-d/ρ) × 100% The change of P and percentage shrinkage (S) of the diameter of the disc shape sample before and after final sintering process, with composition is also presented in Table 1. As expected, porosity and shrinkage behave inversely to each other as a function of concentration(x). The variation of porosity (P) with x shown in Table 1 is the result of the interplay of ρ and d. In order to determine the cation distribution, X-ray intensity calculations were carried out using the formula suggested by Burger5 : 2

Ihkl = Fhkl ⋅ Pm ⋅ Lp

where Ihkl is the relative integrated intensity, Fhkl the structure factor, Pm the multiplicity factor and Lp is the Lorentz polarization factor = (1+cos22θ/sin2 θ cos θ). According to Ohnishi and Teranishi6, the intensity ratios of planes I(220)/I(422) and I(400)/I(422) are considered to be sensitive to the cation distribution parameters (x). The ionic configuration based on site preference energy values proposed by Miller7 for individual cations in Li0.5(1-x)NixFe2.5-0.5xO4 system, suggests that Li1+ and Ni2+ ions preferencially occupy B-sites whereas Fe3+ ions occupy A- and B-sites. The final cation distribution for all the compositions was deduced by simultaneously considering the Bragg plane ratios, the fitting of magnetization data1 at 300K and theoretical determination of lattice constant and Neel temperature values. The best-fitted cation distribution alongwith observed and calculated values of plane ratios are given in Table 2. It is worthwhile

to note that, Ni2+ ions occupy B-site while occupancy of Fe3+ to the B-site varying from 60% for x = 0.0 to 50% for x = 1.0 composition. It is observed that on increasing the Ni-concentration, the occupancy of Li1+ ions decrease from 100% to around 40% for x = 0.8 compositon. The study has revealed the octahedral site preference of Li1+, Ni2+ and Fe3+ as follows: Ni2+>Li1+>Fe3+. Besides using experimentally found values of lattice constant and oxygen11 parameter (u), it is possible to calculate the value of the mean ionic radius per molecule of the tetrahedral and octahedral sites, rA and rB, respectively, using the cation distribution for each composition using the relation9: rA = CALi r (Li) + CAFé r(Fe) rB = ½ [CBLi r (Li+2) + CBFe r(Fe) + CBNi r (Ni+2)] where r(Li1+), r(Ni2+) and r(Fe3+) are the ionic radii of Li1+ (0.68Å), Ni2+ (0.69 Å) and Fe3+ (0.64 Å), respectively, while CALi, CAFé are the concentrations of the Li1+ and Fe3+ ions on A-sites and CBLi, CBNi, CBFe are concentrations of Li1+, Ni2+ and Fe3+ ions on B-sites. Using these formulae, the values of mean tetrahedral and octahedral ionic radii for each composition have been calculated and are listed in Table 3. It can be seen that mean tetrahedral ionic radius shows very slow increase while ionic radius of octahedral site increases rapidly with increasing (x), which in turn causes the lattice constant, a, to increase with x. It can be concluded that the octahedral site substitution plays a dominant role in infuluencing the value of the lattice constant. Moreover, it is found that the average ionic radius r = [(rA + rB)/2)] increases slowly with increasing nickel concentration(x) which is reflected in increase in ‘a’ with x (Table 1). It is known that there is a correlation between the ionic radius and the lattice constant. The lattice constant can be calculated theoretically by the relation suggested by Mazen et al.10.

Table 2Cation distribution and X-ray intensity ratios for Li-Ni-Fe-O system Content (x) 0.0 0.2 0.4 0.6 0.8 1.0

A-site

B-site (cation distribution)

(Fe+31.0) (Li+10.032Fe3+0.968) (Li+10.060Fe+30.940) (Li+10.074Fe+30.926) (Li+10.060Fe+30.940) (Fe+31.0)

[Li+10.5Fe+31.5] [Li+10.368Ni+20.2Fe+31.432] [Li+10.24Ni+20.4Fe+31.360] [Li+10.126Ni+20.6Fe+31.274] [Li+10.040Ni+20.8Fe+31.160] [Ni+21.0Fe+31.0]

I(220)/ obs

I(400) cal

I(400)/ obs

I(422) cal

1.683 1.625 1.503 1.554 1.080 1.00

1.662 1.602 1.483 1.501 1.06 0.99

1.532 3.33 1.408 20.85 1.459 1.296

1.532 1.283 1.386 1.919 1.359 1.329


599

Table 3Ionic radii(r), lattice constant (a) for Li-Ni-Fe-O system

ath =

8 3 3

Content (x)

rA(nm)

rB(nm)

r(nm) ±0.0002nm

ath (nm)

aO (nm)

0.0 0.2 0.4 0.6 0.8 1.0

0.06400 0.06413 0.06424 0.06430 0.06424 0.06400

0.06500 0.06523 0.06548 0.06575 0.06608 0.06650

0.06450 0.06468 0.06486 0.06502 0.06516 0.06525

0.8313 0.8321 0.8330 0.8338 0.8346 0.8353

0.8321 0.8303 0.8283 0.8258 -

[(rA + RO) +

3 (rB + RO)]

where RO is the radius of the oxygen ions (1.32 Å). The agreement between ath and aexp (Table 3) obtained from X-ray data indirectly supports the cation distribution deduced from X-ray intensity calculations. The lattice parameter ‘a’ for each composition was also determined using the relation4:

1  cos 2 θ cos 2 θ  F (θ ) =  +  2  sin θ θ  where, θ is Bragg’s angle. The experimentally found values of lattice constant were plotted against the function F(θ). The precise value of lattice constant for each composition was obtained by extrapolating the straightline with the least square fitting of a F(θ) plot using a computer programme. Typical plots of lattice constant (a) versus F(θ) for x = 0.2, 0.4, 0.6 and 0.8 are given in Fig. 2. The lattice constant values thus obtained are listed in Table 3. The bracketed term is sometimes called the Nelson – Riley function4. The oxygen positional parameter or anion parameter (u) for each composition was calculated using the formula11: 1

u

1 2 2  11 2 1  2 R − + R −  3m 4 3  48 18  = 2 R2 − 2

… (1)

Fig. 2Variation of lattice constant as a function of Nelson-Riley function F(θ)

two end members Li0.5Fe2.5O4 (x=0.0) (u= 0.260Å) and NiFe2O4 (x=1.0) ( u = 0.256 Å) obtained from the literature8,12 and presented in Table 4. Using the experimental value of ‘a’ and anion parameter (u) of each composition in the equations13:

where R = B − O . The bond lengths B-O and A-O are

dxx = a(2)1/2 (2u-1/2) tetrahedral edge dxx = a(2)1/2 (1-2u) shared octahedral edge … (2) dxx = a (4u2 - 3u + 11/16)1/2 unshared octahedral edge

average bond lengths calculated based on the cation distribution listed in Table 2; B-O = and A-O = .The values of oxygen positional parameter obtained from Eq. (1) are in very good agreement to those deduced from the value of u for

The selected interatomic distances were calculated for Li0.5(1-x) Nix,Fe2.5-0.5xO4 system and same are presented in Table 4. Eq. (2) is applicable for oxygen positional parameter determined assuming centre of

A−O

600


Table 4Oxygen positional parameter (u) and edge length (dxx) in nm for Li-Ni-Fe-O system Content (x)

U3m(nm) (cation distribution) (1/4,1/4,1/4)

u (nm) (inter polation) (1/4,1/4,1/4)

uA(nm) (3/8,3/8,3/8)

Tetra. edge (nm)

Shared octa. edge (nm)

Unshared octa edge (nm)

0.0 0.2 0.4 0.6 0.8 1.0

0.02630 0.02635 0.02630 0.02629 0.02597 0.02610

0.02600 0.02592 0.02584 0.02574 0.02568 0.02560

0.03850 0.03842 0.03834 0.03826 0.03818 0.03810

0.3175 0..3159 0.3146 0.3129 0.3112 0.3097

0.2705 0.2726 0.2749 0.2770 0.2791 0.2813

0.2945 0.2946 0.2951 0.2952 0.2954 0.2956

symmetry at (3/8, 3/8, 3/8), for which uAideal = 0.375 (origin at A site) as compared to uBideal =0.250 (origin at B site) for which the center of symmetry is at (1/4,1/4,1/4). For the sake of comparison anion parameter (uA) for all the compositions are also given in Table 4. The length of tetrahedral edge and unshared octahedral edge show small change as a function of nickel substitution while the length of shared octahedral edge slowly increases with increasing x (Table 4). This may be due to the replacement of smaller Li1+and Fe3+ ions by slightly larger Ni2+ ion on the octahedral site (Table 2) of the system. The unshared octahedral edge remains almost constant may be due to overall compensation taken place by decrease in tetrahedral edge and increase in shared octahedral edge. The configuration of ion pairs in spinel ferrites with favourable distances and angles for effective magnetic interactions are shown in Fig. 3. The inter-ionic distances between the cation (b, c, d, e and f) (MeMe) and between the cation and anion (p, q, r and s) (Me-O) were calculated using the experimental values of lattice constants and oxygen positional parameter (u) (Tables 1 and 4) by the relations14,15 . Me-O p = a(1/2-u) q =a(u-1/8)31/2 r =a(u-1/8)111/2 s = a/3(u+1/2)31/2

Me-Me b = (a/4)21/2 c = (a/8)111/2 d = (a/4)31/2 e = (3a/8)31/2 f = (a/4)61/2

From Table 5, it is seen that both inter-ionic distances between the cation-anion and between cations, increase with increasing nickel-concentration (x), but the distance between anion and A-site cation (p,q,r,s) decreases on substitution of Ni2+ in the system. This result is in accordance with decrease in length of

Fig. 3Configuration of the ion pairs in spinel ferrites with favorable distances and angles for effective magnetic interactions Table 5Interatomic distances and bond angles for Li-Ni-Fe-O system. (Distances in nm and angles in degree) x

0.0

0.2

0.4

0.6

0.8

1.0

b c d e f p q r s θ1

0.2940 0.3446 0.3600 0.54.0 0.5092 0.2023 0.190 0.3632 0.3633 122.87

0.2942 0.3450 0.3602 0.5406 0.5095 0.2029 0.1891 0.3621 0.3634 123.27

0.2948 0.3456 0.3610 0.5415 0.5106 0.2037 0.1887 0.3615 0.3638 123.42

0.2950 0.3458 0.3612 0.5418 0.5108 0.2042 0.1881 0.3602 0.3640 123.59

0.2952 0.3461 0.3615 0.5423 0.5113 0.2048 0.1875 0.3592 0.3638 123.77

0.2954 0.3464 0.3618 0.5428 0.5116 0.2054 0.1870 0.3579 0.3640 123.89

θ2

144.00

144.72

145.27

146.10

146.80

147.70

θ3

93.21

92.93

92.70

92.49

92.22

91.95

θ4

125.99

125.84

125.88

125.69

125.78

125.60

θ5

106.16

105.73

105.28

104.82

104.40

103.90

tetrahedral edge (Table 4). The bond angles (θ1, θ2, θ3, θ4 and θ5), (Fig. 3) were calculated by simple trignometry principles using the values of inter – ionic distances. It is seen that angles θ1 and θ2 increase while θ3, θ4 and θ5 decrease with nickel concentration (x) (Table 5). The observed increase in θ1 and θ2 angles suggests weakening of A-B interactions while decrease in θ3, θ4 and θ5 suggests strengthening in


B-B and A-A interactions on nickel substitution in the system. The X-ray diffraction data was further used to calculate the tetrahedral and octahedral site radii (RA and RB). The site radii, RA and RB, were calculated using the relations: RA = a(3)1/2 (δ+1/8) RB = a(3δ2 +1/16-δ/2)1/2

… (2)

where δ = usystem – uideal, usystem=oxygen parameter =(u1+u2)/2 and uideal = 0.250Å, were u1 and u2 are anion parameter of the two end members of the system respectively (Table 4). The values of site radii calculated from Eq. (2) are given in Table 6. It can be seen that RA and RB increase slowly with increase in nickel concentration in the system, which can be attributed to the fact that lattice parameter increases linearly with x (Table 1). The site radius RB is greater than RA. 3.2 Theoretical determination of Neel temperature

The Neel temperatures have also been calculated theoretically for x = 0.0-1.0 compositions by applying molecular field theory and using cation distribution (Table 2). The Neel temperature depends upon the active magnetic linkages per magnetic ion per formula unit16 . The statistical model proposed by Gilleo and Geller16 has been found to hold reasonably well for non-magnetic substitution in spinel ferrites like MgFe2O4, ZnFe2O4, etc. in which magnetic ion (Fe3+) concentration remains unaltered. The calculation became more involved in case of magnetic substitution for magnetic divalent cation. When the magnetic/nonmagnetic substitution is made for Fe3+ [MgAlxFe2-xO4 (Ref.17), MgCrxFe2-xO4 (Ref. 18)], it becomes too complex to estimate the Neel temperature. We have used modified molecular field theory as suggested by Baldha et al.19. for Zn2+

substituted cobalt ferrite based on inter and intrasublattice interactions. The TN(x) for a spinel ferrite doped with magnetic ion (Ni2+(2µ β)) concentration (x) for nonmagnetic (Li1+) and magnetic (Fe3+(5µ β)) can be expressed in terms of Neel temperature of unsubstituted spinel ferrite TN (x=0.0), i.e.Li0.5Fe2.5O4 by the equation;

TN (x ) =

M (x = 0.0)TN ( x = 0.0)n( x ) n(x = 0.0 )M ( x ).

… (3)

where, M(x) is the relative weighted total magnetic ions per formula unit, calculated by considering the weightening of magnetic ion concentration for substitution ferrite to that of un-substituted one. Thus, M(x) for Li0.5(1-x)NixFe2.5-0.5xO4 can be expressed as : M (x ) =

5(2.5 − 0.5 x ) + 2 x 5(2.5)

M ( x) =1−

x 25

… (4)

n(x) is the number of interaction per formula unit expressed as :

 2  n( x) =  ∑ Ai B j µ iµ j  i , j =1 

… (5)

where Ai and Bj are the fraction of magnetic ions on the A- and B- sites, respectively while µ i and µ j are the magnetic moment of the cation involved. Here i, j = 1 stand for magnetic ions (Ni2+) and i, j =2 for that of iron. The Neel temperatures estimated using the Eqs (3)(5) are in good agreement with the Neel temperature obtained from thermal variation of ac susceptibility

Table 6Site radii (R), molecular weight (M), band position (ν) and force constant (k) for Li-Ni-Fe-O system Content (x)

RA

0.0 0.2 0.4 0.6 0.8 1.0

0.1880 0.1886 0.18897 0.1890 0.1892 0.1894

RB

M

M1 (kg) × 10-3

M2

0.2031 0.2033 0.2037 0.2038 0.2039 0.2041

207.09 212.56 218.02 223.48 228.95 234.41

55.85 54.28 52.92 52.23 52.92 55.85

087.24 094.27 101.11 107.25 112.03 114.56

(nm)

601

ν1 ν2 × 10-2 m-1 588.2 592 592 604 606 608

428.7 418 415 412 405 400

kt kO × 102 (N/m) 1.472 1.449 1.413 1.451 1.480 1.520

0.8514 0.8560 0.9245 0.9665 0.9755 0.9700

602


measurements1. The values of M(x), n(x) and TN(x) for all the compositions are listed in Table 7. 3.3 Infrared spectroscopy

Ferrites possess the structure of mineral spinel (MgAl2O4) that crystallizes in the cubic form with space group20 Fd3m – Oh7 . It is generally known that the spinel ferrites exhibit four IR active bands, designated as υ1, υ2, υ3, υ4. The occurrence of these four bands have been rationalized on the basis of group theoretical calculations employing space group and point symmetric both in normal and inverse spinels. The magnetic properties of ferrites are decisively dependent on the precise configuration of the atoms or ions in the structure. Therefore, the non-destructive method of characterization such as infrared spectroscopy is especially suited for such investigations. The IR absorption spectra were recorded at 300 K in the wave number range 4001000 cm-1. The infrared spectra for typical compositions x=0.2, 0.4, 0.6 and 0.8 are shown in Fig. 4. The IR spectrum of nickel substituted lithium ferrite are found to exhibit two major bands in the range 400-610cm-1 (Fig. 4). No absorption bands were observed above 610 cm-1. The high frequency band ν1 is in the range 588 -610 cm-1, and the lower frequency band v2 is in the range 400-470 cm-1. These bands are common features of all the ferrites21. According to Waldron’s classification21 the vibrations of the unit cell of cubic spinel can be construed in the tetrahedral (A-) site and octahedral (B-) site. So, the absorption band ν1 is caused by the stretching vibration of the tetrahedral metal – oxygen bond, and the absorption band ν2 is caused by the metal – oxygen vibrations in octahedral sites. The band positions (ν1, ν2) for all the compositions are given in Table 6. The nature and the position of bands for Li0.5Fe2.5O4 (x=0.0) and NiFe2O4

(1.0) are consistent with literature22 . The change in the band position is expected because of the difference in the Fe3+ - O2- distances for the octahedral and tetrahedral complexes. It is found that Fe-O distance of A-site (1.89Å) is smaller than that of the B-site23 (1.99Å) . This can be interpreted by the more covalent bonding of Fe3+ ions at the A-sites than Bsites. The first principal band ν1 for Li-ferrite is found to shift from 588 cm-1 to 600 cm-1 on increasing Niconcentration. The shift in ν1 band is due to the charge imbalance of A-sites which probably makes

Table 7M(x), n(x) and Neel temperature (TN) for Li-Ni-Fe-O system Ni-content (x)

M(x)

n(x)

Theory

0.0 0.2 0.4 0.6 0.8 1.0

1.000 0.992 0.984 0.976 0.968 0.960

96.00 85.45 84.47 83.84 83.66 84.00

910 890 872 854 837 823

TN(K) Susceptibility (Ref. 1) ±5K 910 890 852 823 786 820

Fig. 4Infrared spectra of Li-Ni-Fe-O system with x = 0.2, 0.4, 0.6 and 0.8 sample


the oxygen ions shift towards Li1+ ions24 . The splitting of the absorption band near ν2 ~ 400 cm-1 has been clearly observed. The splitting of octahedral (Bsite) absorption band is due to the presence of three kinds of cations24 at the B-site: Li1+, Ni2+ and Fe3+ . Thus, the cation distribution determined through the X-ray diffraction and magnetization is also supported by IR spectroscopy. The centre frequency of the ν2 band shows a small variation and they shift towards lower frequency side for x = 0.0-1.0 (Table 6). It is known that increase in site radius (Table 6) hinders the fundamental frequency and therefore the center frequency should shift towards lower frequency side. The ν3 band which arises due to divalent metal ionoxygen complexes in the B-sites and the ν4 band which depends upon the mass of vibrating divalent tetrahedral cation25 appears generally below22,26 400 cm-1 and hence they were not observed in the study. On increasing Ni content in the system, intensity of both the bands does not change. Earlier it has been reported that broadening is commonly observed for normal spinel ferrites and have attributed to the statistical distribution of Fe3+ ions on A- and B-sites. In the present case both the end members: Li0.5Fe2.5O4 (x = 0.0) and NiFe2O4 (x =1.0) possess inverse spinel structure so no such broadening of the bands has been observed. The force constant is a second derivative of potential energy with respect to the site radius (RA and RB), the other independent parameter being kept constant. The force constants, for tetrahedral site (kt) and octahedral site (ko), were calculated employing the method suggested by Waldron20. According to Waldron, the force constants, kt and ko for respective sites are given by: kt = 7.62 × M1 × ν12 × 10-7 N/m ko = 10.62 × M2/2 × ν22 × 10-7 N/m where M1 and M2 are the molecular weight of cations on A- and B-sites, respectively (Table 6). The variation of force constants with Ni-content (x) is presented in Table 6. The force constants are found to increase with x, and this suggests strengthening of interatomic bonding. These results are in support with our earlier results on structural properties of Li0.5(1-x)NixFe2.5-0.5xO4 system. It was found that interionic distances between anion and A-site cations as well as bond angles (θ3, θ4, θ5) decrease with

603

increasing content (x), resulting in enhancement of strength of interatomic bonding. 3.4 Elastic properties

Theory of elasticity sets forth problem of determining internal forces in solid body. These internal forces (body forces) act on the elements of volume or mass inside the body. The action of external forces (surface forces) on the solid body causes deformation resulting in the change in distance also. The actions of external forces produce deformation which gives rise to additional internal forces. The elastic constants are of much importance because they elucidate the nature of binding forces in solids and to understand the thermal properties of the solids. In general, there are 36 elastic constants, but in the case of isotropic and homogeneous materials like spinel ferrites and garnets, the elastic constants can be reducing to three constants. In engineering practice the elastic constants often used are the Young’s modulus, rigidity modulus, bulk modulus and Poisson’s ratio. The most conventional technique for elastic constants and Debye temperature determination is the ultrasonic pulse transmission technique 27. The sample size required for such measurement is around 1 cm in length. In the study of elastic properties of nanoparticles, single crystal, irradiated or specially treated materials, where sample quantity is very small, such technique may not be useful. We have developed a new method to study the elastic properties of spinel ferrites and garnets28-31 through infrared spectroscopy and it is expected to generalize for other materials also. The values of lattice constants (a), X-ray density (ρ) and pore fraction (f = 1-d/ρ), through X-ray diffraction analysis and force constants for tetrahedral (A-) site (kt) and octahedral (B-) site (ko) determined through IR spectral analysis are used to calculate elastic moduli and Debye temperature. The agreement of the results obtained from the present method with the results of the other methods32,33 confirms the validity of the method used. The force constant (k) is a product of lattice constant (a) and stiffness constant27 (C11 = L; longitudinal modulus) . The value of lattice constant obtained from X-ray diffraction analysis (Table 1) and average force constant (k = (kt + ko)/2) (Table 7) value have been used for the determination of C11 = L and

604


under investigation are porous, (pore fraction (f) ≈ 0.17-0.20; Table 1), the values of elastic moduli have been corrected to zero porosity using Hosselman and Fulrath’s formula35 , given by:

same are included in Table 6. On the other hand, the data analysis yielded the following equation for the variation of Poisson’s ratio as a function of Pore fraction34 (f): σ = 0.324(1-1.043f). The Poisson’s ratio however remains constant for different compositions. The value of σ is found to be ≈ 0.27, this value lies in the range from -1 to 0.5 which is in conformity with the theory of isotropic elasticity. Physically, the reason is that for the material to be stable, the stiffness must be positive; the bulk and shear stiffness are interrelated by formula which incorporate Poisson’s ratio. Just by knowing any two elastic moduli, we can able to find other one. The other elastic moduli of the ferrite specimens are evaluated using the following formulae 27,34; for a cubic lattice: Stiffness constant (C12 ) =

1 1  3 f (1 − σ )(9 + 5σ )  = 1 −  E0 E  2(7 − 5σ )  1 1  15 f (1 − σ )  = 1 −  G0 G  (7 − 5σ )  σ0 =

E0 −1 2G0

The corrected values of Young’s modulus (E0), rigidity modulus (G0) and Poisson’s ratio (σ0) for different compositions are given in Table 9. The elastic moduli increase continuously with increasing Ni-content (x) as presented in Tables 8 and 9. The Poisson’s ratio, however, remains constant for different compositions. The value of σ0 is found to be ≈ 0.28 for all the compositions, this value lies in the range -1-0.5 which is in conformity with the theory of isotropic elasticity. Following Wooster’s work [36], the variation of elastic constants with increasing nickel concentration may be interpreted in terms of interatomic bonding. Thus, it can be inferred from the increase of elastic moduli with concentration (x) that the interatomic bonding between various cations is getting strengthened continuously. In the present system, strengthening of interatomic bonding on nickel (Ni2+) substitution for Li1+ and Fe3+

σ C11 (1-σ )

(C11 − C12 )(C11 + 2C12 ) C11 + C12 E Rigidity modulus (G ) = 2(σ +1) 1 4 Bulk modulus (K ) = [C11 + 2C12 ] = L − G 3 3

Young’s modulus (E ) =

In engineering practice, the elastic constants often used are the Young’s modulus, rigidity modulus, bulk modulus and Poisson’s ratio. As the ferrite specimens

Table 8Elastic wave velocity (V), stiffness constant (Cij), Young’s modulus (E), rigidity modulus (G), Poisson’s ratio (σ) and pore fraction (f) for Li-Ni-Fe-O system. Content (x)

Vlo

Vso (m/s)

Vmo

C11

C12

K (GPa)

E

G

σ

f

0.0 0.2 0.4 0.6 0.8 1.0

5404.27 5316.32 5295.53 5338.47 5317.10 5283.91

3691.01 3700.51 3887.38 3865.60 3865.89 3807.59

4022.02 4021.61 4190.12 4175.92 4173.50 4116.41

139.69 138.47 140.16 144.91 147.06 148.96

50.36 49.92 48.23 49.86 50.34 52.07

80.14 79.43 78.87 81.54 82.58 84.36

112.99 112.03 115.48 119.40 121.38 122.00

44.66 44.28 45.97 47.53 48.36 48.45

0.265 0.265 0.256 0.259 0.258 0.259

0.1748 0.1750 0.2006 0.1922 0.1939 0.1918

Table 9Elastic moduli (corrected to zero porosity) and Debye temperature (θ) for Li-Ni-Fe-O system Content (x)

Eo

0.0 0.2 0.4 0.6 0.8 1.0

174.01 172.63 193.22 194.31 198.63 198.28

Go (GPa) 65.16 67.09 75.53 75.98 77.74 77.35

σ0

E*

0.335 0.287 0.279 0.279 0.277 0.282

176.65 183.25 189.85 196.45 203.05 209.67

G*

σ*

(GPa) 69.03 71.23 73.43 75.62 77.82 80.00

0.297 0.297 0.297 0.297 0.297 0.297

θE 549.29 550.62 572.67 569.27 568.69 561.24

(K)

θM 135.36 165.21 165.26 157.80 -


can be explained as : the Li1+ ions with 1s2 and Fe3+ with 3d5 orbital configuration are replaced by Ni2+ with 3d8 orbital configuration. Here, 1s2 is completely filled while 3d5 is half filled orbit. It is well known that completely filled or half filled orbits are stable and they do not contribute to the bond formation while Ni2+ have d-orbit with 8 electrons that form strong bonding with oxygen and other cations. Thus, on increasing nickel substitution interatomic bonding is expected to strengthen. We have also determined the values of longitudinal elastic wave velocity (Vl0) and shear wave velocity (Vs0) using the formula: 1

 C 2 Vl 0 =  11  and  ρ  1

 G 2 Vs0 =  0   ρ 

The values of Vl0 and Vs0 were further used to calculate mean elastic wave velocity (Vm0), which in turn used to calculate Debye temperature, using the relation: 1

  V 3 .V 3  3 Vm 0 = 3 3l 0 s 0 3    Vs 0 + 2Vl 0 

The calculated values of elastic constants and wave velocities for all the compositions (Table 8) are in good agreement to those obtained from ultrasonic pulse transmission32,33 technique (UPT). This validates the present method of elastic moduli determination. The Debye temperature (θE) values of all the ferrites have been calculated using the Anderson’s formula37: 1

h  3N A  3 Debye temperature θ E =   Vm 0 k  4πV A  where h and k are Planck’s and Boltzmann’s constant respectively, NA is Avogadro’s number and VA is mean atomic volume given by (M/ρ)/q, where M, the molecular weight (Table 6) and q is the number atoms (i.e. q = 7) in the formula unit. The value of θE for each composition is presented in Table 9. It is seen that θE increases with increasing Zn- content (x) up to x = 0.4. The Debye temperature is the temperature at

605

which maximum lattice vibrations take place. The observed increase in θE with content (x) suggested that lattice vibrations are hindered due to zinc substitution. This may be due to the fact that strength of inter atomic bonding increases with the replacement of Li+1 and Fe+3 by Ni2+ in Li0.5(1-x) NixFe2.5-0.5xO4 system as supported by our results on variation of elastic moduli with Ni -content (x). The magnitude of Debye temperature in the present work is consistent with those obtained for various ferrite systems 28-33. It is always desirable to have a general idea of the elastic moduli values before synthesis and characterization of the material, in order to tailor the properties. Recently, we have developed and successfully implemented heterogeneous metal mixture rule (MMMR) to estimate elastic constants of various spinel ferrites30 , garnets31 superconductors38,39 and La-based perovskites40 . The aim of the present work is to test the applicability of this model for estimating elastic moduli of various Li-Ni ferrite compositions and to compare the result with experimentally obtained values. The basic idea behind this model is that the density, longitudinal and transverse wave velocities and thus elastic moduli and Debye characteristics temperature of such polycrystalline oxide compositions depend up on the density and elastic wave velocity of individual metallic cation present in the system. According to this model, the elastic constant and Debye temperature value of polycrystalline oxide material (K*pm) is equal to the average stoichiometric compositional addition of elastic constant values of metallic elements present in the material. The elastic moduli such as bulk modulus, Young’s modulus, rigidity modulus, Poisson’s ratio and Debye temperature values of various metallic elements are taken from Refs 41, 42 and are used to estimate K *pm . The elastic constant value, to be estimated, for a given ferrite system can be given as:

K *pm =

1 ∞ ∑ C in K n n i >0 n =1

where K *pm is either Young’s modulus, rigidity modulus, bulk modulus, Poisson’s ratio or Debye temperature of the composition to be estimated, n is the total concentration of metallic cations involved in the chemical formula of the polycrystalline material

606


(n = 3 in the present case), Cin is the concentration of the nth cation in the formula unit while Kn is the corresponding modulus of the metallic elements. The values of elastic moduli (E*, G*, σ*) obtained from MMM rule are summarized in Table 9. The result of our calculations are in conformity with the elastic constant and Debye temperature values obtained from informed and X-ray spectroscopic measurements with adequate accuracy (Table 9). This finding is interesting in the sense that the elastic moduli of such oxide compositions can be estimated from the elastic constants of metallic elements present in the system and surprisingly the oxygen does not seem to play significant role for assigning elastic constants of these oxide compositions. Further investigation in this direction is in progress. The Debye temperature plays an important role in the study of a large number of solid-state problems involving lattice vibrations. A number of physical parameters such as mean square atomic displacements, elastic constants, are known to depend upon the Debye temperature of a solid. It has been shown that the Debye temperature obtained through different physical properties will not, in general, be equal. An attempt has been made to extract the X-ray Debye temperature θM for the typical compositions of Li0.5(1-x)NixFe2.5-0.5xO4 system. The θM values thus obtained are compared with the Debye temperature values obtained from the elastic constant data. A method which depends on the principle of measuring the integrated intensities of a large number of Bragg reflections at a fixed temperature was first outlined by Burger5 . 2

M (T ) =

2

6h T  x  sin θ φ ( x) +  2 2  mkθ M  4 λ

or M (T ) = B

sin2 θ

λ2

… (6)

6h2T  x where B = φ ( x) +  2  mkθ M  4

where m is the mass of atom (taken as the mean mass), θM is the X-ray Debye temperature, T is the temperature, θ is Bragg’s angle, x = θM/T and λ is the wavelength of X-ray; the function (Φ(x) + x/4) is calculated by James et al.43 . The average vibration amplitudes are related to the Bragg intensities, within the quasi-harmonic approximation, through the Debye-Waller theory. The

integrated intensity, I, from a cubic powder sample can be expressed as follows: I = Lp P ІFІ2

… (7)

where Lp is function of the Bragg angle known as the Lorentz-polarization factor; P is multiplicity; and |F| is the modulus of the structure factor. The structure factor for present system (space group Fd3m-Oh-7) can be written as: F(h k l) = x fNi FNi.e-MNi + 0.5(1-x) fLiFLi.e-MLi + 2.50.5xfFeFFe.e-MFe + 4fo Fo.e-Mo … (8) The exponential term in Eq. (6) represents the Debye-Waller factor for the four constituent atoms of lithium, nickel, iron and oxygen; fLi, fNi, fFe and fo are their respective atomic scattering factors; fLi, fNi, fFe and fo are their respective structure factors which are sine and cosine functions of hkl values. As the mass of lithium, nickel, iron and oxygen atoms is different; the respective Debye-Waller factors will also be different. To first approximation let MLi, MNi, MFe and Mo be equal to each other, Eq. (6), therefore, becomes: F(h k l) = ∑ f e-M

… (9)

Eq. (7) may be written using Eqs (6) and (9) as : 2

I = L p P ∑ f e − 2 B sin

2

θ / λ2

… (10)

The experimental structure factor, Fcorr, may be obtained from Eq. (10) using measured integrated intensity, I, as: 2

Fcorr =

I LP P

2

=∑f e

−2 B sin2 θ

λ2

… (11)

It follows from Eq. (11), that the slope In(ІFcorrІ2/ |∑f|2) against sin2θ/λ2 yields the temperature factor B and hence, θM can be obtained. The plots of In(|Fcorr|2/|∑f|2) against sin2θ/λ2 for typical compositions x = 0.2 and 0.4 at T = 300K is shown in Fig.5. The even-even (h + k = 2n, 1 + k = 2n), evenodd (h + k = 2n, 1 + k = 2n + 1), odd-even (h + k = 2n + 1, 1 + k = 2n), odd-odd (h + k = 2n + 1, 1 + k = 2n + 1) reflections lie the same straight line. The solid line


in Fig. 5 is least squares fit to the experimental data points. The Debye temperatures derived from the slope of ln( |Fcorr|2/ | ∑f |2 ) against sin2θ/λ2curves for the compositions with x = 0.2 and 0.4 are given in Table 9. The Debye temperature, θM, obtained at 300K from X-ray lies between 130-165K range, while the Debye temperature, θM from elastic constant data is ranging from 470 to 495K (Table 9) for x = 0.0-1.0. This suggests that the Debye temperature obtained from elastic constant is higher than that of the X-ray diffraction analysis. The discrepancy between the two can be explained on the basis of vibrational spectra. Possibly the elastic constant value θE is higher than the X-ray θM value mainly due to the existence of peaks in the vibrational spectra at low frequencies. The fact that the eveneven, even-odd, odd-even and odd-odd reflections lie on the same straight line (Fig. 5) indicates that MLi ≈ MNi ≈ MFe ≈Mo. Thus, the experiment has not distinguished between the individual Debye temperatures of mean square atomic displacements appropriate to the atoms of lithium, nickel iron and oxygen in the system. 4 Conclusions The results and conclusion drawn from structural properties, infrared spectroscopy and elastic properties studies of Li0.5(1-x)NixFe2.5-0.5xO4 spinel ferrites system can be summarized as follows:

607

The structural parameters and the Neel temperature determined theoretically on the basis of cation distribution are in good agreement with those deduced experimentally. This suggests that properties of ferrites are governed by the distribution of cations among the tetrahedral and octahedral sites of spinel lattice. The shift in ν1 band of IR spectra towards higher frequency side is due to the charge imbalance of A-sites which probably makes the oxygen ions shift towards Li1+ ions while shifting of the centre frequency of the ν2 band towards lower frequency side is due to increase in site radius, that hinders the fundamental frequency. The splitting of octahedral absorption band is due to the presence of three kinds of cations at the B-sites. The method of elastic constants, elastic wave velocities and Debye temperature determination by X-ray and infrared spectral analysis is found easier, valid and suitable for spinel ferrites. The observed increase in magnitude of elastic moduli with nickel substitution suggests strengthening of interatomic bonding and can be explained on the basis of electronic configuration of d-orbit of cations involved. The elastic constant corrected to zero porosity are in good agreement with those determined through metal-mixture rule, confirming the validity of the method. In polycrystalline oxide compositions oxygen anions play an important role for structure formation but less decisive to the elastic constant values. Acknowledgement One of the authors (KBM) is thankful to AICTE, New Delhi, for providing financial assistance in the form of career award for young teachers (2004). Appendix Illustrative calculation for Young’s modulus determination through MMM rules

(i)

Li0.2Ni0.6Fe2.2O4 (x = 0.6) E*pm = 1/3[(0.2)4.9 GPa + (0.6)207 GPa + (2.2)211 GPa] = 196.45 GPa (MMMR) Ref. [41] E0 = 194.31 GPa (IR)

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Fig. 5Plot of ln(|Fcorr|2/|Σf|2) versus sin2θ/λ2 at 300K

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