716 DOI: 10.17344/acsi.2014.1235
Acta Chim. Slov. 2015, 62, 716-720
Short communication
Lattice Enthalpies of Lanthanide Orthoferrites LnFeO3
Dimitar Petrov
Department of Physical Chemistry, Plovdiv University, 24, Tsar Asen Str., 4000 Plovdiv, Bulgaria
* Corresponding author: E-mail: petrov_d_n@abv.bg Tel: +35 932 261206; fax: +35 932 261403
Received: 27-11-2014
Abstract
Lattice enthalpies ALHe of lanthanide orthoferrites, LnFeO3 have been determined by the Born-Haber cycle and compared with those calculated by an empirical equation. Enthalpies of formation of LnFeO3 from two different sources have been employed: from oxides (Ln2O3, Fe2O3), for 12 LnFeO3, and from elements, for 8 LnFeO3, but the differences in ALHe are very small. The Born-Haber cycle in both routes results in close values of ALHe to those obtained by the empirical equation of Glasser and Jenkins. A correspondence in dimension and magnitude has been found between the partial derivative of the lattice enthalpies to the molar volumes and an upper limit of the shear moduli of the lanthanide ort-hoferrites.
Keywords: Lattice enthalpies; lanthanide orthoferrites; shear moduli; Born-Haber cycle
1. Introduction
Lanthanide orthoferrites, LnFeO3 are members of the perovskites group ABX3 and crystallize in an orthor-hombic structure of space group Pbnm (No.62); each unit cell comprises four molecules (z = 4) with lanthanide and iron sites equivalent in symmetry.1 Lanthanide orthoferrites, LnFeO3 are among the most studied lanthanide solids. The continuing research interest is stimulated by a number of applications based on their valuable properties: magnetic, magneto-optical, sensing, catalytic, electrical, thermochemical, etc.
The magnetic properties of hydrothermally grown single-phase LnFeO3 with all lanthanides Ln, except Ce and Pm, have been investigated recently and related to the Ln3+ ionic radii.2 Spontaneous magnetostriction and thermal expansibility have been found in TmFeO3 at low tem-peratures,3 while SmFeO3 exhibits temperature-induced magnetization reversal below the critical low temperatu-re.4 Low-temperature magnetic phase transitions in HoFe-O3 have been related to heat-capacity anomalies.5
Thick films of p-type semiconducting LnFeO3, Ln = La or lanthanides from Pr to Lu, except Pm, have been prepared by polyol synthesis and tested in respect to gas sensing.6 Lanthanide orthoferrites, LnFeO3, Ln = Sm, Nd, for the reaction of formation of LnFeO3 has been related Gd, have been synthesized as nanoparticles with size less to the Madelung energy;17 ab initio calculated energy dif-than 150 nm,7 as ceramic fibres with Ln = La, Sm, Gd, Dy, ferences for LnFeO3 have been assigned to the antiferro-Er, Yb8 or from molten NaOH flux with Ln = La, Pr, Nd.9 magnetic and ferromagnetic alignments between the iron
LnFeO3 have been found effective as pigments, with Ln = La, Gd, Tm, Yb, Lu,10 or as nanosize catalysts (Ln = La, Sm) in the photodegradation of rhodamine B under visible light.11 Pressure and gamma sensing properties of substituted orthoferrites, Ln0 7Ca0 3FeO3 (Ln = La, Gd, Dy, Y, Er) have been related to dc/ac resistivity and magnetic susceptibility.12 Recent study of polycrystalline Mn-do-ped PrFeO3 has been directed to structural, optical and dielectric properties.13
While the studies of the abovementioned properties are extensive and large in number, those on the thermody-namic properties are rare and do not cover the entire lanthanide series of orthoferrites. For example, nine ther-modynamic functions have been generated from differential scanning calorimetry and solid-state electrochemical cells of LnFeO3 (Ln = Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er).14,15 The changes of the standard enthalpies of formation (CSE) for twelve LnFeO3, excluding those of Ce and Pm, and four reactions pertaining to the stability of lanthanide perovskites have been discussed on the basis of high-temperature (977 K) calorimetry using 2PbO.B2O3 flux.16 Various authors relate different energy characteristics of lanthanide orthoferrites to structural stability and physical properties. The standard free-energy change ArGe
ions;18 it has been observed a dependence between M-O bond energies and gas sensitivity of LnMO3 (M = Cr, Fe).6 The energetics of the lanthanide orthoferrites is important for their systematic studies, including thermody-namic stability. The amount of molar energy binding the ions in the crystalline LnFeO3 is a basic quantity for this series of compounds. In this paper we follow our research interest in the lattice enthalpies of various groups of lanthanide compounds, namely: LnAlO3,19 Ln3Ga5O12,20 Ln-VO4,21 and LnPO4.22 The purpose of this work is to determine the lattice energies of LnFeO3 by the Born-Haber cycle and to relate the results to certain mechanical properties.
2. Method
The lanthanide orthoferrites, LnFeO3 exhibit a defined stoichiometry and it is assumed that the lattice is built up of ions with integral charges. Hence, the lattice enthalpies AlH® can be determined by the Born-Haber cycle expressed in Eq. (1) below. The term A corresponds to two different CSE of formation of LnFeO3 - either from oxides or from elements (Eq. (2)):
mation, and potential energy of the lattice according to the formulae, respectively:
AdH® = -AdU® - (5/2)RT, A H® =
= AegU® - 5RT, A.H® = A.U® + 3(5/2)RT,
eg 11
AsH® = AsU® + (5/2)RT;
LnFeO3(s) ^ Ln3+(g) + Fe3+(g) + + 3 O2-(g), An(g) = + 5 mol; ALH® = = -AlU® -3RT.
(3)
(4)
(5)
The necessary data for the calculation of ALH® are presented in Table 2 and Table 3. The equation for each step (physical or chemical change) of the cycle is presented in Table 1. The sign of each CSE must be reversed if the actual process takes place in the opposite direction. The sum of all CSE is equal to zero for a closed route of changes starting and ending at one and the same state. Here, the final process is the formation of lanthanide orhofer-rites in solid phase, LnFeO3(s), from ions in gas phase. This step is reverse to that one in the definition of lattice energy as displayed in Eq. (5).
A + AsH®(Ln) + AsH®(Fe) + (3/2) AdH®(O-O) +
+ AH®(Ln) + A.H®(Fe) + 3A H®(O) -
1 1 eg
- ALH®(LnFeO3) = 0,
(1)
A = [- Af oxH®(LnFeO3) - (1/2)AfH®(Ln2O3) -
(1/2)AfH®(Fe2O3)],
or A = [- Af elH®(LnFeO3)],
(2)
where the notation is as follows:23 the left-hand side subscript to each enthalpy refers to, respectively: L-lattice, f-formation, f,el-formation from elements, f,ox-formation from oxides, s-sublimation, i-ionization, d-dissociation, and eg-electron gain; the superscript (®) designates "standard conditions": temperature T = 298.15 K, pressure P = 101325 Pa. The CSE are related to the corresponding energies of dissociation, electron gain, ionization, subli-
3. Results and Discussion
The lattice enthalpies obtained in this work are presented in Table 4. The values of ALH® of lanthanide orho-ferrite lattice determined by the Born-Haber thermoche-mical cycle vary slightly, 2.1 or 2.4 % within the lanthanide series to the mean value, respectively for those determined with CSE of formation of LnFeO3 from elements or from oxides.
The Born - Haber cycle displayed in Table 1 begins with either steps 1a and 1b (reverse process of the formation of LnFeO3 from oxides) or with step 1 (reverse process of the formation of LnFeO3 from elements) and then to proceed via steps 2 to 8. The reported values of Af oxH® 16 have been used in the first route and of Af elH® 15 in the second one.
Table 1. Born - Haber cycle for lanthanide orthoferrites, LnFeO3
No	Equation of the process in each step	AH®
(1a.	LnFeO3(s) ^ (1/2)Ln2O3(s) + (1/2)Fe2O3(s)	- Af, oxH®
1b.	(1/2)Ln2O3(s) + (1/2)Fe2O3(s) ^ Ln(s) + Fe(s) + (3/2)O2(g)	- (1/2)AfH®
		(Ln2O3), (Fe2O3))
1.	LnFeO3(s) ^ Ln(s) + Fe(s) + (3/2)O2(g)	- Af,elH®
2.	Ln(s) + Fe(s) + (3/2)O2(g) ^ Ln(g) -2 Fe(s) + (3/2)O2(g)	AsH®(Ln)
3.	Ln(g) + Fe(s) + (3/2)O2(g) ^ Ln(g) + Fe(g) + (3/2)O2(g)	AsH®(Fe)
4.	Ln(g) + Fe(g) + (3/2)O2(g) ^ Ln(g) + Fe(g) + 3O(g)	(3/2)AdH®(O2)
5.	Ln(g) + Fe(g) + 3O(g) ^ Ln3+(g) + 3e- + Fe(g) + 3O(g)	AiH®(Ln)
6.	Ln3+(g)+3e-+Fe(g)+3O(g)^Ln3+(g)+3e- +Fe3+(g)+3e-+3O(g)	AiH®(Fe)
7.	Ln3+(g)+Fe3+(g)+3O(g)+6e- ^ Ln3+(g)+Fe3+(g)+3O2-(g)	3AegH®(O)
8.	Ln3+(g) + Fe3+(g) + 3O2-(g) ^ LnFeO3(s)	-AlH®
The range of ALHe variation is small since the formation of LnFeO3 is determined mainly by the change of Ln3+ ionic radii appropriate to the perovskite structure, i.e. by the ionic Ln - O bonds. The accuracies of determination of ALHe have been evaluated with the accuracies of the quantities as included in Eq. (1) where available. Because of the lack of standard deviations for certain quantities in Table 2 and Table 3, the r.m.s. deviations of ALHe determined in the present work should be considered minimal ones and not lower than 0.2% of the respective value. The r.m.s. deviations of ALHe obtained from AfelHe are smaller because of the smaller number of terms included in the summation, Eqs. (1) and (2).
Table 2. Standard enthalpy changes of iron and oxygen
AHe /kJ mol1	Value	Ref.
AiHe(Fe)	5300.4 ± 0.1	[24]
AfHe(Fe2O3)	- 824.2	[24]
AsHe(Fe)	398.6 ± 0.1	[25]
AdHe(O2)	498.36 ± 0.17	[24]
AegHe(O)	715.4	[23]
ALHe determined here are compared in Table 4 with an empirical equation for lattice potential energy ALU, i.e. UPOT 26. The equation is as follows:
X
U.
-4ÜJ
(6)
where A = 121.39 kJ mol-1 nm (an electrostatic factor), I = Ч Z n Zj2 is the ionic strength with n being the number of ions with charge z; per formula, I = 15 for LnFeO3, and vm is the molecular volume in nm3; the corresponding values of vm are given in Table 4.
According to Eq. (5), the quantities ALHe and ALU are related by factor proportional to RT = 2.48 kJ mol-1 at T=298.15 K, or by 7.44 kJ mol-1 with the inclusion of the zero-point energy.27 This value presents about 0.05% of the value of ALHe.
The values in the last column of Table 4 determined by Eq. (6) are about 4% lower than those yielded by the Born - Haber cycle. It has been commented that the empirical formula (6) should yield estimates within ± 7% compared to the known values.26
It should be noted that the ALHe values in the present work are obtained from experimental values of CSE included in Eq. (1) and that they do not depend on structural features or mechanisms of summation of pair interactions.
The plot of lattice enthalpies vs. molar volumes of LnFeO3 (with CSE of formation of LnFeO3 from oxides) is presented in Fig. 1. The straight line has a regression coefficient R2 = 0.981 and a negative slope (dALHe/dVm) = - 127.0 X 106 kJ m-3, or (dALHe/dVm) = - 127.0 x 109 Pa. The negative sign of the slope accounts for the trend of changes of lattice enthalpies within the series of 12 lanthanide orthoferrites.
Hence, an upper limit for the shear modulus of Ln-FeO3, G « 150 GPa, has appeared in this case. The molar volumes Vm of LnFeO3 have been determined here from the reported unit cell volumes.2 Taking the molar volume of PmFeO3 as a mean value between those of NdFeO3 and SmFeO3, 35.365 x 10-6 m3 mol-1, the missing lattice enthalpy of PmFeO3 has been found, ALHe = 14031 kJ mol-1.
The variation of the lattice enthalpies vs. molar volumes of LnFeO3 with CSE of formation of LnFeO3 from elements results in similar straight line with a regression coefficient R2 = 0.9602 and negative slope, -128.1 X 109 Pa.
It is important to note that the novelty of the present study is equally based on the lattice enthalpies and on the
Table 3. Standard enthalpy changes of formation of lanthanide orthoferrites and sesquioxides, and of sublimation and ionization of lanthanide metals (all in kJ mol-1)
LnFeO3	- A, X	- A, He ox	- A,He(Ln2O3)	AsHe(Ln)	AjHe(Ln)
	[15]	[16] a	[24]	[24]	[28]
PrFeO3		48.53	1809.6	355.6	3646.1 ± 9.9
NdFeO3	1357.4	44.35	1807.9	327.6	3715.8 ± 38.6
SmFeO3	1355.2	44.35	1823.0 ± 3.0	206.7	3887.7 ± 38.6
EuFeO3	1285.6	44.35	1651.4 ± 12.1	175.3	4054.3 ± 10.9
GdFeO3	1360.5	44.35	1819.6 ± 12.1	397.5	3768.1 ± 19.3
TbFeO3	1372.4	40.17	1865.2 ± 7.5	388.7	3808.7 ± 19.3
DyFeO3	1369.4	35.98	1863.1 ± 7.5	290.4	3916.3 ± 37.4
HoFeO3	1364.2	35.98	1880.7 ± 4.8	300.8	3941.5 ± 19.3
ErFeO3	1400.5	35.98	1897.9 ± 1.9	317.1	3952.4 ± 19.3
TmFeO3		27.61	1888.7 ± 5.9	232.2	4062.7 ± 17.4
YbFeO3		23.43	1814.6	152.1	4212.6 ± 2.5
LuFeO3		19.25	1878.2	427.6	3905.5 ± 38.7
a all with r.m.s. devs.= ± 12.55 kJ mol ';
Table 4. Molecular (vm) and molar (Vm) volumes and lattice enthalpies of lanthanide orthoferrites
LnFeO3	vm/10-30	Vm/10-6	ALHe/	ALHe/	AlU /
	m3	m3 mol 1	kJ mol-1	kJ mol-1	kJ mol-
			this worka	this work b	c
PrFeO3	59.35	35.74	13960 ± 24		14505
NdFeO3	59.175	35.64	13997 ± 52	13994 ± 40	14519
SmFeO3	58.275	35.09	14055 ± 54	14042 ± 40	14593
EuFeO3	58.00	34.93	14104 ± 31	14108 ± 12	14616
GdFeO3	57.775	34.79	14125 ± 39	14119 ± 20	14635
TbFeO33	56.825	34.22	14175 ± 37	14163 ± 20	14716
DyFeO33	56.75	34.18	14179 ± 55	14169 ± 38	14723
HoFeO33	56.30	33.90	14223 ± 35	14199 ± 20	14762
ErFeO33	56.025	33.74	14259 ± 34	14263 ± 20	14786
TmFeO3	55.475	33.41	14272 ± 34		14835
YbFeO33	55.275	33.29	14300 ±16		14853
LuFeO33	54.90	33.06	14296 ± 52		14886
a obtained with Af oxHe, b obtained with Af elHe, c determined after an empirical equation of Glasser and Jenkins 26
physical meaning, dimension and magnitude obtained from the slope (dALHe/dVm); this slope retains a correct shear- modulus dimension: [J m-3] = [Pa].
Figure 1. Variation of the lattice enthalpies vs. molar volumes of LnFeO3 with enthalpies of formation of LnFeO3 from oxides
The slope (dALHe/dVm) and shear moduli have the same dimension, [Pa]; it is obvious that a displacement of ions can be related to shear modulus. The meaning of the slope is of a critical amount of energy that, after being absorbed, will result in lattice destruction. Recent studies of the mechanical moduli of NdFeO3 by equation of state V(P) in the Birch - Mournaghan form have revealed a value of E = 244 ± 4 GPa for the elastic (Young's) modulus 29 and K = 195.1 GPa 30 for the bulk modulus. From these experimental results we have calculated the value of the shear moduli G using the relationships between E, K, and G;23 the obtained value is G = 94.5 GPa, which is lower than both slopes found in this work. Other reported values of mechanical moduli of LnFeO are close to those for
NdFeO3: for GdFeO3, mean calculated bulk modulus K= 182 GPa,31 and experimental K = 204.2 GPa.30 The ther-modynamic relations between the internal energy and the moduli of a solid have explicit forms only for crystals of simple structure and small molar volumes.23
4. Conclusions
The lattice enthalpies ALHe of LnFeO3 increase linearly with decreasing the molar volumes Vm within the lanthanide series and remain close to those determined after an empirical equation. The negative slope of this dependence corresponds to lattice enthalpy per molar volume and can be considered as an upper limit of the shear moduli for the series of LnFeO3. Similar relationships have been observed in our previous studies on lanthanide complex oxides.19-22
5. Acknowledgments
The author would like to thank Prof. B. M. Angelov for useful comments.
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Povzetek
Z uporabo Born-Haberjevega ciklusa smo določili mrežne entalpije, ALHe, lantanoidnih ortoferitov, LnFeO3, ter dobljene vrednosti primerjali s tistimi, dobljenimi z empiričnimi enačbami. Entalapije smo za 12 spojin določili s pomočjo podatkov za okside (Ln2O3, Fe2O3), za 8 spojin pa s pomočjo podatkov za elemente, vendar je razlika v ALHe zelo majhna. Vrednosti, dobljene s pomočjo Born-Haberjevega ciklusa se dobro ujemajo s tistimi, izračunanimi z empiričnimi enačbami Glasserja in Jenkinsa. Ugotovili smo povezavo med dimenzijami ter parcialnimi odvodi mrežnih entalpij na parcialni molski volumen ter zgornjo limito strižnega modula proučevanih spojin.