Study of cation distributions in spinel ferrites M_xMn_1-xFe₂O₄ (M=Zn, Mg, Al) Open Access

Spinel ferrites have received much attention because of their promising applications in various technologies, such as magnetic refrigerators, microwave devices, color imaging, high-density recording devices, and magnetic fluids, etc.^1–10 

Spinel ferrites have the general formula (A)[B]₂O₄, with each spinel unit cell containing eight formula units. The larger oxygen anions form a close-packed face-centered-cubic structure with the smaller metal cations occupying interstitial sites: the tetrahedral (8a) or (A) sites, and the octahedral (16d) or [B] sites. According to the traditional theory,³ ZnFe₂O₄ has a normal spinel structure and has no magnetic moment. All the Zn²⁺ ions occupy the tetrahedral sites, while the Fe³⁺ ions occupy the octahedral sites.

The effect of Zn-doping on the crystal structure and physical properties of spinel ferrites has been reported in many studies, but the Zn ion distributions have been disputed.^4–12 Jagadeesha Angadi.V et al.⁴ prepared a series of Mn_1-xZn_xFe₂O₄ (x=0, 0.1, 0.3, 0.5, 0.7, 0.9, 1.0) nanocrystalline materials, and claimed that Fe³⁺ ions migrate from the [B] sites to the (A) sites and that the Mn²⁺ concentration decreases in both the (A) and [B] sites as the zinc content increases. P Mathur et al.⁵ prepared a series of Zn_xMn_1-xFe₂O₄ ferrites, with x=0.1, 0.3, 0.5, 0.7 and 0.9, and concluded that Zn²⁺ ions tend to go to the (A) sites at low concentrations, but also go to the [B] sites when x≥0.3. For Mg doping, Mahavir Singh⁶ prepared Mg_xMn_1-xFe₂O₄ (0.0≤ x ≤0.8) ferrites, and thought that Mg²⁺ ions tend to go to the (A) sites, but also go to the [B] sites when x≥0.4. Antic et al.⁷ synthesized MgFe₂O₄ nanoparticles. They investigated the cation distribution using Rietveld refinement, and concluded that according to the expression $( Mg 1 − δ Fe δ ) [ Mg δ Fe 2 − δ ] O 4$ with $δ = 0.69$⁠. Finally, Khot et al.⁸ prepared nanocrystalline samples of Mn_xMg_1-xFe₂O₄ (x=0, 0.2, 0.4, 0.6, 0.8, 1.0). They thought that Mg²⁺ ions entered only the [B] sites when the Mg content was (1-x)≤0.6. To study the effects of Al doping, Rabia Pandit et al.¹⁰ prepared polycrystalline samples of CoAl_xFe_2-xO₄ with x=0.0, 0.2, 0.4, 0.6 and 0.8. They concluded, using an analysis of X-ray diffraction data, that the Al cation distribution ratio at the (A)/[B] sites was approximately 3/7.

In order to resolve these disparities, in this work, we prepared M_xMn_1-xFe₂O₄ (M = Zn, Mg, Al) spinel ferrite samples, and measured the magnetic moments, $μ exp$⁠, of the samples at 10 K. We obtained the cation distributions of the samples by fitting the curves of $μ exp$ versus x using a quantum-mechanical potential barrier model earlier proposed by our group.^11–22 In the fitting process we assumed that the magnetic moment direction of the Mn³⁺ ions was antiparallel to those of the Mn²⁺ and Fe cations, in accord with the O2p itinerant electron model proposed by our group.^23–25 

Ferrite powder samples, M_xMn_1-xFe₂O₄ (M = Zn, Mg, Al), were prepared using the chemical co-precipitation method.^17,18 The initial AR Grade chemicals were Zn(NO₃)₂·6H₂O, Mg(NO₃)₂·6H₂O, Al(NO₃)₂·9H₂O, Mn(NO₃)₂, Fe(NO₃)₃·9H₂O and NaOH. The nitrates in the stoichiometric ratios appropriate for each value of x were dissolved in deionized water with stirring at a constant temperature of 343 K. Precipitates were formed with the slow addition of a 1.1 molar sodium hydroxide solution to 1 molar solutions of the samples, followed by stirring over a period of 40 min, with the final pH value of the solutions being adjusted to about 11-12. The precipitates were washed several times with deionized water in order to remove residual NO₃^- and Na⁺ ions from the solutions, and then filtered. The precipitates were dried in air for 24 h at 363 K, and then for an additional 24 h at 373 K in a drying cabinet. After being ground for 30 min in an agate mortar, the resulting products were calcined in a tube furnace with argon-flow at 1373 K for 2 h, and then cooled to room temperature in the furnace. The final powder samples were obtained after being ground again in an agate mortar.

The crystal structures of the powder samples were analyzed using X-ray diffraction (XRD) patterns measured on an X’Pert Pro X-ray diffractometer (XRD) with Cu Kα radiation $( λ = 1.5406 Å )$⁠, a working voltage of 40 kV and a current of 40 mA. The data were collected in the $2 θ$ range 15–120° with a step size of 0.0167°. Magnetic hysteresis loops were measured using a Physical Properties Measurement System (Quantum Design Corporation, PPMS, USA) at 10 K and 300 K.

Fig.1 (a), (b) and (c) show XRD patterns of the Zn, Mg and Al doped samples, respectively. All samples have a single-phase cubic spinel structure with space group $F d 3 ¯ m$⁠. The X-ray diffraction data were fitted using the X’Pert HighScore Plus software with the Rietveld powder diffraction profile fitting technique.²⁶ In the fitting process, we obtained the crystal lattice constant, a, the distances, d_AO and d_BO, from the O anion to the cations at the (A) and [B] sites, and the distances, d_AB, from the cations at the (A) sites to those at the [B] sites, as listed in Table I. The ideal values of d_AO, d_BO and d_AB are$3 a / 8$⁠, $a / 4$ and $11 a / 8$⁠, respectively, for the spinel structure. However, it can be seen that the observed average values of d_AO and d_BO in Table I are respectively 1.09 and 0.96 times the ideal values for all three series of samples, while the value of d_AB is equal to its ideal value. That is,

Fig.2 shows the dependence of the lattice constants,$a$⁠, of the three series of samples on the doping level x. It can be seen that the lattice parameter of the Mg and Al doped samples decreased more rapidly with increasing x than did that of the Zn doped samples. The variation in the lattice constant is related to the cation radii and cohesive energies of the sample.

The volume averaged crystallite sizes of all samples, estimated using the X’Pert HighScore Plus software, were larger than or close to 100 nm, so that surface effects are expected to be very weak in all the samples.

Fig.3, Fig.4 and Fig.5 show the magnetic hysteresis loops of the Zn, Mg and Al doped samples measured at 10 K and 300 K. The specific saturation magnetization $( σ S )$ measured at 10 K and 300 K and the magnetic moments $( μ exp )$ per formula of the samples at 10 K are given in Table II. It can be seen that the specific saturation magnetization values of all samples at 10 K (⁠$σ S − 10 K$⁠) were higher than those at 300 K (⁠$σ S − 300 K$⁠) for each doping level x. It can be seen from Fig.4 (b) and Fig.5 (b) that all Mg and Al doped samples have ferromagnetic hysteresis loops at 300 K, indicating the Curie temperature of these samples to be higher than 300 K. It can be seen from Fig.3 (b) that the saturation magnetization of the Zn doped samples with x≥0.6 decreases rapidly with increasing x and that the hysteresis loops transit from ferromagnetic to paramagnetic one. Fig.6 (a) shows the temperature dependence of the specific magnetization, $σ$⁠, for Zn doped samples with x≥0.6 at an applied field of 0.05 T from 300 K down to 10 K. Fig.6 (b) shows the temperature dependence of the $d σ / d T$ with the values of the Curie temperature, T_C. We therefore obtained that T_C decreases with increasing x, being 280.0, 187.5, 38.2, and 21.7 K for x=0.6, 0.7, 0.8 and 0.9. The insert of Fig.6 (b) shows that T_C of ZnFe₂O₄ sample to be 10.8 K. Here, T_C is defined as the temperature where $d σ / d T$ reaches its minimum value.

It is easy to see from Table II that the magnetic moments $( μ exp )$⁠, for the Zn doped samples, first increased, then decreased quickly with increasing doping, reaching a maximum when x=0.4. However, for the Mg doped samples, $μ exp$ showed a slight increase for x=0.1, but then gradually decreased with increasing x. For the Al doped samples, $μ exp$ decreased monotonically with increasing x. The curves of $μ exp$ vs x for the three series of samples are shown in Fig.7, where the points are the observed sample magnetic moments, and the curves labeled $μ C$⁠, $μ AT$ and $μ BT$ are the fitted total magnetic moments of the samples, and the moments for the (A) and [B] sublattices separately. Details of the fitting methods are described in the following section.

As in our previous work,^17–25 the cation distributions for all samples were estimated by fitting the dependence on the doping level x of the magnetic moments measured at 10 K, using the quantum mechanical potential barrier model proposed by our group.^13–16 In the fitting process, the following factors were taken into account:

Factor 1: We suppose that there is a square potential barrier between each adjacent cation-anion pair. The height and the width of the potential barrier are related to the cation ionization energy and the distance between the cation and anion. The content ratio (R) of the different cations is therefore related to the probability of the last ionized electrons transmitting through the potential barriers, and takes the form

Here nanometers (nm) and electron-volts (eV) are used as the units of length and energy; P_C (P_D) represents the probability of the last ionized electron of the C (D) cations jumping to the adjacent anions through a potential barrier of height V_C (V_D) and width r_C (r_D). V_C and V_D are the ionization energies of the last ionized electron of the cations C and D, and r_C and r_D are the distances from the cations C and D to the anions. The parameter c is a barrier shape correcting constant related to the different extents to which the shapes of the two potential barriers deviate from a square barrier. It is obvious that parameter c = 1 when V_C = V_D and r_C = r_D. In this work, as an approximation, we let c = 1.

Factor 2: We considered the Pauli repulsion energy of the electron cloud between adjacent cations and anions. This can be taken into account using the effective ionic radius:²⁷ smaller ions should be located at the sites with smaller available space in the lattice. We note that the available space at the (A) sites is smaller than that at the [B] sites in the spinel ferrites.

Factor 3: Due to the fact that cations at the (A) and the [B] sites have four and six adjacent oxygen ions, in the thermal treatment process for the samples, a tendency toward electrical charge density balance forces a portion of the divalent cations (with large effective ionic radius) to enter the (A) sites (with smaller available space) from the [B] sites, jumping an equivalent potential barrier, V_BA. V_BA is related to the ionization energy, ionic radius and the thermal treatment temperature. For the Fe and Mn cations with magnetic moments, we also assumed that V_BA changes following the expression²⁵ 

where r(Mn²⁺) and r(Fe²⁺) are the effective radii²⁷ of the Mn²⁺ and Fe²⁺ cations, and V(Mn³⁺) and V(Fe³⁺) are the third ionization energies of the Mn and Fe cations, as shown in Table III. For the Zn²⁺, Mg²⁺ and Al²⁺ cations, which do not have magnetic moments, V_BA(Zn²⁺), V_BA(Mg²⁺) and V_BA(Al²⁺) were determined in the fitting processes.

Factor 4: Taking into account the fact that due to the ionicity^15,28 of the ions in oxides, some of the oxygen anions are found to be monovalent,^29–31 the total valence and the total number (N₃) of trivalent cations per formula are both less than the traditional values of 8 and 2, respectively, in the (A)[B]₂O₄ spinel ferrites.

Factor 5: According to the O2p itinerant electron model,²⁴ the magnetic moment direction of Mn³⁺ is antiparallel to that of Mn²⁺, Fe³⁺ and Fe²⁺ in the same sublattice whether at the (A) sites or the [B] sites. In following calculations, we therefore set the moment of the Mn³⁺ cations to be - $4 μ B$⁠, as shown in Table III.

Factor 6: Doping with Zn, Mg and Al cations, which have zero magnetic moments, results in the angles between the cation (Fe, Mn) magnetic moments increasing from zero as the doping level of the Zn, Mg and Al cations increases.^11,12

The chemical formulas for the ferrite samples Zn_xMn_1-xFe₂O₄, Mg_xMn_1-xFe₂O₄ and Al_xMn_1-xFe₂O₄, are rewritten here as M_x1Mn_x2Fe_3−x1−x2O₄ (M=Zn, Mg, Al). In this way, the cation distributions can be described by the formula^24,25

It can be seen from Eq.(4) that

where N₃ is the number of trivalent cations per formula,

and f_M=f_Zn, f_Mg,f_Al,, f_Mn and f_Fe represent the ionicities of the Zn, Mg, Al, Mn and Fe ions.¹⁵ From Eq.(4), we have

where R_A1, R_A2, R_A4, R_A5 and R_A6 represent, respectively, the probability ratios of the Zn³⁺(Mg³⁺, Al³⁺), Mn³⁺, Zn²⁺(Mg²⁺, Al²⁺), Mn²⁺ and Fe²⁺ ions taken with respect to the Fe³⁺ ions at the (A) sites, while R_B1 and R_B2 represent the probability ratios of the Zn³⁺(Mg³⁺, Al³⁺) and Mn³⁺ ions with respect to the Fe³⁺ ions at the [B] sites. From Eq.(5) and (8), we have

From Eq.(6) and (9), we can obtain that

According to the above mentioned quantum mechanical potential barrier model for estimating the cation distributions in spinel ferrites, the content ratios R_A1, R_A2, R_A4, R_A5 and R_A6 at the (A) sites, and R_B1 and R_B2 at the [B] sites, can be derived in the forms:

where M=Zn, Mg or Al, and V(M²⁺), V(M³⁺), V(Mn²⁺), V(Mn³⁺), V(Fe²⁺) and V(Fe³⁺) are the second and third ionization energies of Zn, Mg, Al, Mn, and Fe, respectively. See Table III. Likewise, V_BA(M²⁺), V_BA(Mn²⁺), and V_BA(Fe²⁺) are the heights of the equivalent potential barriers, all of width d_AB, which must be jumped by the M²⁺, Mn²⁺, and Fe²⁺ ions as they move from the [B] sites to the (A) sites during the thermal treatment. The values of d_AO, d_BO, and d_AB are shown in Table I.

According to the O2p itinerant electron model, the magnetic moments of the Mn³⁺ cations are antiparallel to those of Mn²⁺, Fe³⁺ and Fe²⁺ in the same sublattice of the spinel ferrites.²⁴ Consequently, we write the magnetic moments of the M²⁺, M³⁺, Mn²⁺, Mn³⁺, Fe²⁺ and Fe³⁺ ions as m₂, m₃, 5, -4, 4 and 5 $μ B$⁠, respectively, where for M = Zn, m₂ = 0 $μ B$⁠, m₃ = 1 $μ B$⁠; for M = Mg, m₂ = 0 $μ B$⁠, m₃ = 0 $μ B$⁠, and for M = Al, m₂ = 0 $μ B$⁠, m₃ = 0 $μ B$⁠, as shown in Table III. Therefore, we can calculate the average magnetic moment per formula in the samples from Eq.(4),

Here, $μ C$ is the calculated magnetic moments of the samples, $μ AT$ and $μ BT$ are the magnetic moments of the (A) and [B] sublattices, and $μ B 1$⁠, $μ B 2$⁠, and $μ B 3$ are the magnetic moments of the Zn (Mg, Al), Mn, and Fe ions at the [B] sites. Due to the fact that Zn, Mg, Al doping results in a cant angle between cation magnetic moments, the factors [1-c₁(y₁+y₄)^1.2] and [1-c₁(x₁−y₁−y₄)^1.2] were used to fit the average angles, $φ A$ and $φ B$⁠, between cation magnetic moments at (A) and [B] sites.¹⁶ Therefore the cant angles are

There are altogether 19 independent equations, including Eq.(3), Eq.(5)–(9), and Eq.(12)–(19), where Eq.(8) contains 5 equations and Eq.(9) contains 2 equations. There are altogether 22 parameters: y₁ − y₆, z₁ − z₃, N₃, R_A1, R_A2, R_A4, R_A5, and R_A6, R_B1 and R_B2, V_BA(M²⁺), V_BA(Mn²⁺) and V_BA(Fe²⁺), and $μ C$ and c₁, for each value of the doping level x. Therefore, for a given sample, we need to obtain the values for at least 3 independent parameters in order to fit the observed magnetic moment $μ exp$⁠.

Applying the above parameters and equations, we fitted the dependence of the magnetic moments at 10 K on the doping level x for the three series of samples, as shown in Fig.7 (a), (b) and (c), where the points represent the observed sample magnetic moments, $μ exp$⁠, and the curves, labeled $μ C$⁠, $μ AT$ and $μ BT$⁠, are respectively, the calculated total magnetic moments per formula sample, and the moments for the (A) and [B] sublattices. For the Zn doped samples, it can be seen that $μ AT$ decreases more rapidly than does $μ BT$ when x<0.4, while $μ BT$ decreases far more rapidly than does $μ AT$ when x>0.4. This results in $μ C$ having a maximum value at x = 0.4. For the Mg and Al doped samples, we made the approximation that $μ C$ decreases linearly with increasing x, and it can be seen that the fitted curves are very close to the experimental results. This results in $μ BT$ decreasing more rapidly than $μ AT$ for all doping levels. The data for the fitted results are shown in Tables IV and, V, and VI, and the dependence of the parameters V_BA(M²⁺) and V_BA(Mn²⁺) on the doping level x is shown in Fig.8 (a) and (b), where it may be seen that V_BA(Mn²⁺) decreases linearly with increasing x. In addition, the cant angle parameters are c₁=2.0, 0.5 and 0.5 for the Zn, Mg and Al doped samples.

The cation distributions of the Zn, Mg and Al doped samples are shown in Fig.9, 10, and 11, respectively. The cant angles $φ A$ and $φ B$_, calculated using Eq.(20) and (21), as functions of x, are shown Fig.12 (a), (b) and (c). The cation distributions and the cant angles are discussed in the next section.

The third ionization energies, V(M³⁺), of Al, Fe, Mn, Zn and Mg are 28.54, 30.65, 33.67, 39.72 and 80.14 eV (see Table III). We can see from Table IV, V and VI that their trivalent cation contents decrease with increasing V(M³⁺). Consequently, there are only Mg²⁺ and no Mg³⁺ cations whatever at the (A) sites or the [B] sites in the Mg doped samples (see Table V), and there are relatively few Zn³⁺ cations (see Table IV) at either the (A) sites or the [B] sites in the Zn doped samples. Such as, for our calculated results for ZnFe₂O₄ in Table IV, there are only 1.0% and 3.4% of Zn³⁺ cations in the (A) and [B] sites, respectively. Such small Zn³⁺ content can be neglected. These calculated results are very close to the conventional view, in which there are no Mg³⁺ and Zn³⁺ cations in spinel ferrites,³ indicating that the cation distributions in Table IV, V and VI are reasonable, because it is difficult for oxygen ions to obtain electrons from cations with high ionization energies.

For all three series of samples, the portion of Fe (Fe²⁺ and Fe³⁺) cations occupying the [B] sites lies between 62% and 74% of the total Fe content, and the content of Mn²⁺ cations occupying the [B] sites lies between 55% and 65% of the total Mn content. This results in the magnetic moment direction of the samples being the same as that of the [B] sublattice.

For Mg_xMn_1-xFe₂O₄ (0.0≤ x ≤1.0) and Al_xMn_1-xFe₂O₄ (0.0≤ x ≤0.5), it can be seen from Fig.10 (d) and Fig.11 (d) that the contents of Mg²⁺ and Al (including Al²⁺ and Al³⁺) cation increases approximately linearly at both the (A) and [B] sites, with the content at the [B] sites being greater than at the (A) sites for every doping level. This may be the underlying reason why the magnetic moments of the [B] sublattices decrease more rapidly than those of the (A) sublattices, while the total magnetic moments decrease approximate linearly with increasing x. It is noticed that all the magnetic moments of Mg²⁺, Al²⁺ and Al³⁺ cations are zero.

For the cation distributions of the Zn doped samples, shown in Fig.9, it maybe seen that when x<0.4, the Zn²⁺ content at the (A) sites increases rapidly and the Fe²⁺, Fe³⁺ and Mn²⁺ contents at the (A) sites decrease gradually with increasing x, resulting in the magnetic moment of the (A) sublattice decreasing rapidly and the total magnetic moment of the samples increasing. When x>0.4, the Zn²⁺ content at the [B] sites increases rapidly and the Mn²⁺ content at the [B] sites decreases gradually with increasing x, resulting in the magnetic moment of the [B] sublattice decreasing rapidly and the total magnetic moment of the samples also decreasing rapidly. The cation distributions at both the (A) and [B] sites resulted in that the samples with x<0.7 have higher magnetization at 10 K. For the reason why Zn cations have the distributions in Fig.9, it may result from the lattice energy: (a) It can be seen from Eq.(1) that the distance between an O anion and an adjacent cation at the (A) sites, d_AO, is greater than its ideal value, $3 a / 8$⁠, while the distance between an O anion and an adjacent cation at the [B] sites, d_BO, is smaller than its ideal value, $a / 4$⁠. Therefore, the actual repulsion energy between A-O ions is higher than that when d_AO has the ideal value, and that actual repulsion energy between B-O ions is lower than that when d_BO has the ideal value. This repulsion energy may include Pauli exclusion energy of electron cloud between A-O (B-O) ions and the magnetic repulsion energy between A-A (B-B) cations with ferromagnetic order. (b) When x<0.4, the Zn²⁺ content at the (A) sites increases rapidly, maybe due to that the doping Zn cations having no magnetic moment can decrease the magnetic repulsion energy in the (A) sublattice. (c) It can be seen from Fig.9 that the contents of Fe³⁺ and Mn³⁺ cations at the (A) sites decrease with Zn²⁺ content increasing. The radius of Zn²⁺ (0.074 nm) is larger than the radii of Fe³⁺ (0.0645 nm) and Mn³⁺ (0.0645 nm).²⁷ Zn²⁺ cations substituting for Fe³⁺ and Mn³⁺ cations result in Pauli exclusion energy in (A) sublattice to increase. When x>0.4, the (A) sublattice can therefore be no longer to have more Zn²⁺ cations. This results in Zn²⁺ content at the [B] sites increasing rapidly.

For the same doping level, it can be seen from Fig.12 and Tables IV, V and VI that for Zn doping the average cant angles between the cation magnetic moments are larger than for Mg and Al doping. This may be due to the effective radius of Zn²⁺ (0.074 nm) being larger than the radii of Mg²⁺ (0.072 nm) or Al²⁺ (0.060 nm), as shown in Table III. This may be the underlying reason why the magnetic moment of ZnFe₂O₄ is close to zero, while the magnetic moment of MgFe₂O₄ is $1.68 μ B / formula$⁠.

As mentioned in Section C, a tendency toward electrical charge density balance forces a portion of the divalent cations to enter the (A) sites from the [B] sites, jumping an equivalent potential barrier, V_BA. It can be seen from Tables IV, V and VI and Fig.8 that the values of V_BA(Zn²⁺) are between 0.10 and 0.29 eV, while V_BA(Fe²⁺), V_BA(Mn²⁺), V_BA(Mg²⁺) and V_BA(Al²⁺) are between 0.31 and 1.40 eV for the samples studied here. These results explained why the Zn²⁺ cation content at the (A) sites in Zn_xMn_1-xFe₂O₄ (0.0≤ x ≤0.4) increases more rapidly with increasing x than does the content of Mg²⁺ and Al²⁺ at the (A) sites in Mg and Al doped samples. The reason why the content of Zn²⁺ cations at the [B] sites in Zn_xMn_1-xFe₂O₄ (0.4≤ x ≤1.0) increases rapidly with increasing x, may lie with the fact that the Pauli repulsion energy of the electron cloud does not allow excess Zn²⁺ cations (with large effective ionic radius) to enter the (A) sites (with small available space). Therefore, the values of V_BA obtained in the fitting processes are reasonable.

Using the ion contents at the (A) and [B] sites as given in Table IV, V and VI, the XRD patterns of the Zn_xMn_1-xFe₂O₄ (0.0≤ x ≤1.0), Mg_xMn_1-xFe₂O₄ (0.0≤ x ≤1.0) and Al_xMn_1-xFe₂O₄ (0.0≤ x ≤0.5) samples were fitted with the FullProf-Suite software.³² Because that there is no parameters for Zn³⁺ cation in the FullProf-Suite software, we substitute Zn²⁺ for Zn³⁺ in the fitting process. All the ion contents were kept constant in the fitting process. The Rietveld fitting parameters as well as the profile factor, R_p, the weighted profile factor, R_wp and the goodness of fit indicator, s, for the samples are listed in Table VII. It can be seen that the various fitting parameters are all acceptable, which indicates that the cation distributions provided by our method are reasonable. The fitted patterns are shown in Fig.13 (a), (b) and (c). The crystal lattice constant, a, and the cell volume, V, obtained in the fitting process, are listed in Table VII. It can be seen that the values of a for all samples are very close to those in Table I obtained by the X’Pert HighScore Plus software.