Structural Phase Transitions in the Potential Hydrogen Storage

May 7, 2008 - Department of Physics & Astronomy and High Pressure Science and Engineering Center (HiPSEC), University of Nevada Las Vegas, Las ...
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J. Phys. Chem. C 2008, 112, 8452–8457

Structural Phase Transitions in the Potential Hydrogen Storage Compound KBH4 under Compression Ravhi S. Kumar,* Eunja Kim, and Andrew L. Cornelius Department of Physics & Astronomy and High Pressure Science and Engineering Center (HiPSEC), UniVersity of NeVada Las Vegas, Las Vegas, NeVada 89154 ReceiVed: August 13, 2007; ReVised Manuscript ReceiVed: March 10, 2008

Angle-dispersive high-pressure powder X-ray diffraction, high-pressure Raman, and ab initio density functional theoretical (DFT) calculations were performed on KBH4 up to 20 GPa. We demonstrate that KBH4 exhibits structural phase transitions from the ambient R-KBH4 phase (cubic Fm3jm) to β-KBH4 (tetragonal P421c) at 3.8 GPa and to the γ-KBH4 phase (orthorhombic Pnma) at 6.8 GPa, which is similar to the phase transition sequence observed for NaBH4 earlier. The transition pressure and bulk modulus obtained for KBH4 in our experiments are in good agreement with theoretical calculations. The results further show that the ambient pressure phase of this hydride is reversible. 1. Introduction In many laboratories intense research on hydrogen storage materials is in progress to find out a suitable hydrogen storage material that can store 6 wt % of hydrogen at ambient conditions, which is the benchmark specified by the U.S. Department of Energy (DOE).1 Complex borohydrides are viable candidates which can store a mass percent of hydrogen as high as 18.9% in LiBH4.2,3 Portable Proton Exchange membrane fuel cells (PEMFC) and direct borohydride fuel cells (DBFC) are alternate devices to the conventional batteries and use NaBH4 successfully for hydrogen storage in commercial fuel cell applications.4–6 In the PEM cells, the hydrolysis reaction of NaBH4 yields NaOH and H3BO3.7 The hydrolysis reaction is highly exothermic. With a proper hydrolysis catalyst the reaction conversion can be raised to 100%8 and more recently highpower densities have been achieved in a borohydride-hydrogen peroxide fuel cell that is suitable for portable applications.8 Recent experiments performed at high pressures to understand the structural stability of the borohydrides have shown several phase transitions in these systems. LiBH4, which crystallizes in the orthorhombic structure, was reported to undergo a pressure-induced structural transition from the ambient R-LiBH4 to a high-pressure β-LiBH4 between 0.8 and 1 GPa.9 Our recent synchrotron X-ray diffraction experiments performed on NaBH4 at high pressures show pressure-induced structural transitions from the cubic Fm3jm structure to a tetragonal (P421c) structure at 6.3 GPa and then to an orthorhombic (Pnma) structure at 8.9 GPa.10 Such pressure-induced changes were also reported in NaBH4 by Raman experiments.11 On studying the low-temperature structures of ABH4 (A ) Li-Cs) borohydrides, Vajeeston et al.12 predicted lower to higher symmetric atomic arrangements while moving from Li to Cs. Thermal conductivity measurements on NaBH4 show the transition temperature to the lowtemperature structure increases with application of pressure.13 Due to its high stability, nonvolatility, and high hydrogen content, KBH4 is considered as a potential system for fuel cell applications similar to NaBH4.14 Despite the experimental and theoretical studies made in the past, full understanding of the high-pressure phases of these ternary borohydrides has not yet * Corresponding author. E-mail: [email protected].

been achieved. The aim of the present investigation is to examine the crystal structure-stability of KBH4 under pressure, which is a part of our ongoing effort to understand the effect of pressure on borohydrides following our earlier work on NaBH4. At ambient temperature and pressure, NaBH4, KBH4, RbBH4, and CsBH4 possess similar fcc-lattice structures and hence these compounds are expected to follow similar pressure-induced structural sequences. We have used high-pressure synchrotron X-ray diffraction, Raman scattering, and density functional theoretical calculations for studying the pressure-induced changes of KBH4. The experimental details and results are described in the following sections. 2. Experimental and Theoretical Techniques 2.1. High-Pressure X-ray Diffraction. High-purity reagent grade (97%) powders of KBH4 were obtained from Sigma Aldrich. High-pressure experiments were performed at the high resolution powder diffraction beam line at Sector 16 ID-B of HPCAT at the Advanced Photon Source. A monochromatic X-ray beam from a double crystal branching monochromator consisting of diamond (111) and silicon (220) crystals was focused to a size of 30 × 30 µm2 by using Kickpatrick-Baez mirrors. The sample was loaded in a 150 µm hole of a Merrill-Bassett type diamond anvil high-pressure cell in a stainless steel gasket chamber with ruby grains for pressure measurement. The culet size of the diamonds was 300 µm and the stainless steel gaskets were indented to 50 µm for drilling. The diamond anvil cell was moderately heated in an oven before sample loading to avoid moisture in the gasket chamber and sample handling was done in an Ar inert gas atmosphere to prevent oxidation. We have conducted different independent experimental runs for KBH4 with and without pressure transmitting medium (silicone fluid). We did not observe any evidence of reaction between the sample and media both visually and also in the X-ray diffraction patterns collected prior to pressurization. Unlike other hydrides where the sample reaction with the media is considerable, KBH4 was found to be quite stable. A Mar 345 imaging plate was used in the experiments to obtain high-quality powder diffraction patterns from the samples. The distance between the sample and the detector and the inclination angle of the imaging plate were calibrated by using a CeO2

10.1021/jp0765042 CCC: $40.75  2008 American Chemical Society Published on Web 05/07/2008

Structural Phase Transitions in KBH4 standard. We have used monochromatic X-ray beams of wavelengths λ ) 0.3931 Å and 0.40165 Å for the experiments. Due to the weak X-ray scattering signal from the light elements, a longer exposure time of about 10 min was necessary to improve the signal-to-noise ratio. The Debye-Scherrer rings recorded by the imaging plate were then integrated by using the Fit2D software to obtain a one-dimensional intensity versus diffraction angle plot.15 We have analyzed the diffraction patterns with the JADE 7.0 commercial software program and LHPM-Reitica Rietveld package for obtaining structural information.16 The pressure at the sample site was determined by measuring the shift of the R1 fluorescence line with the offline system at HPCAT and fitting with the standard ruby pressure scale.17 All experiments were conducted at room temperature. 2.2. High-Pressure Raman Spectroscopy. The Raman spectrum of a flat sample of KBH4 was collected at UNLV at ambient temperature with an Ar-ion laser, using an excitation wavelength of 514.5 nm operating below 100 mW. The spectrum was recorded on an ISA Jobin-Yvon U100 doublegrating spectrometer with a resolution of 1 cm-1 and a thermoelectrically cooled CCD detector (ISA SpectrumOne). High-pressure Raman experiments were also performed with a similar system at HPCAT with a Merrill-Bassett-type diamond anvil cell, using no pressure medium. Care is taken to avoid the local heating effect by attenuating the laser power and minimizing the exposure time. 2.3. Theoretical Calculations. Ab initio calculations were performed by using the CPMD code to examine the highpressure phases of KBH4.18 The local density approximations (LDA) were adopted in the exchange and correlation interactions. The Troullier-Martins pseudopotentials are used in the calculations with sp nonlocality for both K and B atoms and s nonlocality for H atoms.19 The standard generalized gradient approximation (GGA) with use of the BLYP functional is adopted for the exchange-correlation interaction.20 The energy cutoff for the plane-wave expansion is 130 Ry. The k-points in the Brillouin-zone are sampled by using a 2 × 2 × 2 MonkhorstPack grid.21 3. Results and Discussion 3.1. X-ray Diffraction and Raman Spectra of KBH4. Phase analysis on a powder sample of KBH4 was carried out with a PANalytical X-ray diffraction system operating with Cu KR radiation before sample loading in the DAC. The sample was inserted in an inert gas chamber with a continuous Ar flow during data collection. Analysis of the diffraction patterns showed a single phase for KBH4. KBH4 crystallizes in the NaCl type cubic (Fm3jm) structure at ambient conditions and the cell dimension increases with the increasing atomic number of the alkali atom in the unit cell. The [BH4]- anions are tetrahedrally surrounded by the alkali cations. The cell parameter obtained experimentally for KBH4, a ) 6.7557(3) Å, agrees well with the reported literature value.22 The assignments of Raman vibration bands for KBH4 are shown in the Table 1 and also in Figure 1 for the powder sample outside the DAC before loading. Bending modes were observed at the lower spectral region with the asymmetric bending mode observed at 1121 cm-1 and symmetric bending mode at 1246 cm-1. The bond-stretching modes were observed from 2100 to 2500 cm-1. The Raman vibration bands observed and mode assignment in our experiments are in excellent agreement with previous Raman studies.23 During high-pressure application, monitoring the bending modes was difficult due to the shift of the Raman peaks (corresponding to the bending modes) toward

J. Phys. Chem. C, Vol. 112, No. 22, 2008 8453 TABLE 1: Raman Mode Assignment for KBH4 powder Raman (Renaudin powder peak DAC et al.) (this work) intensity (as-loaded) 1121 1124 1247 2180 2217 2231 2288 2308 2378 2492

w w-sh m m m w-sh w-sh s w w

1121 1126 1247 2180 2217 2231 2288 2308 2378 2492

1119 10 ν4 1246 2181 2217 2231 2288 2312 2384 2505

mode assignment ν4

bending

ν2 2ν4 (A1) stretching 2ν4 (F2) 210ν4 (F2) ν3 ν1 ν2+ν4 2ν2

the first-order diamond peak from the DAC, which appears around 1331 cm-1. 3.2. High-Pressure Phases of KBH4. In situ angle dispersive X-ray diffraction data measured at different pressures for KBH4 are shown in Figure 2. At 0.5 GPa, the diffraction pattern can be indexed to the Fm3jm cubic structure. On increasing pressure to 4 GPa, an additional diffraction line appeared around 2θ ) 11.4°. At this pressure the diffraction pattern is well indexed with a tetragonal P4j21c space group. On increasing pressure to 7 GPa we inferred the (101) and (110) lines corresponding to the tetragonal structure started splitting. Further at 11 GPa, a number of new diffraction lines emerged in between 2θ ) 6°

Figure 1. Raman spectrum of KBH4 collected at ambient conditions. The bending modes and stretching modes are labeled with their corresponding Raman frequency.

Figure 2. Representative X-ray diffraction patterns of KBH4 at various pressures up to 20 GPa. Background was subtracted for all the patterns. The spectrum labeled as 4.4 d denotes the diffraction pattern collected during decompression. The cubic structure of the material is retained on releasing the pressure in the cell as shown in the diagram.

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Figure 4. Raman shift as a function of pressure (run 1). The shaded area indicates the tetragonal to orthorhombic transition region. The markers shown with lines represent the new modes observed above 6.8 GPa.

Figure 3. Raman spectra of KBH4 stretching modes as a function of pressure (run 2). The inset shows the changes in the bending modes as a function of pressure.

to 10°. These changes indicated another structural phase transition occurring in this compound. On close analysis of the diffraction pattern collected at 7 GPa, we found that the pattern represents a mixture of both tetragonal and the new highpressure phase. We have indexed the high-pressure phase to an orthorhombic Pnma structure very similar to that of NaBH4. The tetragonal to orthorhombic transition is clearly evidenced in the high-pressure Raman experiments. Examining the stretching modes of the Raman spectra shown in Figure 3 and comparing the spectra collected at 5.7 and 7.9 GPa, it can be seen that three additional Raman modes emerge around 7.9 GPa due to splitting of the 2323 cm-1 stretching mode. On increasing the pressure the intensity of the mode at 2327 cm-1 increases while the Raman mode at 2271 cm-1 corresponding to the tetragonal phase diminishes. The transition pressure agrees well with the tetragonal-orthorhombic transition pressure observed in the X-ray diffraction experiments. The cubic-tetragonal transition observed in KBH4 is very similar to the one observed for NaBH4, which takes place around 6.3 GPa as reported in our previous high-pressure X-ray diffraction measurements (Figure 4).10 More recently we have performed high-pressure neutron diffraction experiments, density functional theoretical calculations on NaBD4, and confirmed the existence of the high-pressure intermediate tetragonal phase.24 Subsequent high-pressure diffraction studies reported by Filinchuk also showed such cubic-tetragonal transformation occurring around 6 GPa.25 However, it is noticed that apart from the diffraction experiments, previous reports on high-pressure Raman spectra of NaBD4 did not show any significant changes

Figure 5. Rietveld full profile refinement for three different phases of KBH4. The solid line in the spectra is observed data and the symbols represent the refined pattern. The difference and phase markers are shown below the diffraction patterns.

during cubic-tetragonal transition.11 The reason may be the transition occurs due to a slight distortion of the cubic unit cell and does not accompany any volume collapse.10 We have observed a similar scenario in the high-pressure Raman experiments on KBH4, which is consistent with NaBH4. Below 5 GPa, no significant changes in the stretching modes were noticed. However, vanishing of the bending mode at 1121 cm-1 at low

Structural Phase Transitions in KBH4

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Figure 6. Total energy versus volume plot for different phases of KBH4.

Figure 7. Pressure versus volume plot for KBH4 (solid symbols). Arrows indicate the phase transition pressures. The continuous line is the fit for the equation of state and the open symbols represent theoretical simulation.

pressures below 4 GPa with a decrease in intensity of the 1247 cm-1 (Figure 3) mode indicates that the cubic-tetragonal transition involves a slight distortion of atomic positions of the alkali, boron, and hydrogen atoms from the ideal cubic symmetry. To obtain more details on the structural behavior,

we have performed density functional calculations on three different phases in addition to the low-temperature tetragonal phase that will be discussed in a later section and the theoretical results support the phase transition observed in the diffraction experiments. We have attempted to obtain detailed structural information regarding the ambient and high-pressure phases from the X-ray diffraction patterns. Specifically we have chosen the diffraction patterns which correspond to cubic (P ) 0.5 GPa), tetragonal (P ) 4 GPa), and orthorhombic (P ) 20 GPa) phases for full profile Rietveld refinements. The Fm3jm (space group 225) cubic symmetry with K atoms residing at 4a positions, B at 4b positions, and H atoms at 32f positions has been used as a starting model for the first diffraction pattern. For refinement of the X-ray diffraction pattern measured at 4 GPa we have adopted the P421c tetragonal symmetry (space group 114), where the K atoms are located at 2a, B atoms at 2b, and H atoms at 8e positions. Further neutron diffraction experiments to obtain accurate atomic positions are in progress. Finally for structural refinements of the high-pressure phase above 6.8 GPa, we have adopted the Pnma symmetry (space group 62) with K and B atoms at 4c and H atoms at 4c and 8d, respectively. We have obtained good agreement between the calculated and observed spectra with small R factors as shown in the Figure 5. A phase transition from the ambient Fm3jm structure to the tetragonally ordered P42/nmc structure was reported in KBH4 at low temperatures, below 100 K.23 We have also attempted to fit the 4 and 20 GPa data with other possible space group symmetries such as P42/nmc and P21/c for explaining the two high-pressure phases, 11,12 but they could not reproduce the experimental diffraction patterns. The orthorhombic Pnma phase persisted up to 20 GPa, the highest pressure achieved in this experiment. The total energy curves from our calculations as a function of volume for KBH4 are shown in Figure 6. The pressure-volume data obtained from the theoretical calculations match well with the observed experimental data as shown in Figure 7. The highpressure tetragonal phase (P421c) is a metastable structure,

TABLE 2: Structural Parameters for the Ambient and High-Pressure Phases of KBH4 cubic (Fm3j m) (P ) 0.5 GPa) expt

tetragonal (P421c) (P ) 4 GPa)

theory

expt

theory

a (Å) b (Å) c (Å) B0 (GPa)

6.6897(3)

6.90 (1)

4.4754(2)

4.55(1)

16.8(4)

15.4(1)

6.363(2)

6.37(1)

K

4a (0,0,0)

(0,0,0)

2a

(0,0,0)

B

4b (0.5,0.5,0.5)

(0.5,0.5,0.5)

2b

(0,0,0.5)

H(1)

32f (x,x,x) x ) 0.6053a x ) 0.6029b x ) 0.6116c

x ) 0.6023

8e x)0 y ) 0.79061 z ) 0.41501

x)0 y ) 0.78433 z ) 0.39148

orthorhombic (Pnma) (P ) 20 GPa) expt 6.976(6) 4.933(4) 5.111(3)

atomic positions (x,y,z)

H(2) H(3) Rwp a

1.4% b

1.4% c

Present work. Experimental from ref 23. Theory from ref 12.

4c: x ) 0.2052 y ) 0.25 z ) 0.24555 4c: x ) 0.4559 y ) 0.25 z ) 0.43208 4c: x ) 0.3920 y ) 0.25 z ) 0.7070 4c: x ) 0.1490 y ) 0.25 z ) 0.8200 8e: x ) 0.2070 y ) 0.04590 z ) 0.5400 1.5%

theory 7.05(1) 4.685(1) 5.613(1)

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Figure 8. Crystal structure representation of cubic, tetragonal, and orthorhombic phases of KBH4. The atomic coordinates provided (at the same pressures) in the table are used for illustration.

which appears between 3 and 4 GPa and it eventually evolves into the low-temperature tetragonal phase (P42/nmc) at low temperatures. These two tetragonal structures share similar structural properties. However, energetics is slightly different due to the change in atomic positions of K, B, and H atoms. Above 7 GPa the calculations show a structural transition from tetragonal to orthorhombic. The bulk modulus was estimated by fitting the pressurevolume data of cubic and tetragonal phases together to a thirdorder Birch-Murnaghan equation of state as there are no discontinuities observed in the volume. The experimentally obtained bulk modulus value of B0 ) 16.8(4) GPa with a fixed pressure derivative B0′ ) 4 agrees very well with the theoretically estimated value of B0 ) 15.4(1) GPa and B0′ ) 4. The bulk modulus value of this compound is further comparable to the bulk modulus of the cubic phase of NaBH4 (B0 ) 19.9(7) GPa with B0′ ) 3.9) reported in our previous work.10 The bulk modulus value of KBH4 mainly depends on the compression of K-H bonds as discussed by Errandonea et al., as the BH4 clusters have a very small Gruneissan parameter.26 On using the empirical relation and the K-H bond length, the bulk modulus value of 24 ( 4 GPa estimated compares well with our results. The calculated volume compression data based on GGA for the cubic, tetragonal, and orthorhombic phases are in good agreement with theoretical values provided in Table 2. The crystal structures of ambient and high-pressure phases are shown in Figure 8. We have collected diffraction patterns on decompression at 4.4 GPa and also on releasing the pressure from the cell to ambient (denoted as 4.4 d in Figure 2). The diffraction pattern at 4.4 GPa showed the coexistence of both tetragonal and orthorhombic phases. The ambient cubic phase was found to be reversible on pressure release both in diffraction and Raman experiments. 4. Conclusions We have performed high-pressure X-ray diffraction experiments on KBH4 up to 20 GPa. High-pressure Raman experiments and first principle calculations were also performed on KBH4. From the experiments and theoretical calculations we have revealed a phase transition from the ambient Fm3jm cubic phase to a high-pressure tetragonal (P421c) phase at 4 GPa and subsequently to an orthorhombic (Pnma) phase at 11 GPa, which is very similar to the pressure behavior of the iso-structural compound NaBH4. We speculate that the continuous cubictetragonal phase transition observed in the pressure-volume plots without a large volume collapse in KBH4 and NaBH4 is a consequence of a slight distortion of the cubic symmetry. We have recently carried out high-pressure X-ray diffraction measurements on RbBH4 where we found pressure induces a

direct transition to the orthorhombic (Pnma) phase. The reduction in the cubic to tetragonal transition pressure and the absence of a tetragonal high-pressure phase in RbBH4 suggest that the transition pressure is influenced by the size of the alkali atom during the structural rearrangement of [BH4]- anions and alkali cations.27,28 Better insight of the pressure-induced changes may be obtained if the intermolecular bonding in the [BH4]- units for these high-pressure phases is studied with suitable experiments. To shed more light, high-pressure inelastic X-ray scattering experiments on boron K-edge are planned on NaBH4 and KBH4. Acknowledgment. The authors thank Stanislav Sinogeikin for technical help at HPCAT in X-ray and Raman measurements. Work at UNLV is supported by DOE award DE-FG3605GO08502. HPCAT is a collaboration among the UNLV High Pressure Science and Engineering Center, The Lawrence Livermore National Laboratory, the Geophysical Laboratory of the Carnegie Institution of Washington, and the University of Hawaii at Manoa. The UNLV High Pressure Science and Engineering Center was supported by the U.S. Department of Energy, National Nuclear Security Administration, under Cooperative agreement number DE-FC52-06NA26274. Use of APS was supported by the U.S. Department of Energy, Office of Science, and Office of Basic Energy Sciences under contract number DE-AC02-06CH11357. References and Notes (1) U.S. Department of Energy Hydrogen Program [http://www. hydrogen.energy.gov]. (2) Ritter, J. A.; Ebner, A. D.; Wang, J.; Zidan, R. Mater. Today 2003, l6, 18. (3) Zuttel, A.; Wenger, P.; Rentsch, S.; Sudan, Ph.; Mauron, P.; Emmenegger, Ch. J. Power Sources 1999, 84, 130. (4) Aiello, R.; Sharp, J. H.; Mathews, M. A. Int. J. Hydrogen Energy 1999, 24, 1123. (5) Amendola, S. C.; Onnerud, P.; Kelly, M. T.; Petillo, P. J.; SharpGoldman, S. L.; Binder, M. J. Power Sources 1999, 84, 130. (6) Wee, J. H. J. Power Sources 2006, 155, 329. (7) Wee, J. H.; Lee, K. Y.; Kim, S. H. Fuel Process. Technol. 2006, 87, 811. (8) Raman, R. K.; Prashant, S. K.; Shukla, A. K. J. Power Sources 2006, 162, 1073. (9) Talyzin, A. V.; Anderson, O.; Sundqvist, B.; Kurnosov, A.; Dubrovinsky, L. J. Solid State Chem. 2007, 180, 510. (10) Kumar, R. S.; Cornelius, A. L. Appl. Phys. Lett. 2005, 87, 261916. (11) Araujo, C. M.; Ahuja, R.; Talyzin, A. V.; Sundqvist, B. Phys. ReV. B 2005, 180, 510. (12) Vajeeston, P.; Ravindran, P.; Kjekshus, A.; Fjellvag, H. J. Alloys Compd. 2005, 387, 97. (13) Sundqvist, B.; Andersson, O. Phys. ReV. B 2006, 73, 092102. (14) Jasinksi, R. Electrochem. Technol. 1965, 3, 40. (15) Hammersley, A. P.; Svensson, S. O.; Hanfland, M.; Fitch, A. N.; Ha¨usermannn, D. High Press. Res. 1996, 29, 301. (16) Howard, C. J.; Hunter, B. Unpublished.

Structural Phase Transitions in KBH4 (17) Mao, H. K.; Xu, J.; Bell, P. M. J. Geophys. Res. 1986, 91, 4673. (18) Car, R.; Parrinello, M. Phys. ReV. Lett. 1985, 55, 2471. (19) Troullier, N.; Martins, J. L. Phys. ReV. B 1991, 43, 1993. (20) Lee, C.; Yang, W.; Parr, R. G. Phys. ReV. B 1988, 37, 785. (21) Monkhorst, H. J.; Pack, J. D. Phys. ReV. B 1976, 13, 5188. (22) Abrahams, S. C.; Kalnajs, J. J. Chem. Phys. 1954, 22, 434. (23) Renaudin, G.; Gomes, S.; Hagemann, H.; Keller, L.; Yvon, K. J. Alloys Compd. 2004, 375, 98. (24) Kim, E.; Kumar, R. S.; Wecke, P. F.; Cornelius, A. L.; Nicol, M.; Vogel, S. C.; Zhiang, J.; Hurtl, M.; Stowe, A. C.; Daemen, L.; Zhao, Y. J. Phys. Chem. B 2007, 111, 13873.

J. Phys. Chem. C, Vol. 112, No. 22, 2008 8457 (25) Filinchuk, Y.; Talyzin, A. V.; Chernyshov, D.; Dmitriev, V. Phys. ReV. B 2007, 76, 092104. (26) Errandonea, D.; Manjon, F. J.; Somayazulu, M.; Hausermann, D. J. Solid State Chem. 2004, 177, 1087. (27) Kumar, R. S.; et al. Unpublished. (28) Errandonea, D.; Pellicer-Porress, J.; Manjon, F. J.; Segura, A.; Ferrer-Roca, Ch.; Kumar, R. S.; Tschauner, O.; Rodriguez-Hernandez, P.; Lopez-Solano, J.; Radescu, S.; Mujica, A.; Munoz, A.; Aquilanti, G. Phys. ReV. B 2005, 72, 174106.

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