Scientific paper
Lattice Enthalpies, Polarizabilities and Shear Moduli of Lanthanide Orthophosphates LnPO4
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: 31-07-2013
Abstract
Lattice energies ALHe of lanthanide orthophosphates, LnPO4 (Ln=Ce-Lu, excluding Pm) have been determined from the Born-Haber cycle and compared with those calculated by other methods. The Born-Haber cycle results in close values of ALHe to those obtained after an empirical equation proposed by Glasser and Jenkins. It has been found that: (i) the partial derivative of the lattice enthalpies to the molar volumes corresponds by dimension and magnitude to the shear moduli of these crystals; (ii) these moduli differ for the monazite- and xenotime-type structures of LnPO4. Molar polarizabilities have been calculated for three LnPO4 with monazite structure, Ln=Ce, Nd, Sm, and for three LnPO4 with xe-notime structure, Ln=Tb, Dy, Yb.
Keywords: Lanthanide orthophosphates, lattice enthalpies, shear moduli, molar polarizabilities
1. Introduction
Lanthanide orthophosphates, LnPO4 (Ln = Ce-Lu), are naturally occurring lanthanide compounds within the rare earth series (REPO4). LnPO4 are crystals with the highest thermal stability among the compounds in the binary system Ln2O3 - P2O5 and valuable fundamental properties. For example, the first half of the lanthanide orthophosphates have melting points in the temperature range 2173-2373 K and five groups of applications: optical devices based on the 4f-4f transitions of Ln3+, ionic conductors, geochrono-logy, coatings, and matrices for radioactive wastes.1-3
LnPO4 crystallize in two closely related structure types: monoclinic monazite (with space group P 21/n), for Ce to Eu, and tetragonal zircon (xenotime) (I41/amd), for Ho to Lu; GdPO4, TbPO4, and DyPO4 exist in both structure types.1,2 Each unit cell contains four formula units LnPO4 (Z = 4). The transformations of certain xenotime-type LnPO4 (Ln = Tb, Ho, Er, Tm) to the scheelite (141/a) or monazite structure have been studied by high-pressure high-temperature and inelastic neutron scattering.4
LnPO4 have been also synthesized on nanoscale, e.g. Ln3+: GdPO4 nanorods as bioprobes for optical and magnetic imaging5 or one-dimensional nanostructures Eu3+: LnPO4.6
The changes of the standard enthalpy (CSE) of formation of lanthanide orthophosphates from lanthanide oxides have been estimated by electronic structure calculations.7 The results, however, needed a scaling factor in order to achieve a systematic adequacy compared to the alkaline earth carbonates, silicates, and sulfates. The discrepancy has been explained with overestimated electronic total energy derived from the density functional calculations. It has been concluded in the same study that the simplest ionic model and localized charges should reproduce more accurately the mentioned energy characteristics.
A confirmation of the ionic model follows from a study of the electronic structure of LnPO4 by near edge x-ray absorption fine structure spectroscopy which reveals that all lanthanide ions are in (3+) - valence state.8 The Ln3+-state in LnPO4 has been also achieved for the particular cases of cerium and praseodymium by appropriate stepwise decomposition and reduction in the solid-state syntheses.3
The energetics of LnPO4 is important for the systematic studies of this series of compounds, including their thermodynamic stability. The thermodynamic properties of the monazite-type series of LnPO4 are still not very well studied9 and the stability limits of the same series have been correlated with models based only on geometric
criteria.
The amount of energy per mole that binds the ions in the crystal lattice of LnPO4 is among those basic characteristics of these compounds. The purpose of the present work is to determine the lattice enthalpies of LnPO4 by the Born-Haber thermochemical cycle and to relate these quantities to certain mechanical properties and molar po-larizabilities.
2. Method
2. 1. Lattice Enthalpies
The lanthanide orthophosphates LnPO4 have a defined stoichiometry and structure with assumed integral charges of the ions in the lattice. Hence, the lattice enthalpies ALHe can be determined by the Born-Haber cycle after eq. (1) below. The CSE of formation of LnPO4 from oxides, Af oxH0(LnPO4), has been included to the cycle:
- Af oxHe(LnPÜ4) - (1/2)AfHe(Ln2Ü3) -(1/2)AfH0(P2O5) + AsH0(Ln) + A sH0(P) + + 2 AdH 0(O-O) + AjH 0(Ln) + A;H 0(P) + + 4A H0(O) - A H0(LnPO4) = 0,
(1)
where the following notation has been used:10 the left-hand side subscript to each enthalpy pertains to, respectively: d-dissociation, eg-electron gain, f-formation, i-ionization, L-lattice, and s-sublimation; the right-hand side superscript (0) designates "under standard conditions": temperature T = 298.15 K, pressure P = 101325 Pa. The CSE are related to the corresponding energies of dissociation, electron gain, ionization, sublimation, and potential energy of the lattice according to the formulae, respectively:
AdH0 = -AdU0 - (5/2)RT, AegH0 =
= - [AegU0 - 5RT], A.H0 = A.U0 + n(5/2)RT, 11 0 n = 3 for Ln, or n = 5 for P, AsH0 =
= AsU0 + (5/2)RT;	'
LnPO4(s) ^ Ln3+(g) + P5+(g) + 4 O2-(g), An(g) = + 6 mol; ALH0 = -ALU0 + 6 (5/2) RT.
(2)
(3)
The equations of each step (physical or chemical change) of the cycle are presented in Table 1. The data used in the calculations of ALH0 are given in Table 2 and Table 3. The sign of the CSE must be reversed if the actual process takes place in the opposite direction. The overall sum of CSE is equal to zero for a closed path of changes starting and ending at the same state. The final step of the cycle is the formation of lanthanide orthophosphate in solid phase, LnPO4(s), from a gas of separated ions. This process is reverse in direction to that one in the definition of lattice energy as displayed in the first of eqs. (3).
2. 2. Molar Polarizations
The molar polarization Pm tained by the Debye equation:
of a substance can be ob-
P	v = —
rm	:vin> vin —•
£r +2 p
(4)
Experimental relative permittivity er = eleo, molar mass Ml10-3 kg mol-1, and density pl103 kg m-3 are needed in the determination of a molar polarization.
Pm has a dimension of m3 mol-1, i.e. the same as the molar volume Vm. Polarization of ionic solids arises almost exclusively from ionic polarizabilities which in turn originate from displacements of cations and anions in opposite directions by the applied electric field.
3. Results and Discussion
3. 1. Lattice Enthalpies
The lattice enthalpies obtained in this study are presented in Table 4. The values of the CSE of LnPO4 lattice determined by the Born-Haber cycle vary in a narrow range, less than ± 0.8 % to their mean value within the lanthanide series. The deviations quoted in the same table should be considered as minimal since four out of nine quantities included in eq. (1) for determination of ALH0 have been reported with standard deviations.
Table 1. Born - Haber cycle for lanthanide orthophosphates, LnPO4
No	Equation of the process in each step	AH0
1.	LnPO4(s) ^ (1/2)Ln2O3(s) + (1/2)P2O5(s)	- Af, 0xHe
2.	(1/2)Ln2O3(s) + (1/2)P2O5(s) ^ Ln(s) + P(s) + 2O2(g)	- (1/2)AfH0
		(Ln2O3), (P2O5>
3.	Ln(s) + P(s) + 2O2(g) ^ Ln(g) + P(s) + 2O2(g)	AsH0(Ln)
4.	Ln(g) + P(s) + 2O2(g) ^ Ln(g) + P(g) + 2O2(g)	AsH0(P)
5.	Ln(g) + P(g) + 2O2(g) ^ Ln(g) + P(g) + 4O(g)	2AdH0(O2)
6.	Ln(g) + P(g) + 4O(g) ^ Ln3+(g) + 3e- + P(g) + 4O(g)	AlH0(Ln)
7.	Ln3+(g) + 3e- + P(g) + 4O(g)^Ln3+(g) + 3e- + P5+(g) + 5e- + 4O(g)	A1H0(P)
8.	Ln3+(g) + P5+(g) + 4O(g) + 8e- ^ Ln3+(g) + P5+(g) + 4O2-(g)	4^egH0(O)
9.	Ln3+(g) + P5+(g) + 4O2-(g) ^ LnPO4(s)	- ^lH0
Table 2. Changes of standard enthalpies of phosphorus and oxygen
AHe / kJ mol1	Value	Reference
AiHe (P)	17101.9	[11]
AsHe (P-white)	12.4	[12]
AfHe (PA)	- 1493.0	[13]
AdHe (O)	498.36 ± 0.17	[11]
AeSHe (O)	715.4	[10]
ALHe obtained in this work are compared in Table 4 with an empirical equation for lattice potential energy
A, U, i.e. UP
in the following form:
1/3
Upor — A I
m
(5)
where A = 121.39 kJ mol-1 nm is an electrostatic factor, I = xh X n z2 is the ionic strength with n being the number of ions with charge z; per formula and vm is the molecular volume in nm3; the values of vm are given in Table 4.
According to eq. (3), both quantities ALH0 and ALU are related by factors proportional to RT = 2.48 kJ mol-1 at T = 298.15 K, i.e. by 37.2 kJ mol-1. The latter presents only 0.14% of the mean value of ALH0 in the series. The values derived here by the Born - Haber cycle are less than 1.5% higher than those in the last column of Table 4 as determined by the empirical eq. (5), except for GdPO4(m), TbPO4(m), and LuPO4(x) which are lower by 0.15 %, 0.5%, and 0.12%, respectively. It has been pointed out that eq. (5) used to obtain the last column in Table 4 results in estimates with certainty ± 7% compared to the known values.14
Table 3. Standard enthalpy changes of formation of lanthanide orthophosphates and sesquioxides, and of sublimation and ionization of lanthanide metals (all in kJ mol-1)
LnPO4 a	- A- He ox [2]	- AfHe(Ln2O3) [11]	AsHe(Ln) [11]	AjHe(Ln) [16]
CePO4(m)	317.2 ± 4.8	1796.2	422.6	3541.7 ± 9.9
PrPO4(m)	312.2 ± 5.0	1809.6	355.6	3646.1 ± 9.9
NdPO4(m)	312.0 ± 2.2	1807.9	327.6	3715.8 ± 38.6
SmPO4(m)	301.8 ± 2.1	1823.0 ± 3.0	206.7	3887.7 ± 38.6
EuPO4(m)	286.8 ± 1.8	1651.4 ± 12.1	175.3	4054.3 ± 10.9
GdPO4(m)	296.2 ± 1.3	1819.6 ± 12.1	397.5	3768.1 ±19.3
TbPO4(x)	286.1 ± 1.9	1865.2 ± 7.5	388.7	3808.7 ± 19.3
TbPO4(m)	283.5 ± 1.8			
DyPO4(x)	283.9 ± 1.7	1863.1 ± 7.5	290.4	3916.3 ± 37.4
HoPO4(x)	278.8 ± 2.4	1880.7 ± 4.8	300.8	3941.5 ±19.3
ErPO4(x)	275.6 ± 1.9	1897.9 ± 1.9	317.1	3952.4 ± 19.3
TmPO4(x)	268.0 ± 2.0	1888.7 ± 5.9	232.2	4062.7 ± 17.4
YbPO4(x)	269.6 ± 2.4	1814.6	152.1	4212.6 ± 2.5
LuPO4(x)	263.9 ± 1.9	1878.2	427.6	3905.5 ± 38.7
a (m) - monazite structure, (x) - xenotime structure
Table 4. Molecular (vm) and molar (Vm) volumes and lattice energies of lanthanide orthophosphates
LnPO4	v /10-30 m3 a m	Vm/10-6	ALHe/kJ mol-1	UPOT /
		m3 mol-1	this work b	kJ mol-1
CePO4(m)	75.156	45.260	26899 ±15	26493
PrPO4(m)	74.114	44.632	26938 ±15	26616
NdPO4(m)	73.026	43.977	26978 ± 41	26748
SmPO4(m)	71.384	42.988	27027 ± 42	26951
EuPO4(m)	70.539	42.480	27061±12	27058
GdPO4(m)	69.975	42.140	27091 ± 27	27131
TbPO44(x)	73.180	44.070	27135 ± 25	26729
TbPO4(m)	68.964	41.531	27133 ± 25	27263
DyPO4(x)	72.318	43.551	27141±43	26835
HoPO4(x)	71.500	43.058	27181± 24	26937
ErPO4(x)	70.715	42.586	27213 ± 22	27036
TmPO4(x)	70.051	42.186	27226 ± 23	27121
YbPO4(x)	69.353	41.765	27242 ± 5	27212
LuPO4(x)	68.773	41.416	27255 ± 41	27288
a values calculated with data from Ref. [2] ; b the minimal deviations are indicated (see text);c determined after an empirical equation proposed by Glasser and Jenkins.14
The molecular and molar volumes of LnPO4 given in Table 4 have been calculated in this study by means of the structural data obtained from recent X - ray diffraction studies.2 Unit cell volumes of LnPO4 have been also given15 or compiled for the monazite-type structure only.1
A comparison of the lattice enthalpies of LnPO4 obtained in the present work with the lattice energies for the same compounds found by the density functional calculations reveals that the latter are 8-9 % higher than the former; this comparison is based on the graph "lattice energy - lanthanide ionic radius".7
It is noteworthy that the values of the lattice enthalpies ALHe presented here are determined from experimental values of CSE included in eq.(1) and independent on the types of pair interactions, mechanisms of summation, structural properties, etc.
The graphs of lattice enthalpies vs. molar volumes of LnPO4 (with CSE of formation of LnVO4 from oxi-
Figure 1. Variation of the lattice enthalpies vs. molar volumes of LnPO4 with monazite structure (Ln = Ce to Tb in LnPO4 follow consecutively from right to left)
des) are presented in Figure 1 (monazite structure) and Figure 2 (xenotime structure). The straight lines have regression coefficients: R2 = 0.9984 and R2 = 0.9935, for LnPO4(m) and LnPO4(x) respectively. Their slopes are negative, (dALHe/dVm) = - 60.6 x 106 kJ m-3, or (dALHe/dVm) = - 60.6 x 109 Pa for the monazite structure of LnPO4, and (dALH0/dVm) = - 49.1 x 106 kJ m-3, or (dALH0/dVm) = - 49.1 x 109 Pa for the xenotime structure of LnPO4. The negative sign of the slope accounts for the trend of changes of lattice enthalpies vs. molar volumes within the light and heavy lanthanide orthophosphates. Hence, lower approximate limits have resulted for the shear moduli of LnPO4(m), G « 61 GPa, and G « 49 GPa for LnPO4(x).
The novelty of the present study is based on the ther-modynamic evaluation of lattice energies, on one side, and on the physical meaning and dimension obtained from the slope (dALH0/dVm), on the other. Only this slope preserves a correct shear- modulus dimension: [J m-3] = [Pa]. Any other linear plot, e.g. ALH0 = f (vm1/3) would result in dimension [N mol-1] and unclear physical meaning. Therefore, a possible improvement of the regression coefficient R2 from a plot other than ALH0 = f (Vm) would be incorrect.
The slope (dALH0/dVm) and shear moduli have the same dimension [Pa]. The meaning of the slope is of a lower limit of energy per volume of formula unit that, after being absorbed, would result in lattice deformation.
Measurements of the shear moduli of monazite LnPO4 have yielded the following values of G in GPa: 64 ± 1 (PrPO4), 63 ± 0.5 (NdPO4), and 79 ± 7 (EuPO4).20 These results are close to the value of G « 61 GPa for the monazites found in the present work. It should be also noted that the brittle behavior of the monazites affects the accuracy of micro-indentation experiments.20 The ther-modynamic relations between the internal energy and the moduli of a solid are known at least for crystals of simple structure and small molar volumes.10
Figure 2. Variation of the lattice enthalpies vs. molar volumes of LnPO4 with xenotime structure (Ln = Tb to Lu in LnPO4 follow consecutively from right to left)
3. 2. Molar Polarizations
The relative dielectric permittivities, molar volumes and molar polarizations of LnPO4 are presented in Table 5. Microwave dielectric properties of LnPO4 as sintered ceramics have been measured at frequency of 8-12 GHz.17 The electric field at such frequencies is considered static and uniform. The molecular volume is a portion of that of the unit cell vm = (1/4)Vuc. The values of the molar polarizations differ for the monazite- and xenotime-type structures due to the different molar volumes and different dielectric permittivities, i.e. Pm do not change consecutively within the lanthanide series. It has been suggested that the values of er below 10 for the xenotime LnPO4 are due to decreased P - O bond length and stronger covalent bonding in the PO4 tetrahedra compared with the monazi-te group.18,19
Table 5. Molar polarizations Pm of lanthanide orthophosphates
LnPO4	Er [17]	V /10-6 m m3 mol-1	P /10-3 m m3 mol-1
CePO4(m)	11.6	45.260	3.53
NdPO4(m)	10.3	43.977	3.32
SmPO4(m)	11.1	42.988	3.31
TbPO4(x)	8.5	44.070	3.15
DyPO4(x)	9.1	43.551	3.18
YbPO4(x)	8.7	41.765	3.00
4. Conclusions
The lattice enthalpies ALHe of LnPO4 increase slightly with increasing the atomic number in the lanthanide series and remain close to those determined by an empirically derived equation. The dependence between AlH0 and molar volumes Vm is linear with different negative slopes for monazite and xenotime structure types. Both correspond to lattice enthalpy per molar volume and can be considered as a lower limit of the shear moduli for the series of LnPO4(m) and LnPO4(x), respectively. The results indicate that the change from monazite- to xenoti-me structure type distinctly affects the values of the shear moduli. The difference in the molar volumes has been also reflected in the molar polarizations of certain members of both groups of lanthanide orthophosphates.
5. Acknowledgments
The author would like to thank Prof. B. M. Angelov for helpful discussions.
6. References
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Povzetek
Z uporabo Born-Haberjevega ciklusa smo določili mrežne entalpije, ALHe, lantanidnihorto fosfatov, LnPO4 (Ln=Ce-Lu, razen Pm) ter dobljene vrednosti primerjali s tistimi, izračunanimi z drugimi metodami, predvsem z empirično zvezo, ki sta jo predlagala Glasser in Jenkins. Ugotovili smo, da (i) dimenzija in velikost parcialnega odvoda mrežne entalpije na molski volumen ustreza strižnemu modulu teh kristalov; (ii) ti moduli se razlikujejo za monazitnein ksenotimske tipe struktur LnPO4. Izračunali smo molske polarizabilnosti treh LnPO4 z monazitno strulturo, Ln=Ce, Nd, Sm, in za tri LnPO4 sksenotimsko strukturo, Ln=Tb, Dy, Yb.