# IR Spectra, Elastic and Dielectric Properties of Li–Mn Ferrite

Authors: S. A. Mazen; N. I. Abu-Elsaad
Journal: ISRN Condensed Matter Physics
 (2012)

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## Abstract

Polycrystalline ferrites,Li0.5–0.5xMnxFe2.5–0.5xO4, (where x=0.0,0.1,0.3,0.5,0.7,0.9, and
1.0), were prepared by using ceramic method. Single phase cubic structure was confirmed by X-ray diffractometer. The lattice parameter “a” was found to increase with increasing Mn2+ ion substitution. IR spectra of the samples were recorded from 200 to 1000 cm−1. The two primary bands corresponding to tetrahedral υA and octahedral υB were observed at about 575 cm−1 and 370 cm−1, respectively. Elastic properties of these mixed ferrites were estimated as a function of composition. Young's modulus (E), rigidity modulus (G), bulk modulus (B), Debye temperature (θD), and mean sound velocity (Vm) were calculated from the transverse (Vt) and longitudinal (V1) wave velocities. The variation of elastic moduli with composition was interpreted in terms of binding forces between the atoms of spinel lattice. AC conductivity σ~ and dielectric properties of the samples were measured at room temperature over 100 Hz–1 MHz. The electrical conduction mechanism could be explained with the electron hopping model. Frequency exponential factor (s) was calculated and it was found between 0.4 and 0.8.


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## Body

## 1. Introduction

Polycrystalline ferrites have very important structural, magnetic, electrical, and dielectric properties that are dependent on various factors, such as method of preparation, substitution of cations, and microstructure [1, 2]. Introduction of a relatively small amount of foreign ions can change the properties of ferrites [3]. It can provide us with information for obtaining a high-quality ferrite for particular applications. Modification in electric and magnetic properties of lithium ferrites by substitution of different ions has been extensively studied [4–6].Infrared spectroscopy was used to determine the local symmetry of crystalline and noncrystalline solids and to study the ordering phenomenon in ferrites [6]. IR absorption bands mainly appear due to the vibrations of oxygen ions with cations at various frequencies. The frequencies depend upon cation masses, lattice parameter, and cation-oxygen bonding, and so forth. [7].Ultrasonic pulse transmission (UPT) is a very common technique for studying elastic constants. However, a new technique based on infrared spectroscopy has been developed by Modi et al. [8] for studying the elastic properties of spinel and garnet ferrites. The elastic constants are of much importance because they reveal the nature of binding forces in solids and help to understand thermal properties of the solids [9]. Elastic properties of spinel ferrites have not been studied so systematically as their magnetic and electrical properties. Moreover, there is a need for a thorough study of the elastic behaviour of these ferrites with new compositions possessing certain desired elastic properties.The present work was aimed to estimate elastic properties of Li-Mn ferrite using IR technique. Also, dielectric properties of this ferrite were investigated at room temperature.


## 2. Experimental Techniques

Polycrystalline Li-Mn ferrites,Li0.5–0.5xMnxFe2.5–0.5xO4; 0.0≤x≤1.0, were prepared by the standard ceramic method. Experimental details were reported previously [4, 10, 11]. A single phase spinel structure was recorded for all compositions using X-ray diffractometer, XRD, employing Cu Kα radiation (λ=1.5405Å) (type PHILIPS X’PertDiffractometer). IR spectra over 200–1000 cm−1 were recorded at room temperature by using an infrared spectrometer, (model 1430, PerkinElmer). Dielectric measurements were conducted at room temperature over 100 Hz–1 MHz by using a RLC (Model PM6306 FLUKA). Disk samples of 13 mm in diameter and 3–5 mm in thickness were used for dielectric measurement with silver electrode.The AC conductivity measurements were carried out using the complex impedance technique and calculated from this equation:(1)σ~*=Z-1tA.The dielectric constant(ε′) and dielectric loss (ε′′) were calculated from AC conductivity (σ~) and the dielectric loss tangent (tanδ) using the following formulae [12]:
(2)ε′′=εoσ~f=1.8×1010σ~f,ε′=ε′′tanδ,
where f is the frequency of the applied field.


## 3. Results and Discussion

### 3.1. X-Ray Analysis and IR Absorption

Figure1 shows the X-ray diffraction patterns for the above-mentioned composition of Li-Mn ferrite. The main reflection planes of the spinel structure of ferrites are shown in the X-ray patterns where these planes are (111), (210), (211), (220), (311), (222), (400), (422), (333), (440), and (533). The samples of x=0.7,0.9 and 1 contain a little of second phase of  α-Fe2O3. In general, the behaviour of diffraction pattern proved that the prepared samples were formed in the cubic spinel structure according to JCPDS card. The estimated values of the experimental lattice parameter “aexp” as a function of composition are presented in Figure 2. It is found that the lattice parameter increases with the increase of Mn content. This increment of the lattice parameter could be explained on the basis of ionic radii, where the radius of Mn2+ ion (0.08 nm) is larger than that of Li+ (0.073 nm) and Fe3+ (0.064 nm) ions. The Mn2+ ions successively replace the Li+ and Fe3+ ions. The unit cell expands to accumulate the larger ion, so it is expected to increase the lattice parameter. The uniform increase in lattice constant with manganese substitution indicates that lattice expands without disturbing the symmetry of lattice [13]. A similar variation has also been reported by Ravinder et al. [9].Figure 1
X-Ray diffraction pattern ofLi0.5-0.5xMnxFe2.5-0.5xO4(x=0.0,0.1,0.3,0.5,0.7,0.9,  and 1.0).Figure 2
Variation of experimental lattice parameter with Mn substitution.The IR spectra of the investigated composition for Li-Mn ferrite are shown in Figure3. The absorption bands are summarized in Table 1. No absorption bands were observed above 1000 cm−1. The IR spectrum of pure Li ferrite (x=0.0) has been analyzed in detail by Mazen et al. [14]. It is noticed that the primary band  υ1* and the second absorption band υ2* are in the ranges of 585–546 cm−1 and 379–372 cm−1, respectively. The values of  ν1* are higher than those of ν2*,  indicating that the normal mode of vibration of the tetrahedral complexes is higher than that of the corresponding octahedral site. This may be due to the shorter bond length of the tetrahedral site (RA=1.89Å) than that of the octahedral site (RB=1.99Å) [15]. From Table 1 it is obvious that by introducing Mn2+, the band υ1* shifts slightly towards lower frequency meanwhile the band υ2* has no shift or change with frequency. The third band υ3 appears at 325 cm−1 for x=0.0 and changes to weak shoulder for x≥0.1. This band could be attributed to the divalent metal ion-oxygen complexes at octahedral sites. Hence, it indicates the presence of a small amount of Fe2+ ions [16]. The fourth vibrational band υ4 (265 cm−1) is due to the lattice vibration [17].Table 1
IR absorption bands of the ferrites.
x
Tetrahedral site (A-site)
Octahedral site (B-site)
υ
1 (1)
υ
1 (2)
υ
1
*
υ
1 (3)
υ
2 (1)
υ
2
*
υ
3
υ
4
0.0
707
667
585
543
457
379
325
267
0.1
704 (w, sh)
—
578
530 (w, sh)
450
375
330w
265
0.3
—
—
584
—
—
372
327w
267
0.5
—
—
557
—
—
374
310w
268
0.7
—
—
553
—
—
375
324w
265
0.9
—
—
547
—
—
376
327w
265
1
—
—
546
—
—
376
325w
—
*
the primary band, w: weak, and sh: shoulder.Figure 3
IR absorption spectra of the mixed Li-Mn ferrites.The spectrum indicates a splitting in the absorption bands; that is, the first primary band (υ1*) consists of two shoulders υ1(1) at 710 and  υ1(2) at 670 cm−1 and a small band  υ1(3) at 535 cm−1. It has been shown that the presence of Fe2+  ions in ferrites can cause splitting of or produce shoulders on the absorption bands [18]. This is attributed to the Jahn-Teller distortion caused by Fe2+, which cause local deformations in the crystal field potential and hence lead to the splitting of the band υ1*. These bands appeared in the samples with low content of Mn (x=0.0 and 0.1) and completely disappeared at high concentration of Mn.According to Tarte [19] and Mazen et al. [14], the high frequency band ν2(1) recorded only for x=0.0 and x=0.1 in the range of 450–458 cm−1 could be assigned to  Li+-O2-  complexes at the octahedral site. The intensity of this band goes on decreasing with increasing x since the Li+ content decreases with increasing  x, so it persists only up to x=0.1. It completely disappeared for x≥0.3.The threshold frequencyυth for the electronic transition can be determined from the maximum point of the absorption spectra [20]. These threshold values are illustrated in Figure 4. It is found that the threshold frequency (υth) decreases with increasing Mn content for x≥0.1, that is, with decreasing both Fe3+ and Li+ concentrations.Figure 4
Dependence of threshold frequencyνth with composition x.The force constant can be calculated for tetrahedral site(kt) and octahedral site (ko) by using the method suggested by Waldron [20]:
(3)Kt=7.62×MA×υ12×10-7N/m,(4)Ko=10.62×MB2×υ22×10-7N/m,
where  MA and MB are the molecular weights of cations at A- and B-sites, respectively. Both MA and  MB  were calculated from the cation distribution formula suggested in Table 2. The calculated values of the force constants Kt and  Ko are listed in Table 3. It can be seen that Kt decreases while Ko increases with increasing Mn content. Furthermore the calculated values of Kt are greater than those of  Ko. However, the values of the bond length of A-site (RA) are smaller than those of B-site (RB). This is due to the inverse proportionality between the bond length and the force constants [21].Table 2
Cation distribution of the Li-Mn ferrite samples (with0.0≤x≤1.0).
x
Cation distribution
0.0
(
Fe
3
+
)
A
[
L
i
0.5
+
F
e
1.5
3
+
]
B
O
4
2
-
0.1
≤
x
≤
0.5
(
F
e
1
-
x
3
+
M
n
x
2
+
)
[
L
i
0.5
-
0.5
x
+
F
e
1.5
+
0.5
x
3
+
]
O
4
2
-
0.5
<
x
≤
1.0
(
F
e
1
-
x
+
y
3
+
M
n
x
-
y
2
+
)
[
L
i
0.5
-
0.5
x
+
M
n
y
F
e
1.5
+
0.5
x
-
y
3
+
]
O
4
2
-
(  )A: A-site and []B: B-site.Table 3
Force constants and elastic constants of the Li-Mn ferrite samples.
x
K
t
K
o
V
L
V
t
B
E
G
P
V
m
V
A
Θ
D
'
Θ
D
(N/m)
(m/s)
GPa
m/s
(10−6 m3)
K
K
0.0
145.41
66.54
5171
2985
127.22
114.49
42.41
0.35
3314
6.22
454
693
0.1
142.03
66.97
5119
2956
125.22
112.7
41.74
0.35
3281
6.26
448
685
0.3
144.35
69.62
5137
2965
127.82
115.04
42.61
0.35
3293
6.32
448
687
0.5
130.78
74.01
4982
2876
121.89
109.71
40.63
0.35
3193
6.37
434
669
0.7
128.83
78.04
4969
2869
122.69
110.43
40.89
0.35
3186
6.43
431
668
0.9
125.44
82.06
4943
2853
122.78
110.51
40.93
0.35
3168
6.49
428
664
1.0
124.87
83.85
4961
2864
122.78
110.49
40.93
0.35
3179
6.61
427
663Also, from IR spectra, the values of half bandwidth for each site are calculated and the ratioΓA/ΓB is presented in Figure 5. It can be noticed that the ratio ΓA/ΓB decreases with increasing Mn content. The half bandwidth depends on the statistical distribution of various cations over the two sites [16]. This dependence is related to the replacement process that occurs between the smaller Fe3+ (0.064 nm) and Li+ (0.073 nm) with the larger Mn2+ (0.08 nm).Figure 5
Change in half band width ratio(ΓA/ΓB) against composition x.The Debye temperatureθD can be calculated using the following relation [20]:
(5)θD=ℏCνavk,
where νav=(νA+νB)/2, νA is the frequency of the primary band of A-site, νB is the frequency of the primary band of B-site, ℏ=h/2π, h is the Plank constant, k is the Boltzmann’s constant and C=3×1010 cm/s, C is the velocity of light. The calculated values of the Debye temperature are listed in Table 3. It can be noticed that all values of the Debye temperature are varying between 660 and 690 K. These values have a great importance to determine the conduction mechanism of these ferrites. It can be seen that θD decreases with increasing Mn concentration. This behaviour can be discussed on the basis of a specific heat theory. According to this theory, electrons absorbed part of the heat and θD may decrease with increasing Mn concentration; this suggests that the conduction for these samples is due to electrons (i.e., n-type) [22].


### 3.2.  Elastic Properties

To study the elastic properties of the spinel and garnet ferrites, a new technique based on infrared spectroscopy was developed by Modi et al. [8]. The elastic moduli can be evaluated by using the following relations.The bulk modulus(B) of solids is defined as
(6)B=13[C11+2C12],
where  C11  and C12 are the stiffness constants. But according to Waldron [20], for isotropic materials with cubic symmetry like spinel ferrites and garnets, C11≈C12, therefore, B=C11. Also, the force constant (k) is related to the stiffness constant by [k=aC11] [23], where k  is the average force constant (k=(kt+ko)/2). Further, the values of the longitudinal elastic wave (Vl) and the transverse elastic wave (Vt) have been determined as follows [24, 25]:
(7)Vl=(C11dx)1/2,Vt=Vl3.The variation in longitudinal(Vl) and transverse (Vt) sound velocity as a function of Mn composition x is depicted in Table 3. It can be seen that both Vl and Vt decrease upon substitution of Mn. The values of Vl and Vt are used to calculate elastic moduli of the ferrite specimens by using the following formulae [26]:
(8)meanelasticwavevelocityVm=13[2Vl3+1Vt3]-1/3,rigiditymodulus(G)=dxVt2,Poisson’sratio(P)=3B-2G6B+2G,Young’smodulus(E)=(1+P)2G.The calculated values of different elastic moduli for the ferrites are listed in Table3. From this table, it can be seen that, B, E, and G slightly decrease with increasing Mn-content (x). According to Wooster [24], the behaviour of elastic moduli is attributed to the strengthening of the interatomic binding between various atoms of the spinel lattice with increasing Mn content. The interatomic binding between various atoms is weakened continuously and therefore elastic moduli decreases with manganese content x. Similar results were observed in Ni-Zn ferrite [27] and Co-Zn ferrite [28], where the interatomic binding between the various atoms decreases with increasing Zn content x.The Debye temperatureθD′ was calculated by employing Anderson’s formula [29]:
(9)θD′=hkB[3NA4πVA]1/3·Vm,
where VA  is mean atomic volume given by (M/dx)/q, M is the molecular weight, q  is the number of atoms in the formula unit (equals 7), and NA is Avogadro’s number. The calculated values are listed in Table 3. It is observed that the Debye temperature θD calculated according to Waldron equation (5) is higher than that calculated according to Anderson’s formula θD′ (9), but both of them decrease with increasing manganese substitution.A plot of average sound velocity(Vm) against Debye temperature (θD′) is shown in Figure 6. It is interesting to note from the figure that the average sound velocity increases linearly with the Debye temperature. A similar variation was also reported by Reddy [30] in Mn-Mg mixed ferrites and by Ravinder [9] in Li-Mn mixed ferrites. This behaviour clearly indicates the direct relationship between the acoustic parameter (average sound velocity) and the important thermodynamic parameter (Debye temperature).Figure 6
Debye temperatureθD′ against average sound velocity  Vm.


### 3.3. Dielectric Properties

AC conductivityσ~(ω)  with the dielectric properties (dielectric constant ε′ and dielectric loss ε′′) as a function of frequency (f=102–106Hz) at room temperature for Li0.5-0.5xMnxFe2.5-0.5xO4  (where x=0.0→1.0) is shown in Figure 7. Ac conductivity increases slowly at lower frequencies, but after a certain frequency (≈104Hz) it increases rapidly and reaches the highest value at 106 Hz. The dispersion in σ~ with frequency has been explained by Koops theorem [31], which supposed that the ferrite compact acts as a multilayer capacitor. In this model, the ferrite grain and grain boundaries have different properties. The effect of the multilayer condenser rises with frequency; as a result, the conductivity increases. The conductivity is a complex quantity [32]:
(10)σ~=σ′+iσ′′.ε
′, ε′′, and σ versus frequency at room temperature for x=0.0,0.5, and 1.
(a)
(b)
(c)The impedance of this model can be represented by the following equation:(11)Z-1=R-1+j(2πfc),
where f is the applied frequency and R, C are the parallel equivalent resistance and capacitance of the material, respectively.In general, the dielectric can be denoted by a complex dielectric constantε*, hence
(12)ε*=ε′-jε′′,
where ε′ is the true permittivity, the “dielectric constant” (which describes the stored energy), while ε′′  is the imaginary permittivity, the “dielectric loss” (which describes the dissipation energy).The behaviour of bothε′ and ε′′ against frequency at room temperature is shown in Figure 7. The general trend for ε′ and ε′′ is decreasing with increasing frequency for all values of x. However, ε′′ decreases faster than  ε′  over the same range of frequency. In the high-frequency range the value of  ε′  becomes closer to the value of  ε′′. This behaviour of a dielectric may be explained qualitatively by the supposition that the mechanism of the polarization process in ferrite is similar to that of the conduction process. Iwauchi [33] has pointed out that there is a strong correlation between the conduction mechanism and the dielectric behaviour of ferrites. The electrical conduction mechanism can be explained by the electron hopping model proposed by Heikes and Jonston [34]. The electron hopping (Fe2+↔Fe3+ or Mn2+↔Mn3+) occurs by electron transfer between adjacent octahedral sites (B-sites) in the spinel lattice [35]. Thus by the electronic exchange:
(13)Fe2++Mn3+⟷Fe3++Mn2+.One can obtain local displacements of electrons in the direction of the applied electric field. These displacements determine the polarization of the ferrite. It is known that the effect of the polarization is to reduce the field inside the medium. Therefore, the dielectric constant of the substance may decrease substantially as the frequency is increased. Also, such a decrease can be attributed to the fact that the electric exchange between Fe2+ and Fe3+ ions cannot follow the external applied field beyond a certain frequency.Figure8 shows the variation of loss tangent (tanδ) with frequency of the samples (where  tanδ=ε′′/ε′). A maximum in tanδ at a certain frequency  (fo) can be observed when  ε′  has a minimum value. The condition for observing a maximum tanδ of a dielectric material is given by [36]
(14)ωoτ=1turn,
where  ωo=2πfo and τ is the relaxation time. This relaxation time is related to the jumping (hopping) probability per unit time, p, by the equation:
(15)τ=12porfoαp.
This equation shows that fo is proportional to the hopping probability. It was found that τ=6.95×10-4,7.47×10-4 and 2.50×10-4 s for x=0.0,0.5 and 1.0, respectively.Variation in the dielectric loss tangent(tanδ) with frequency at room temperature for x=0.0,0.5, and 1.
(a)
(b)
(c)


### 3.4. Determination of Frequency Exponential Factor(s)

Because hopping conduction mechanism was assumed for the Li-Mn ferrites, the ACconductivityσ~(ω)  can be represented by a power law [32]:
(16)σ~(ω)αAωs,
where A is slightly dependent on temperature, ω is the applied frequency at which the conductivity σ~ was measured, and the power s, which is a weak function of frequency, is determined to be 0.4–0.8 [32]. Pike [37] and Elliott [38] considered s≤1 and it is expressed by the following relation [32, 38]:
(17)s=1-4ln(υph/ω).In the present study, the values of s were calculated from the relation oflogσ~ versus logω as shown in Figure 9. This relation shows a straight line over 104 Hz up to 1 MHz. The estimated values of the power s were listed in Table 4. It is found that the s-factor is composition dependent, where s decreases up to x=0.5 by 55% and then increases by 35% with further increase in  x.Table 4
Calculateds-values and νph of the samples.
x
s
ν
ph (Hz) from(logω-logσ) plots
ν
ph (1013 Hz) from IR spectra
A-site
B-site
0.0
0.70
2.67
×
10
12
1.75
1.14
0.1
0.64
6.21
×
10
11
1.73
1.13
0.3
0.51
1.03
×
10
10
1.75
1.12
0.5
0.31
1.00
×
10
9
1.67
1.12
0.7
0.28
7.99
×
10
8
1.66
1.13
0.9
0.38
2.09
×
10
9
1.64
1.13
1.0
0.48
6.68
×
10
9
1.64
1.12Variation in exponential factor (s) with angular frequency ω at room temperature for x=0.0,0.5, and 1.
(a)
(b)
(c)Also, the phonon frequencyνph for each Li-Mn ferrite was calculated at f=5×105  Hz and compared with those calculated from IR spectra [4]. The values of the phonon frequency were tabulated in Table 4. It was found that the values of νph calculated from the (logσ~-logω) plots are lower than those calculated from IR absorption spectra by a factor of about 102–105 Hz (Table 4). This difference may be related to the nature of polarization in each case. At AC field, the polarization is due to ionic and electronic polarizations, while, in the IR spectra, the polarization is due to electronic polarization only. Also, the value of υph at A-site is slightly higher than that of υph at B-site. This may be attributed to the fact that at A-site is characterized by covalent bond only, while the B-site is characterized by both ionic and covalent bonds.


## 3.1. X-Ray Analysis and IR Absorption

Figure1 shows the X-ray diffraction patterns for the above-mentioned composition of Li-Mn ferrite. The main reflection planes of the spinel structure of ferrites are shown in the X-ray patterns where these planes are (111), (210), (211), (220), (311), (222), (400), (422), (333), (440), and (533). The samples of x=0.7,0.9 and 1 contain a little of second phase of  α-Fe2O3. In general, the behaviour of diffraction pattern proved that the prepared samples were formed in the cubic spinel structure according to JCPDS card. The estimated values of the experimental lattice parameter “aexp” as a function of composition are presented in Figure 2. It is found that the lattice parameter increases with the increase of Mn content. This increment of the lattice parameter could be explained on the basis of ionic radii, where the radius of Mn2+ ion (0.08 nm) is larger than that of Li+ (0.073 nm) and Fe3+ (0.064 nm) ions. The Mn2+ ions successively replace the Li+ and Fe3+ ions. The unit cell expands to accumulate the larger ion, so it is expected to increase the lattice parameter. The uniform increase in lattice constant with manganese substitution indicates that lattice expands without disturbing the symmetry of lattice [13]. A similar variation has also been reported by Ravinder et al. [9].Figure 1
X-Ray diffraction pattern ofLi0.5-0.5xMnxFe2.5-0.5xO4(x=0.0,0.1,0.3,0.5,0.7,0.9,  and 1.0).Figure 2
Variation of experimental lattice parameter with Mn substitution.The IR spectra of the investigated composition for Li-Mn ferrite are shown in Figure3. The absorption bands are summarized in Table 1. No absorption bands were observed above 1000 cm−1. The IR spectrum of pure Li ferrite (x=0.0) has been analyzed in detail by Mazen et al. [14]. It is noticed that the primary band  υ1* and the second absorption band υ2* are in the ranges of 585–546 cm−1 and 379–372 cm−1, respectively. The values of  ν1* are higher than those of ν2*,  indicating that the normal mode of vibration of the tetrahedral complexes is higher than that of the corresponding octahedral site. This may be due to the shorter bond length of the tetrahedral site (RA=1.89Å) than that of the octahedral site (RB=1.99Å) [15]. From Table 1 it is obvious that by introducing Mn2+, the band υ1* shifts slightly towards lower frequency meanwhile the band υ2* has no shift or change with frequency. The third band υ3 appears at 325 cm−1 for x=0.0 and changes to weak shoulder for x≥0.1. This band could be attributed to the divalent metal ion-oxygen complexes at octahedral sites. Hence, it indicates the presence of a small amount of Fe2+ ions [16]. The fourth vibrational band υ4 (265 cm−1) is due to the lattice vibration [17].Table 1
IR absorption bands of the ferrites.
x
Tetrahedral site (A-site)
Octahedral site (B-site)
υ
1 (1)
υ
1 (2)
υ
1
*
υ
1 (3)
υ
2 (1)
υ
2
*
υ
3
υ
4
0.0
707
667
585
543
457
379
325
267
0.1
704 (w, sh)
—
578
530 (w, sh)
450
375
330w
265
0.3
—
—
584
—
—
372
327w
267
0.5
—
—
557
—
—
374
310w
268
0.7
—
—
553
—
—
375
324w
265
0.9
—
—
547
—
—
376
327w
265
1
—
—
546
—
—
376
325w
—
*
the primary band, w: weak, and sh: shoulder.Figure 3
IR absorption spectra of the mixed Li-Mn ferrites.The spectrum indicates a splitting in the absorption bands; that is, the first primary band (υ1*) consists of two shoulders υ1(1) at 710 and  υ1(2) at 670 cm−1 and a small band  υ1(3) at 535 cm−1. It has been shown that the presence of Fe2+  ions in ferrites can cause splitting of or produce shoulders on the absorption bands [18]. This is attributed to the Jahn-Teller distortion caused by Fe2+, which cause local deformations in the crystal field potential and hence lead to the splitting of the band υ1*. These bands appeared in the samples with low content of Mn (x=0.0 and 0.1) and completely disappeared at high concentration of Mn.According to Tarte [19] and Mazen et al. [14], the high frequency band ν2(1) recorded only for x=0.0 and x=0.1 in the range of 450–458 cm−1 could be assigned to  Li+-O2-  complexes at the octahedral site. The intensity of this band goes on decreasing with increasing x since the Li+ content decreases with increasing  x, so it persists only up to x=0.1. It completely disappeared for x≥0.3.The threshold frequencyυth for the electronic transition can be determined from the maximum point of the absorption spectra [20]. These threshold values are illustrated in Figure 4. It is found that the threshold frequency (υth) decreases with increasing Mn content for x≥0.1, that is, with decreasing both Fe3+ and Li+ concentrations.Figure 4
Dependence of threshold frequencyνth with composition x.The force constant can be calculated for tetrahedral site(kt) and octahedral site (ko) by using the method suggested by Waldron [20]:
(3)Kt=7.62×MA×υ12×10-7N/m,(4)Ko=10.62×MB2×υ22×10-7N/m,
where  MA and MB are the molecular weights of cations at A- and B-sites, respectively. Both MA and  MB  were calculated from the cation distribution formula suggested in Table 2. The calculated values of the force constants Kt and  Ko are listed in Table 3. It can be seen that Kt decreases while Ko increases with increasing Mn content. Furthermore the calculated values of Kt are greater than those of  Ko. However, the values of the bond length of A-site (RA) are smaller than those of B-site (RB). This is due to the inverse proportionality between the bond length and the force constants [21].Table 2
Cation distribution of the Li-Mn ferrite samples (with0.0≤x≤1.0).
x
Cation distribution
0.0
(
Fe
3
+
)
A
[
L
i
0.5
+
F
e
1.5
3
+
]
B
O
4
2
-
0.1
≤
x
≤
0.5
(
F
e
1
-
x
3
+
M
n
x
2
+
)
[
L
i
0.5
-
0.5
x
+
F
e
1.5
+
0.5
x
3
+
]
O
4
2
-
0.5
<
x
≤
1.0
(
F
e
1
-
x
+
y
3
+
M
n
x
-
y
2
+
)
[
L
i
0.5
-
0.5
x
+
M
n
y
F
e
1.5
+
0.5
x
-
y
3
+
]
O
4
2
-
(  )A: A-site and []B: B-site.Table 3
Force constants and elastic constants of the Li-Mn ferrite samples.
x
K
t
K
o
V
L
V
t
B
E
G
P
V
m
V
A
Θ
D
'
Θ
D
(N/m)
(m/s)
GPa
m/s
(10−6 m3)
K
K
0.0
145.41
66.54
5171
2985
127.22
114.49
42.41
0.35
3314
6.22
454
693
0.1
142.03
66.97
5119
2956
125.22
112.7
41.74
0.35
3281
6.26
448
685
0.3
144.35
69.62
5137
2965
127.82
115.04
42.61
0.35
3293
6.32
448
687
0.5
130.78
74.01
4982
2876
121.89
109.71
40.63
0.35
3193
6.37
434
669
0.7
128.83
78.04
4969
2869
122.69
110.43
40.89
0.35
3186
6.43
431
668
0.9
125.44
82.06
4943
2853
122.78
110.51
40.93
0.35
3168
6.49
428
664
1.0
124.87
83.85
4961
2864
122.78
110.49
40.93
0.35
3179
6.61
427
663Also, from IR spectra, the values of half bandwidth for each site are calculated and the ratioΓA/ΓB is presented in Figure 5. It can be noticed that the ratio ΓA/ΓB decreases with increasing Mn content. The half bandwidth depends on the statistical distribution of various cations over the two sites [16]. This dependence is related to the replacement process that occurs between the smaller Fe3+ (0.064 nm) and Li+ (0.073 nm) with the larger Mn2+ (0.08 nm).Figure 5
Change in half band width ratio(ΓA/ΓB) against composition x.The Debye temperatureθD can be calculated using the following relation [20]:
(5)θD=ℏCνavk,
where νav=(νA+νB)/2, νA is the frequency of the primary band of A-site, νB is the frequency of the primary band of B-site, ℏ=h/2π, h is the Plank constant, k is the Boltzmann’s constant and C=3×1010 cm/s, C is the velocity of light. The calculated values of the Debye temperature are listed in Table 3. It can be noticed that all values of the Debye temperature are varying between 660 and 690 K. These values have a great importance to determine the conduction mechanism of these ferrites. It can be seen that θD decreases with increasing Mn concentration. This behaviour can be discussed on the basis of a specific heat theory. According to this theory, electrons absorbed part of the heat and θD may decrease with increasing Mn concentration; this suggests that the conduction for these samples is due to electrons (i.e., n-type) [22].


## 3.2.  Elastic Properties

To study the elastic properties of the spinel and garnet ferrites, a new technique based on infrared spectroscopy was developed by Modi et al. [8]. The elastic moduli can be evaluated by using the following relations.The bulk modulus(B) of solids is defined as
(6)B=13[C11+2C12],
where  C11  and C12 are the stiffness constants. But according to Waldron [20], for isotropic materials with cubic symmetry like spinel ferrites and garnets, C11≈C12, therefore, B=C11. Also, the force constant (k) is related to the stiffness constant by [k=aC11] [23], where k  is the average force constant (k=(kt+ko)/2). Further, the values of the longitudinal elastic wave (Vl) and the transverse elastic wave (Vt) have been determined as follows [24, 25]:
(7)Vl=(C11dx)1/2,Vt=Vl3.The variation in longitudinal(Vl) and transverse (Vt) sound velocity as a function of Mn composition x is depicted in Table 3. It can be seen that both Vl and Vt decrease upon substitution of Mn. The values of Vl and Vt are used to calculate elastic moduli of the ferrite specimens by using the following formulae [26]:
(8)meanelasticwavevelocityVm=13[2Vl3+1Vt3]-1/3,rigiditymodulus(G)=dxVt2,Poisson’sratio(P)=3B-2G6B+2G,Young’smodulus(E)=(1+P)2G.The calculated values of different elastic moduli for the ferrites are listed in Table3. From this table, it can be seen that, B, E, and G slightly decrease with increasing Mn-content (x). According to Wooster [24], the behaviour of elastic moduli is attributed to the strengthening of the interatomic binding between various atoms of the spinel lattice with increasing Mn content. The interatomic binding between various atoms is weakened continuously and therefore elastic moduli decreases with manganese content x. Similar results were observed in Ni-Zn ferrite [27] and Co-Zn ferrite [28], where the interatomic binding between the various atoms decreases with increasing Zn content x.The Debye temperatureθD′ was calculated by employing Anderson’s formula [29]:
(9)θD′=hkB[3NA4πVA]1/3·Vm,
where VA  is mean atomic volume given by (M/dx)/q, M is the molecular weight, q  is the number of atoms in the formula unit (equals 7), and NA is Avogadro’s number. The calculated values are listed in Table 3. It is observed that the Debye temperature θD calculated according to Waldron equation (5) is higher than that calculated according to Anderson’s formula θD′ (9), but both of them decrease with increasing manganese substitution.A plot of average sound velocity(Vm) against Debye temperature (θD′) is shown in Figure 6. It is interesting to note from the figure that the average sound velocity increases linearly with the Debye temperature. A similar variation was also reported by Reddy [30] in Mn-Mg mixed ferrites and by Ravinder [9] in Li-Mn mixed ferrites. This behaviour clearly indicates the direct relationship between the acoustic parameter (average sound velocity) and the important thermodynamic parameter (Debye temperature).Figure 6
Debye temperatureθD′ against average sound velocity  Vm.


## 3.3. Dielectric Properties

AC conductivityσ~(ω)  with the dielectric properties (dielectric constant ε′ and dielectric loss ε′′) as a function of frequency (f=102–106Hz) at room temperature for Li0.5-0.5xMnxFe2.5-0.5xO4  (where x=0.0→1.0) is shown in Figure 7. Ac conductivity increases slowly at lower frequencies, but after a certain frequency (≈104Hz) it increases rapidly and reaches the highest value at 106 Hz. The dispersion in σ~ with frequency has been explained by Koops theorem [31], which supposed that the ferrite compact acts as a multilayer capacitor. In this model, the ferrite grain and grain boundaries have different properties. The effect of the multilayer condenser rises with frequency; as a result, the conductivity increases. The conductivity is a complex quantity [32]:
(10)σ~=σ′+iσ′′.ε
′, ε′′, and σ versus frequency at room temperature for x=0.0,0.5, and 1.
(a)
(b)
(c)The impedance of this model can be represented by the following equation:(11)Z-1=R-1+j(2πfc),
where f is the applied frequency and R, C are the parallel equivalent resistance and capacitance of the material, respectively.In general, the dielectric can be denoted by a complex dielectric constantε*, hence
(12)ε*=ε′-jε′′,
where ε′ is the true permittivity, the “dielectric constant” (which describes the stored energy), while ε′′  is the imaginary permittivity, the “dielectric loss” (which describes the dissipation energy).The behaviour of bothε′ and ε′′ against frequency at room temperature is shown in Figure 7. The general trend for ε′ and ε′′ is decreasing with increasing frequency for all values of x. However, ε′′ decreases faster than  ε′  over the same range of frequency. In the high-frequency range the value of  ε′  becomes closer to the value of  ε′′. This behaviour of a dielectric may be explained qualitatively by the supposition that the mechanism of the polarization process in ferrite is similar to that of the conduction process. Iwauchi [33] has pointed out that there is a strong correlation between the conduction mechanism and the dielectric behaviour of ferrites. The electrical conduction mechanism can be explained by the electron hopping model proposed by Heikes and Jonston [34]. The electron hopping (Fe2+↔Fe3+ or Mn2+↔Mn3+) occurs by electron transfer between adjacent octahedral sites (B-sites) in the spinel lattice [35]. Thus by the electronic exchange:
(13)Fe2++Mn3+⟷Fe3++Mn2+.One can obtain local displacements of electrons in the direction of the applied electric field. These displacements determine the polarization of the ferrite. It is known that the effect of the polarization is to reduce the field inside the medium. Therefore, the dielectric constant of the substance may decrease substantially as the frequency is increased. Also, such a decrease can be attributed to the fact that the electric exchange between Fe2+ and Fe3+ ions cannot follow the external applied field beyond a certain frequency.Figure8 shows the variation of loss tangent (tanδ) with frequency of the samples (where  tanδ=ε′′/ε′). A maximum in tanδ at a certain frequency  (fo) can be observed when  ε′  has a minimum value. The condition for observing a maximum tanδ of a dielectric material is given by [36]
(14)ωoτ=1turn,
where  ωo=2πfo and τ is the relaxation time. This relaxation time is related to the jumping (hopping) probability per unit time, p, by the equation:
(15)τ=12porfoαp.
This equation shows that fo is proportional to the hopping probability. It was found that τ=6.95×10-4,7.47×10-4 and 2.50×10-4 s for x=0.0,0.5 and 1.0, respectively.Variation in the dielectric loss tangent(tanδ) with frequency at room temperature for x=0.0,0.5, and 1.
(a)
(b)
(c)


## 3.4. Determination of Frequency Exponential Factor(s)

Because hopping conduction mechanism was assumed for the Li-Mn ferrites, the ACconductivityσ~(ω)  can be represented by a power law [32]:
(16)σ~(ω)αAωs,
where A is slightly dependent on temperature, ω is the applied frequency at which the conductivity σ~ was measured, and the power s, which is a weak function of frequency, is determined to be 0.4–0.8 [32]. Pike [37] and Elliott [38] considered s≤1 and it is expressed by the following relation [32, 38]:
(17)s=1-4ln(υph/ω).In the present study, the values of s were calculated from the relation oflogσ~ versus logω as shown in Figure 9. This relation shows a straight line over 104 Hz up to 1 MHz. The estimated values of the power s were listed in Table 4. It is found that the s-factor is composition dependent, where s decreases up to x=0.5 by 55% and then increases by 35% with further increase in  x.Table 4
Calculateds-values and νph of the samples.
x
s
ν
ph (Hz) from(logω-logσ) plots
ν
ph (1013 Hz) from IR spectra
A-site
B-site
0.0
0.70
2.67
×
10
12
1.75
1.14
0.1
0.64
6.21
×
10
11
1.73
1.13
0.3
0.51
1.03
×
10
10
1.75
1.12
0.5
0.31
1.00
×
10
9
1.67
1.12
0.7
0.28
7.99
×
10
8
1.66
1.13
0.9
0.38
2.09
×
10
9
1.64
1.13
1.0
0.48
6.68
×
10
9
1.64
1.12Variation in exponential factor (s) with angular frequency ω at room temperature for x=0.0,0.5, and 1.
(a)
(b)
(c)Also, the phonon frequencyνph for each Li-Mn ferrite was calculated at f=5×105  Hz and compared with those calculated from IR spectra [4]. The values of the phonon frequency were tabulated in Table 4. It was found that the values of νph calculated from the (logσ~-logω) plots are lower than those calculated from IR absorption spectra by a factor of about 102–105 Hz (Table 4). This difference may be related to the nature of polarization in each case. At AC field, the polarization is due to ionic and electronic polarizations, while, in the IR spectra, the polarization is due to electronic polarization only. Also, the value of υph at A-site is slightly higher than that of υph at B-site. This may be attributed to the fact that at A-site is characterized by covalent bond only, while the B-site is characterized by both ionic and covalent bonds.


## 4.  Conclusions

(1)
IR spectra confirmed the formation of spinel structure and gave information about the distribution of ions between the two sites, tetrahedral (A-site) at 575 cm−1 and octahedral (B-site) at 370 cm−1.
(2)
The behaviour of Debye temperatureθD showed that electrons should make a significant contribution to the specific heat.
(3)
The elastic bulk modulus(B) and Young’s modulus (E) were dependent on Mn concentration.
(4)
AC conductivity can be explained using the hopping model.
(5)
The behaviour of dielectric constant and dielectric loss (ε′ and ε′′) is due to the effect of polarization process on the reduction of the field inside the medium.
(6)
The phonon frequencyυph calculated from IR spectra was higher than that calculated from AC conductivity by a factor of about 102–105 Hz, which was due to the different nature of polarization.



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*Source: 907257-2012-10-18.xml*