Alexander S Côté

John P Lidunkavočadlo

Brodholt

Lidunka Vočadlo

John P Brodholt

The effect of silicon impurities on the phase diagram of iron and possible implications for the Earth's core structure

Keywords: silicon, iron, ab initio, inner core, light elements

The hexagonal close-packed structure (hcp) is the accepted stable form of pure iron (Fe) under Earth's core conditions. Recently, however, a body-centred-cubic (bcc) phase of iron alloyed with lighter elements has been proposed. At relatively modest conditions, experiments have shown that small amounts of silicon (Si) can stabilise the bcc phase with respect to the hcp phase. This result has motivated our present study examining the effect of silicon on the bcc-hcp phase transition. We have performed ab initio calculations on both phases at zero Kelvin; our results are in good agreement with experiment.

Extending our results to core pressures and taking into account previous studies of pure iron at core temperatures, we conclude that the bcc phase of iron alloyed with silicon is likely to be the stable crystalline phase in the inner core.

A c c e p t e d m a n u s c r i p t 1. Introduction

The Earth's inner core consists largely of iron. Density measurements suggest also the presence of other low atomic weight elements [START_REF] Birch | Elasticity and Constitution of the Earths Interior[END_REF]. Quantifying those light elements and determining their role, however, remains a matter of controversy as present experiments cannot simultaneously reproduce the high pressure and temperature conditions of the core. Under ambient conditions, the iron crystal has a body-centred-cubic (bcc) structure.

At pressures above ~15 GPa, (at room temperature), it adopts a hexagonal-close-packed (hcp) structure which persists to inner-core pressures. Temperature, however, has been shown to act in the opposite sense to pressure. For instance, Vočadlo et al. [START_REF] Vočadlo | Possible Thermal and Chemical Stabilization of Body-Centred-Cubic Iron in the Earth's Core[END_REF] used ab initio molecular dynamics (MD) simulations under inner-core pressures and temperatures to show that, while the hcp structure is still marginally more stable, the difference in free energy between the hcp and bcc phases becomes very small. Classical potential MD simulations of Belonoshko et al [START_REF] Belonoshko | Stability of the Body-Centred-Cubic Phase of Iron in the Earth's Inner Core[END_REF] show an even more pronounced effect of temperature and predict that the bcc phase of Fe actually becomes more stable than the hcp structure in the inner-core. Regardless of which is correct, it appears that the free energies of the bcc and hcp phases under core conditions are very similar. The preference of small concentrations of light elements for either phase may thus be the defining factor in determining which Fe structure is stable in the inner core. In particular, it has been shown theoretically [START_REF] Vočadlo | Crystal Structure, Compressibility and Possible Phase Transitions in Epsilon-Fesi Studied by First-Principles Pseudopotential Calculations[END_REF] and experimentally [START_REF] Dobson | A New High-Pressure Phase of Fesi[END_REF] that, at high pressures, FeSi stabilises in the CsCl structure, which has identical atomic coordinates to bcc-Fe, rather than in the hcp structure. This suggests that Si may act to stabilise the bcc phase in the core. Indeed, this is supported by low pressure experiments which have shown that, at low concentrations (7.9 weight %), Si has a large effect on the stability field of bcc Fe, increasing the

A c c e p t e d m a n u s c r i p t

3 transition pressure to hcp from 10-15 GPa up to 36 GPa at 300 K [START_REF] Lin | Iron-Silicon Alloy in Earth's Core?[END_REF]. It is possible, therefore, that with an appropriate concentration of Si and at the high temperatures of the core, the bcc phase could be stabilised with respect to the hcp and the formation of a bccstructured Fe-Si alloy would be favoured.

In order to test whether Si can stabilise the bcc phase sufficiently to make it the stable phase in the core, we used ab initio calculations to investigate the effect of silicon concentration on both the bcc/hcp phase transition and the equation of state of Fe-Si up to core pressures. In the next section we describe the simulation techniques used in our calculations and the way we obtained the equations of state for the different impurity concentrations in the different iron crystal structures. We then show how the relative enthalpy curves can be used to deduce the stable structure. We calculate the pressure of the bcc-hcp phase transition and compare it to experimental data. Finally we extend these results to Earth's core conditions, taking previous ab initio calculations [START_REF] Vočadlo | Possible Thermal and Chemical Stabilization of Body-Centred-Cubic Iron in the Earth's Core[END_REF] further by investigating different impurity concentrations and at different volumes, and we discuss possible implications for the inner core structure. Our conclusions give further validity to those of Vočadlo et al. [START_REF] Vočadlo | Possible Thermal and Chemical Stabilization of Body-Centred-Cubic Iron in the Earth's Core[END_REF].

Quantum mechanical simulation techniques

The calculations in this work were carried out using Density Functional Theory (DFT) [START_REF] Hohenberg | Inhomogeneous Electron Gas[END_REF] as implemented in the VASP code [START_REF] Kresse | Efficient Iterative Schemes for Ab Initio Total-Energy Calculations Using a Plane-Wave Basis Set[END_REF]. The exchange-correlation energy was represented by the generalised gradient approximation (GGA) [START_REF] Wang | Correlation Hole of the Spin-Polarized Electron-Gas, with Exact Small-Wave-Vector and High-Density Scaling[END_REF], which achieves the highest accuracy for transition metals. We made use of the projected augmented wave (PAW)

A c c e p t e d m a n u s c r i p t 4 method, where the wavefunction combines an all-electron core region with a pseudovalence region. The PAW method was introduced by Blöchl [START_REF] Blöchl | Projector Augmented-Wave Method[END_REF], and it combines the accuracy of all-electron methods with the efficiency of pseudopotentials. With a relatively small number of plane waves, it can give results that agree accurately with all electron methods. Indeed, it has been previously shown that the use of this method gives accurate results for iron both under ambient and core conditions [START_REF] Mao | Phonon Density of States of Iron up to 153 Gigapascals[END_REF][START_REF] Vočadlo | The Properties of Iron under Core Conditions from First Principles Calculations[END_REF][START_REF] Alfè | Thermodynamics of Hexagonal-Close-Packed Iron under Earth's Core Conditions[END_REF][START_REF] Vočadlo | First Principles Calculations on Crystalline and Liquid Iron at Earth's Core Conditions[END_REF][START_REF] Price | The Earth's Core: An Approach from First Principles[END_REF].

Unit cells of pure iron were constructed in the bcc and hcp structures. For the bcc case we used the 2-atom cubic unit cell. For the hcp case, the unit cell used was that of the antiferromagnetic II (afmII) structure which contains 8 atoms (see Figure 1), as this was

shown to be the stable hcp phase at low pressures [START_REF] Steinle-Neumann | Magnetism in Dense Hexagonal Iron[END_REF][START_REF] Vočadlo | An Ab Initio Study of Nickel Substitution into Iron[END_REF].

VASP calculates the ground state (T=0) for each set of ionic positions and the electronic free energy is taken as the quantity to be minimised. Care was taken when choosing the kpoint sampling grid and plane wave cutoff energies in order to have an energy convergence of no more than 0.004 eV/atom. A 14x14x14 grid (equivalent to 84 k-points in the irreducible Brillouin zone (IBZ)) was used for bcc-Fe and a 9x9x9 grid (125 kpoints in the IBZ) for hcp-Fe, with a cutoff energy of 400eV. The same cutoff energy was used throughout the calculations.

Substitutional defects of silicon were created by replacing Fe with Si in hcp and bcc supercells. Three different concentrations were investigated which bracket the experimental value of 7.9 wt%, namely ~3.2, ~6.7 and ~10.4 wt. %. Specifically, for both structures, that meant a substitutional defect of one, two and three Si atoms respectively were then allowed to relax and the total energy was calculated. In the supercells containing more than one Si defect, a number of trials were performed with the different possible configurations of the defect atoms. In the hcp supercells the Si atoms were placed at symmetric positions as far away from each other as possible, taking into account the periodic boundary conditions. This proved to be the most stable configuration. In the bcc case, two different substitutional configurations were found to be favoured at high and low pressures respectively, so both were taken into account.

Finding the ground state free energy E for a set of different volumes V allowed us to fit the E-V data to a 3 rd order Birch-Murnaghan equation of state (EOS) [START_REF] Poirier | Introduction to the Physics of the Earth's Interior[END_REF] of the following form:

( ) ( ) 6
- ′ - + ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - ′ + ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = K V K E V V V V V V V V V K K V V V V V V V K V E
The resulting EOS parameters were then used to obtain the enthalpies H at those volumes, and to plot the enthalpy as a function of pressure for the bcc and hcp phases of the different Si concentrations.

A c c e p t e d m

a n u s c r i p t 6

Results

The equation of state parameters for each silicon concentration are shown in Table 1. The parameters for pure iron [START_REF] Guinan | Pressure Derivatives of Elastic Constants of Alpha-Iron to 10 Kbs[END_REF][START_REF] Dewaele | Quasihydrostatic Equation of State of Iron above 2 Mbar[END_REF][START_REF] Mao | Static Compression of Iron to 300 Gpa and Fe0.8ni0.2 Alloy to 260 Gpa -Implications for Composition of the Core[END_REF][START_REF] Steinle-Neumann | First-Principles Elastic Constants for the Hcp Transition Metals Fe, Co, and Re at High Pressure[END_REF] are also displayed, where the volume underestimation due to the GGA functional is apparent. For the higher concentrations of Si, where different Si configurations are favoured at low and high pressures, both sets of parameters are displayed. For the 6.7% concentration, this change of stability between the two configurations in bcc takes place at around 210 GPa, whereas for the 10.4% concentration, it takes place at around 80 GPa. The equation of state parameters were used to plot the enthalpy curves as a function of pressure for pure Fe and for the three different concentrations of Si considered here.

The enthalpy-pressure curves are plotted in Figure 2. Also indicated are the transition pressures for the pseudo-univariant reaction (i.e. the transition from one phase to another with the same composition, ignoring any possible solid-solutions). The calculated transition pressures are ~7 GPa, ~11 GPa, ~16 GPa and ~41 GPa, for concentrations of 0%, 3.2%, 6.7% and 10.4% Si respectively. In Figure 3 we plot the change in the transition pressure for the pseudo-univariant reaction as a function of composition. As can be seen, the effect of Si on the transition pressure is strongly non-linear. Also shown is the pressure range of the room temperature solid solution as found experimentally by Lin et al. [START_REF] Lin | Iron-Silicon Alloy in Earth's Core?[END_REF] for a bulk composition containing 7.9% Si. Although the experimental solid solution cannot be used to establish the exact pressure of the pseudo-univariant reaction, for a particular composition the pseudo-univariant reaction must be bracketed by the minimum and maximum pressure where both phases are present. The experimental two-

A c c e p t e d m a n u s c r i p t

7 phase region at that composition starts at about 4 GPa above the pure Fe transition and persists for about 20 GPa [START_REF] Lin | Iron-Silicon Alloy in Earth's Core?[END_REF]. As can be seen in the figure, we calculate that 7.9% Si will increase the pressure of the pseudo-univariant transition by ~16 GPa, a value that falls within the experimental solid solution range. This adds some confidence to our results and confirms that the presence of Si can strongly increase the stability of the bcc phase over hcp.

We repeated the 0 K calculations at the pressure of the inner core boundary (ICB) (330 GPa). It is worth noting that the magnetic moments at such high pressures greatly diminish in the bcc phase, and disappear completely in the hcp phase. The results are given in Table 2. For pure iron, the calculated difference in enthalpies between the bcc and hcp structures of pure iron is 660 meV in favour of the hcp structure. As with the low pressure results, we find that Si strongly prefers the bcc phase, and in fact, at 10.4 wt. %, the enthalpy difference has dropped to only 115 meV. We also find, again consistent with the low pressure results, that the enthalpy difference drops almost linearly with increasing concentration of Si. These results show that at 330 GPa, Si defects are about 2.5eV per defect atom more stable in the bcc structure than in the hcp phase. This means that, for example, the suggested ~5mol.% concentration of Si in the inner core [START_REF] Lin | Iron-Silicon Alloy in Earth's Core?[END_REF][START_REF] Vočadlo | Possible Thermal and Chemical Stabilization of Body-Centred-Cubic Iron in the Earth's Core[END_REF][START_REF] Stixrude | Composition and Temperature of Earth's Inner Core[END_REF] would lower the enthalpy difference between the two phases by 125 meV.

It is known that bcc-Fe is dynamically unstable at 0 K and high pressures [START_REF] Stixrude | Constraints on the Crystalline Structure of the Inner Core: Mechanical instability of BCC iron at high pressure[END_REF][START_REF] Söderlind | First-Principles Theory of Iron up to Earth-core Pressures: Structural, Vibrational and Elastic Properties[END_REF]. Our results, however, show that the addition of Si decreases the free energy difference between the two phases; the implication of this is that the magnitude of the instability is

A c c e p t e d m a n u s c r i p t

8 also reduced. Indeed it may even be the case that the incorporation of Si removes the instability altogether, even at 0 K.

Discussion and Conclusion

Even though there is no doubt from our results that the inclusion of Si tends to stabilise bcc-Fe, the calculated enthalpy differences between the two pure Fe structures at T=0 still favour hcp stability at inner core pressures, at least for the concentrations considered here.

However, the high temperatures of the core alter this significantly and recent ab initio results show that the difference between the free energies of the two phases of pure iron at ICB pressures and temperatures is only ~33 meV [START_REF] Vočadlo | Possible Thermal and Chemical Stabilization of Body-Centred-Cubic Iron in the Earth's Core[END_REF]. Since, as shown above, Si stabilises bcc over hcp by 2.5 eV per defect atom, less than 1.5 mol.% Si is needed to overcome the 33 meV required to make bcc stable. The higher but still modest concentration of 5 mol.% would provide a stabilisation enthalpy of 125 meV. This shows that even the relatively small concentrations of Si suggested for the inner core [START_REF] Allègre | Chemical Composition of the Earth and Constraints on Its Early History and Chronology[END_REF] are more than enough to stabilise bcc-Fe in the inner core. We conclude, therefore, that the presence of the bcc phase in the inner core cannot be ruled out. ) for the pseudo-univariant reaction as a function of composition. The gray bar represents the experimental pressure range of the 300K solid solution [START_REF] Lin | Iron-Silicon Alloy in Earth's Core?[END_REF] for a bulk composition containing 7.9% Si. We can see that our result on the pseudo-univariant reaction at 7.9% Si falls well within that range. The solid line is a weighted fit through the ab initio results.

A c c e p t e d m a n u s c r i p t

Figure and table captions

Table 1:

The parameters of the Birch-Murnaghan 3 rd order equation of state, for the different concentrations of silicon. For pure iron, the experimental parameters are also displayed (b,d).

A c c e p t e d m a n u

  s c r i p t for each of the above concentrations. The supercell contained 16 atoms both in the bcc (a 2x2x2 multiplication of the 2-atom bcc unit cell) and in the hcp structure (a 1x1x2 multiplication of the afm II unit cell). A 5x5x5 k-point grid (18 k-points in the IBZ) was used for the bcc and a 9x9x5 (123 k-points in the IBZ) for the hcp phase. The structures

Figure 1 :

 1 Figure 1: The afmII structure. The arrows on the atoms represent the spin direction.

Figure 2 :

 2 Figure 2: Calculated enthalpy curves for bcc and hcp structures at 3 different concentrations of silicon. The arrows indicate the transition pressures for the pseudounivariant reactions.

Figure 3 :

 3 Figure 3: Change in the transition pressure (ΔP=P tr FeX -P tr Fe

A c c e p t e d m a n u s c r i p tFiguresFigure 1 :A c c e p t e d m a n u s c r i p t 15 Figure 2 :

 1152 Figures

A c c e p t e d m a n u s c r i p t 16 Figure 3 :

 163 Figure 3: Change in the transition pressure (ΔP=P tr FeX -P tr Fe ) for the pseudo-univariant reaction as a function of composition. The gray bar represents the experimental pressure range of the 300K solid solution [4] for a bulk composition containing 7.9% Si. We can see that our result on the pseudo-univariant reaction at 7.9% Si falls well within that range. The solid line is just a weighted fit through the ab initio results.

Table 2 :

 2 The enthalpy differences between the bcc and hcp structures at P=330GPa and T=0K. At T=0K the hcp phase is always more stable, but one can see that the difference drops almost linearly as the percentage of Si increases.

	Tables
	wt. %Si	Fe structure	V 0 (Å 3 )	K 0 (GPa)	K'
		bcc	11.39(1)	203(3)	4.36(3)
		ref. [17]	11.355(7)	201.502(8)	4.29(1)
	0	ref. [19]	11.78	166.4	5.29
		hcp-afmII	10.48(3)	229(7)	5.12(13)
		ref. [22]	10.55	209	5.2
		ref. [21]	11.176(17)	165(4)	5.33(90)
	A c c e p t e d bcc 11.34(2) hcp-afmII 10.55(8) bcc 1: bcc 2: 11.15(3) 11.28(3) hcp-afmII 10.55(4) bcc 1: bcc 2: 11.12(4) 11.31(5) hcp-afmII 10.61(3) 1: Derived from the Si configuration most stable at low pressures. m 201(3) ~3.2 216(12) 216(6) 198(6) ~6.7 238(9) 212(7) 156(7) ~10.4 239(8) a n u s c r i p t 4.39(3) 4.95(19) 4.35(8) 4.41(9) 4.84(14) 4.47(9) 5.25(12) 4.67(11) 2: Derived from the Si configuration most stable at core pressures.

Table 1 :

 1 The parameters of the Birch-Murnaghan 3 rd order equation of state, for the

	different concentrations of silicon. For pure iron, the experimental parameters are also
	displayed (b,d).

Table 2 :

 2 The enthalpy differences between the bcc and hcp structures at P=330GPa and T=0K. At T=0K the hcp phase is always more stable, but one can see that the difference drops almost linearly as the percentage of Si increases.

Acknowledgements

The work of ASC was supported by NERC grant NE/C519662/1. We would also like to thank the Royal Society for the University Fellowship awarded to LV.


Daniele Di Gioacchino

Paolo Tripodi

Augusto Marcelli

Mariangela Cestelli Guidi

Massimo Piccinini

Paolo Postorino

EmanueleArcangeletti Daniele Di Castro

Emanuele Arcangeletti

PRESS-MAG-O: A new facility to probe materials and phenomena under extreme conditions

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

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Abstract-In the recent years several experiments performed under high magnetic fields [HMF], at high pressure [HP] and/or at low temperature [LT] have led to spectacular discoveries in condensed matter. In many new systems, although challenging, it is strategic to perform a Magneto-Optical analysis, to investigate the phonon behavior in the far infrared (IR) domain. By combining HMF and HP in a wide temperature (T) range to perform simultaneously IR magneto-optics and ac-magneto-dynamic experiments, it will be possible to achieve unique information on systems and/or new phenomena, almost impossible to obtain with standard spectroscopic methods. Here we present PRESS-MAG-O, a new facility under construction that will perform HP experiments under HMF in a wide T range. The system is expected to be operational by the end of 2007 and will be tested at SINBAD, the IR synchrotron radiation beamline operational since 2001 at DAΦNE, the storage ring of the INFN Frascati National Laboratory (LNF). While for IR experiments an interferometer will be used, for the magneto-dynamic experiments a SQUID magnetometer in the 10 Hz -2 KHz frequency range will be utilized. HP will be applied to samples by a Cu-Be diamond anvil cell (DAC), so that the device will be able to collect FTIR spectra and high harmonic ac susceptibility data in a dc magnetic field up to 8 T and to about 20 GPa in a wide temperature range (4.2-200 K).

Phase Transition, High Pressure, High Harmonic Magnetic ac susceptibility measurements, IR spectroscopy.

I. INTRODUCTION

In the last years experiments performed on materials under extreme conditions of high magnetic field, high pressure, at ultra low temperature have led to the spectacular discovery of a great deal of new phenomena as: fractional Quantum Hall Effect (HMF + LT), ferromagnetic superconductivity in UGe 2 (LT, HP), superconductivity in S and Li with largest T c (HP) among the elemental superconductors K [1][2][3], second triple point in liquid phosphorus, metal to insulator transition in a Mott insulator BaVS 3 [4] and a Kondo Insulator FeSi 1-x Ge x [5]. By tuning crystal lattice structure, HP allows changing electronic and magnetic properties of materials [6]. The possibility to follow magnetic properties under pressure, using ac magnetic multi-harmonic susceptibility, allows measuring the critical temperature of new phases and electron dynamic changes. Moreover, IR spectroscopy may investigate different material properties such as phonons and thermo-chemical characteristics, collective modes in correlated systems, etc. In this contest the high brilliance of the synchrotron IR radiation is ideal for HP investigations because with its small beam it may fit at the best the small sample area inside a DAC. Here we present the status of a new apparatus we are developing, called PRESS-MAG-O, where the simultaneous collection of both ac multi-harmonic susceptibility and FTIR data will be performed under extreme conditions of HP, magnetic field and temperature. PRESS-MAG-O is an acronym correlated to the sentence: "Collective phenomena in solid state physics can be trigged by PRESSure, high magnetic dc field and temperature and can be characterized by MAGneto dynamic and magneto Optical techniques" used at the presentation of the project to the 5 th Scientific Committee of the INFN held in a national meeting held in Assisi in 2004. This unique apparatus will allow accurate investigations of exotic materials and quantum phenomena.

II. PRESS-MAG-O APPARATUS

Using the superconducting magnet and the high-pressure cell severe experimental conditions can be applied to a sample. The induced physical changes will be measured by combining magneto-optical spectroscopy and multi harmonic magneto-Daniele Di Gioacchino 1 , Paolo Tripodi 1,2 , Augusto Marcelli 1 , Mariangela Cestelli Guidi A) Cryostat and split magnet An 8 T superconducting magnet with a split coil and four radial access ports has been manufactured by American Magnetics Inc. (USA). The magnet (showed in Fig. 1) will be placed in a cryostat, manufactured by DG-Technology Service (Parma, Italy), equipped by 'ultra flex GARLOK' gaskets showed in Fig. 2. The system is designed with the magnetic field perpendicular to the light propagation vector (k IR ) (Voight geometry). The instrument has a liquid helium (LHe) reservoir around the magnet and an internal 'sample-insert' device, containing the DAC and the magnetometer, capable to perform static sample cooling via a cold finger on the bottom side. Two cold shields, one at liquid nitrogen (LN 2 ) temperature and one at LHe vapor temperature, surround the LHe reservoir to prevent thermal radiation. In order to allow transmission of the IR radiation and to limit the absorption of the IR radiation, the entire device has been designed with only two thin wedged diamond windows of 15 mm of diameter set along the light path in addition to the two 'anvils' of the diamond cell.

B) Diamond Anvil Cell

To produce high pressure we manufactured a miniature DAC [7] (Fig. 3). AC susceptibility measurements of extremely small volume samples (~1x10 -7 cm 3 ) at high pressure are a complex assignment also because of the presence of a possible magnetic background inside the cell. To reduce the background and any additional magnetic noise, a non-magnetic 2% Be-Cu alloy for the cell and the gaskets has been selected. Moreover, to interface the Cu-Be and the anvils backing cylinders with conic holes made by non-magnetic SiC-δ phase has been manufactured. SiC will replace the standard superconducting tungsten carbide that exhibits a T c ~4K and superconducting fluctuations up to 6 K [8], making difficult or impossible investigations of superconducting or other quantum transition in this temperature range.

C) ac-gradiometer and SQUID

In our case a very weak magnetic signal induced in a sample by the superconducting magnet in the presence of a high background noise has to be detected. In fact the extremely small volume of the DAC cell respect to the large pick-up coil dimensions gives an extremely low fill factor of the pick-up coils. A SQUID device coupled to a gradiometer, employed as a flux transformer in the high magnetic field probing the sample response, is the best instrument available to fulfill the challenging experimental conditions of our system. In Fig. 4, we show both the layout of the planar ac micro-gradiometer designed to work inside the DAC and a photograph of the pick-up coils patterned on a Si wafer. In the present design the pick-up coil will be set above the gasket and two Nb thin film loop sensors in the integrated planar micro-gradiometer have been realized by lithography technique on a mono-crystal silicon wafer. The geometrical configuration has been optimized to fit the gasket position around the diamonds of the DAC.

D) DAC/gradiometer sample insert

The ac magnetic multi-harmonic susceptibility response of the sample excited by an ac drive field will be detected by the SQUID micro-gradiometer. For this reason a Cu-Be removable sample insert showed in Fig. 5 has been manufactured. The insert will have a thermal contact with the cryostat bottom cold finger via Cu-Be springs on the SQUID holder (see Fig. 5a). Moreover, the excitation coil of the ac magnetic field is located on the movable DAC holder that is allowed to move with a micrometer accuracy along the three axes plus a rotation around the vertical directions to guarantee the alignment of the DAC respect to the light beam (Fig. 5b). The whole sample insert is placed inside the cryostat by a load-lock system, to allow changing the pressure outside the cryostat while keeping it at LHe temperature. The sample between the anvils will be set at the center of the dc magnetic field along the light path while the gradiometer, connected by a flexible superconducting wire to the SQUID, is in thermal contact with the insert and the ac drive coil placed around the DAC. The system has been designed in order to host the SQUID, thanks also to a magnetic Nb 3 Sn shield cylinder, in a low magnetic field region (<100 Gauss) in the bottom section of the insert. The read-out of the ac signals from the SQUID electronics will be performed by a multi-harmonic lock-in. To be able to control and adjust in an easy and accurate way the sample temperature inside the DAC we will use an external laser (λ= 532 nm).

E) SINBAD beamline and FTIR spectroscopy

Nowadays it is well recognized that because of its high brilliance, IR synchrotron radiation sources are ideal to perform high-pressure investigations on small samples such as those contained inside a DAC. SINBAD (Synchrotron Infrared Beamline At DAΦNE) is an IR beamline operational since 2001 in the DAΦNE-Light laboratory at DAΦNE (Double Annular Φ-factory for Nice Experiments), the Frascati electron-positron collider working in topping up mode at an energy of 0.51 GeV per beam with a maximum beam current > 2 A. With a bending magnet radius of 1.4 m, this source is characterized by a critical energy of 208 eV really optimized for low energy applications. As a consequence SINBAD was designed, assembled and commissioned to work at IR wavelengths from about 10 up to 12000 cm -1 . SINBAD has been designed by ray-tracing simulations and its optical layout consists of six mirrors that first focus the radiation and than transfer the source image as a collimated beam to the experimental area. Therein a BRUKER Equinox 55 interferometer working in a vacuum chamber coupled to a BRUKER Hyperion 3000 microscope are available for FTIR experiments [9]. The SINBAD performances are suitable to work with small samples and or DACs and recently, in addition to other HP

A c c e p t e d m a n u s c r i p t

experiments, the first far-infrared absorption spectra of La 1-x Ca x MnO 3-x manganite samples at pressures up to 10 GPa have been collected using the IR synchrotron radiation emission. Data showed that pressure promotes partial metallization at room temperature through a strong reduction of the insulating gap [10].

The PRESS-MAG-O apparatus was designed to perform FTIR spectroscopy in transmission or reflection mode. Indeed, a parallel beam coming out by the interferometer is focused to the sample inside the DAC, using one of the four lateral ports equipped by a wedged diamond window to allow transmission of both visible and IR light. Focusing will be achieved by a ZnSe or by KBr lenses to work in the mid-IR or in the far-IR range, respectively. Simulations are in progress to replace lenses with Cassegrain objectives designed to this purpose. The transmitted beam will exit through a second diamond wedged window placed on the opposite lateral port and then focused to the detector position by a symmetrical optical component. The focusing lenses have a central hole to allow the beam of the laser to reach the sample following the same path of the IR light. As outlined in Fig. 6, the laser will be highly focused by a lens concentric to the IR focusing lens and aligned independently to achieve its optimal focusing condition. For experiments performed in the reflection geometry, as in the microscope optical layout, the beam will be collected by half of the optical components used for the incoming beam in the transmission geometry.

III. CONCLUSION

When completed, PRESS-MAG-O will represent a unique and extremely versatile instrument capable to cover wide ranges of dc magnetic field (≤8 T), temperature (4.2-200 K) and pressure (0-20 GPa), allowing concurrent or simultaneous FTIR spectroscopy and ac magnetic multi-harmonic susceptibility measurements (10 Hz -20 KHz). This instrument is based on a cryostat for temperature measurements and an ac micro-gradiometer to probe the magnetic response of a sample contained inside a Cu-Be DAC. Both magneto-optical experiments vs. pressure and vs. magnetic field in a dc high field magnet will be possible using the IR synchrotron radiation of SINBAD, the beamline operational at Frascati. We expect this device will be operational in 2008 as an invaluable tool to perform systematic high pressure, high magnetic field experiments in a wide temperature range for forefront condensed matter researches.

IV. ACKNOWLEDGMENTS

A c c e p t e d m a n u s c r i p t

1 , Massimo Piccinini 1 , 4 ,A c c e p t e d m a n u s c r i p t

 114 Paolo Postorino 3 , Daniele Di Castro 3 , Emanuele Arcangeletti 31-INFN-LNF, National Institute of Nuclear Physics, Via Enrico Fermi, 40 -00044 Frascati (Rome), Italy 2-H.E.R.A. Hydrogen Energy Research Agency, Corso della Repubblica, 448 -00049 Velletri (Rome), Italy 3-INFM-CNR-Coherentia, Univ. di Roma 'La Sapienza', Dip. di Fisica, P.le Aldo Moro, 4 -00100 Rome, Italy 4-Università Roma Tre, Dip. di Scienze Geologiche, L.go S. Leonardo Murialdo, 1 -00146 Rome, Italy PRESS-MAG-O: a new facility to probe materials and phenomena under extreme conditions dynamic experiments. The control of the experiments is a demanding task that required the development of a novel apparatus whose characteristics are summarized below.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Photo of the 8 T superconducting magnet with its radial access ports before the installation.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Photos of two different views of the amagnetic CuBe DAC.

Fig. 5 .

 5 Fig. 5. a) the fixed component of the sample insert with the SQUID/magnetic screen system at the bottom; b) layout of the X-Y-Z-Θ movable part of the DAC holder.

Fig. 6 .

 6 Fig. 6. Optical layout for light spectroscopies.

A c c e p t e d m a n u s c r i p tFig. 1 .Fig. 2 .

 12 Fig. 1. Photo of the 8 T superconducting magnet with its radial access ports before the installation.

A c c e p t e d m a n u s c r i p tFig. 3 .Fig. 4 .

 34 Fig. 3. Photos of two different views of the amagnetic CuBe DAC.

A c c e p t e d m a n u s c r i p tFig. 5 .

 5 Fig. 5. a) The fixed component of the sample insert with the SQUID/magnetic screen system at the bottom; b) Layout of the X-Y-Z-Θ movable part of the DAC holder.

A c c e p t e d m a n u s c r i p tFig 6 .

 6 Fig 6. Optical layout for light spectroscopies.

We thank all members of the Solid State Laser Laboratory of ENEA at Frascati for their hospitality and invaluable support during the laser heating tests. A special thank is also due to F. Tabacchioni (INAF) and the technical staff of the DAΦNE-Light laboratory of the LNF.


W Luo

J Souza De Almeida

J M Osorio-Guillen

R Ahuja

Electronic Structure of a Thermoelectric Material: CsBi 4 Te 6

We have calculated the electronic structure of CsBi 4 Te 6 by means of first-principles self-consistent total-energy calculations within the local-density approximation using the full-potential linearmuffin-tin-orbital method. From our calculated electronic structure we have calculated the frequency dependent dielectric function. Our calculations shows that CsBi 4 Te 6 a semiconductor with a band gap of 0.3 eV. The calculated dielectric function is very anisotropic. Our calculated density of state support the recent experiment of Chung et al. [Science 287, 1024 (2000)] that CsBi 4 Te 6 is a high performance thermoelectric material for low temperature applications.

A c c e p t e d m a n u s c r i p t

There is a continued quest to find better thermoelectric materials in the material science community. Thermoelectric materials have a wide range of application and recent interest is in thermoelectric refrigeration. These materials are also used for conversion of heat into electrical power and cryogenic microsurgery. A good thermoelectric material should have basically three important properties; high electrical conductivity, high thermoelectric power and low thermal conductivity. In principal all of the mentioned three properties depend on the electronic structure of the material. Since the last three decades, the thermoelectric industry is using traditional semiconducting chalcogenide compounds for thermoelectric materials. Recently the work has become focused on ternary bismuth chalcogenides doped with alkali metals. The addition of alkali metals reduces the thermal conductivity of these materials.

Very recently Chung et al. [1] have reported on the synthesis of a new high performance thermoelectric material, CsBi 4 Te 6 , for low temperature applications. These authors have measured the thermoelectric power, conductivity and carrier concentration and hole mobility as a function of temperature. They have also demonstrated that when this material is doped it shows a very high thermoelectric figure of merit below room temperature. The thermoelectric properties of this material matches traditional materials at cryogenic temperatures. This very recent finding has motivated us to look into the electronic structure of this material. Up to now there is no calculation available regarding the electronic structure for this compound. The main aim of this manuscript is to study the electronic structure and the optical properties of this material to get a better insight into the fundamental aspects of this important material.

In order to study the electronic structure of CsBi 4 Te 6 we have used the full-potential linear muffin-tin-orbital (FPLMTO) method [2]. The calculations were based on the localdensity approximation and we used the Hedin-Lundqvist [3] parametrization for the exchange and correlation potential. Basis functions, electron densities, and potentials were calculated without any geometrical approximation [2]. These quantities were expanded in combinations of spherical harmonic functions (with a cut-off max = 6) inside non-overlapping spheres surrounding the atomic sites (muffin-tin spheres) and in a Fourier series in the interstitial region. The muffin-tin sphere occupied approximately 35% of the unit cell. The radial basis functions within the muffin-tin spheres are linear combinations of radial wave functions and their energy derivatives, computed at energies appropriate to their site and principal as

A c c e p t e d m a n u s c r i p t

well as orbital atomic quantum numbers, whereas outside the muffin-tin spheres the basis functions are combinations of Neuman or Hankel functions [4,[START_REF] Skriver | The LMTO Method[END_REF]. In the calculations reported here, we have used a pseudo-core for Cs composed of 5s, 5p and valence band 6s, 6p and 5d basis functions, and a pseudo-core for Bi made up from 5d and valence band 6s, 6p and 6d basis functions and for Te, finally a pseudo-core of 4d and valence band 5s, 5p and 5d basis functions. In addition the corresponding two sets of energy parameters, one appropriate for the semi-core 5s, 5p, 5d and 4d states, and the other appropriate for the valence states.

The resulting basis formed a single, fully hybridizing basis set. This approach has previously proven to give a well converged basis [2]. For sampling the irreducible wedge of the Brillouinzone we used the special k-point method [START_REF] Chadi | [END_REF]. In order to speed up the convergence we have associated each calculated eigenvalue with a Gaussian broadening of width 20 mRy.

Calculation of the dielectric function

The (q = 0) dielectric function was calculated in the momentum representation, which requires matrix elements of the momentum, p, between occupied and unoccupied eigenstates.

To be specific the imaginary part of the dielectric function, 2 (ω) ≡ Im (q = 0, ω), was

calculated from[7] ij 2 (ω) = 4π 2 e 2 Ωm 2 ω 2 knn σ knσ|p i |kn σ kn σ|p j |knσ f kn (1 -f kn ) δ(e kn -e kn -hω) (1)
In Eq. ( 1), e is the electron charge, m its mass, Ω is the crystal volume and f kn is the Fermi distribution. Moreover, |knσ is the crystal wave function corresponding to the n th eigenvalue with crystal momentum k and spin σ. With our spherical wave basis functions, the matrix elements of the momentum operator can be conveniently calculated in spherical coordinates and for this reason the momentum is written as p = µ e * µ p µ , [START_REF] Edmonds | Angular Momentum in Quantum Mechanics[END_REF] where µ is -1, 0, or 1, and

p -1 = 1 √ 2 (p x -ip y ), p 0 = p z , and p 1 = -1 √ 2 (p x + ip y ).[9]
The evaluation of the matrix elements in eq.( 1) is done over the muffin-tin and the interstitial region separately. The integration over the muffin-tin spheres is done in a way similar to what Khan [START_REF] Khan | [END_REF] and Gasche[7] did in their calculations using the atomic sphere approximation (ASA). A full detailed description of the calculation of the matrix elements will be presented somewhere else. [11] The summation over the Brillouin zone in eq. ( 1) is calculated using linear interpolation on a mesh of uniformly distributed points, i.e., the tetrahedron method. Matrix elements, The correct symmetry for the dielectric constant was obtained by averaging the calculated dielectric function. Finally, the real part of the dielectric function, 1 (ω), is obtained from 2 (ω) using the Kramers-Kronig

1 (ω) ≡ Re( (q = 0, ω)) = 1 + 1 π ∞ 0 dω 2 (ω ) 1 ω -ω + 1 ω + ω . ( 2
)
For a monoclinic structure, we need to calculate three components of the total dielectric function, [START_REF] Khan | [END_REF] corresponding to light polarized along a-axis, b-axis or c-axis.

In Fig. 1 investigations on Na-doped Si clathrate [12] show that high efficiency thermoelectric materials should have high electronic density of states near the highest occupied levels and free electron like density profile for the empty levels. In this regard, the CsBi 4 Te 6 , as shown in Fig. 2, has a very high density of states at the top of the valence band and conduction band is more or less is free electron like. So our calculated density of states suggests that CsBi 4 Te 6 is a very high performance thermoelectric material. It has in fact been shown by Chung et al. [1] that CsBi 4 Te 6 is a high thermoelectric material. Our calculations also show that by adding a Cs atom, i.e. one electron per two units of Bi 2 Te 3 does not produce an intercalation compound, rather than that it reorganize the Bi 2 Te 3 network.

In Fig. 3, we show our calculated dielectric function ( 2) along three different directions,

i.e. parallel to the a, b and c axis. Our calculated dielectric function is very anisotropic.

Since the x-axis is very long in comparison to the y and z axis, also the dielectric functions look very different along x-axis. The interband transition in the dielectric function along the

x-axis shows a sharp increase near the band edge. This peak is more and less dominating around 1.0 eV. For the dielectric function parallel to the y and z-axis, there is no sharp increase near the band edge and the dielectric function has a broader peak when compared with the peak along x-axis. The main broad peak in the dielectric function along the y and z-axis is located around 2.0 eV. The peak in the dielectric function along the z-axis is much broader than the peak along the y and z-axis. We have also tried to find the origin of these two different peaks in the calculated dielectric function. Since the different peaks are due to interband transitions it is useful for our discussion to first analyze the electronic structure of CsBi-4Te 6 which was already discussed above. The origin of the peaks (Fig. 3) is due to interband transitions from valence-to conduction-band states. According to the selection rule only the transitions changing the angular momentum quantum number l by 1 (∆l = ±1) are allowed. Further, this transition is only allowed within in same type of species. Having looked at the density of states (Fig. 2) one can suggest that the first peak in 2 parallel to x-axis, around 1 eV, is due to the transition from Te-p to Te-s states. The peak in 2 parallel to the y and z-axis, around 2.0 eV, corresponds to the Bi-s → Bi-p transition.

In summary we have shown that modern electronic structure theory is sufficiently developed to provide an accurate description of CsBi

A c c e p t e d m a n u s c r i p t

  eigenvalues, and eigenvectors are calculated in the irreducible part of the Brillouin-zone.

A c c e p t e d m a n u s c r i p t

  we show the crystal structure of CsBi 4 Te 6 . It has a monoclinic symmetry and contains 88 atoms in the unit cell. The x-axis of CsBi 4 Te 6 is very long. It is 12 times larger than the b axis and 4 times larger than c axis. Experimental lattice constants[1] a,b,c and angle(β) are 51.92 Å, 4.40 Å, 14.51 Åand 101.48 o , respectively. This structure is very anisotropic and is composed of Bi 4 Te 6 slabs alternating with layers of Cs ions.In Fig.2, we show the calculated density of states (DOS) for CsBi 4 Te 6 . The lowest lying states between 8 and 13 eV originate from the Bi and Te s states and pseudo core 5p states of Cs. A sharp peak around 8.0 eV is mainly 5p states of Cs suggesting the importance of these pesudo-core states and these states are not very far from the Fermi level. Usually people neglect these states. The peak at 11.0 eV is mainly Te s states and states between 11-13 eV are Bi s states. The states between 8-13 ev below the Fermi level is separated from a group of states between 0-5 eV below the Fermi level by a gap of around 3.0 eV.The states just below are mainly dominated by Te p and Bi p states with some mixture of Bi and Te's s and d states. The valence band is separated from the conduction band by 0.3 eV suggesting that this material is a small gap semiconductor. At present there are no measurement about the band gap in this material. Chung et al.[1] have estimated the band gap from the maximum value of the thermopower and the procedure gave a band gap in between 0.05 to 0.11 eV. Our calculated value of the band gap is higher than this value.LDA is known to underestimate the band gap and in some cases it can be underestimated by 50% This means that the real value of the calculated band gap is larger than present value.So values for the real bandgap measurements for example optical measurement are needed to get a precise information about the band gap. As regards the conduction band, the states between 0-5 eV above the Fermi level consist mainly of Te p and Bi p states with a mixture of Te s and d states. Above 5.0 eV, the Cs d states gave important contributions. Theoretical

A c c e p t e d m a n u s c r i p t is a highA c c e p t e d m a n u s c r i p t FIG. 1 :FIG. 2 :FIG. 3 :

 t123 FIG. 1: Crystal structure of CsBi 4 Te 6 .

4

  Te 6 . The calculated value of the band gap is 0.3 eV which is larger than what Chung et al.[1] have obtained using a crude approxima-

tion. Our calculated density of states support the experimental finding that this material

We wish to thank the Swedish Natural Science Research Council as well as the ATOMICS funded by SSF for financial support.


D Y Kim

S K Kwon

R Ahuja

Ab initio study of the pressure effects on R 2 Mo 2 O 7

Keywords: numbers: 31.15.Ar, 61.50.Ks, 71.15.Dx

Structural parameters and Mo 4d bands in geometrically frustrated pyrochlore R 2 Mo 2 O 7 (R: Nd, Sm, Gd, Tb, Dy) have been studied using ab initio methods. Both of chemical pressure and external pressure give linear behaviors of structural parameters. The chemical pressure affects more strongly on Mo-O-Mo bond angle, oxygen coordinate, and lattice constant, whereas one electron band width varies more rapidly with external pressure in good agreement with experimental results.

The change of one electron band width W follows double exchange interaction, negative pressure dependency behavior of the transition temperature, dT c /dP , can not be explained by double exchange 10 . Therefore, to understand the magnetic interactions, it is essential to investigate the structural parameter's behaviors against pressures. Recently, it suggested that chemical pressure effect with external pressure gives equivalent effects on structural parameters and magnetic properties 10 . However, it is questionable that chemical pressure gives the same size of effects on the structure as external pressure does.

In this paper, we have performed the total energy calculations. We have used a plane wave basis set with the projector augmented wave (PAW) method 11 as implemented in VASP 12

package. The calculations were based on the generalized gradient approximation given by with R ions at 16d, Mo at 16c, O1 at 48f and O2 at 8b. For chemical pressure, we replaced a lanthanide and made the system relaxed fully. Also, we made each system be relaxed with constant volume for external pressure effect.

Figure 1 shows the relations between 48f oxygen coordinate u and lattice constant a. Figure 2 shows θ dependence on lattice constant a, in which chemical pressure takes more Thus, we would address that there is no superexchange interactions in Mo-O-Mo bonds.

We show in Fig. 3 pressure dependences of lattice constant a and oxygen coordinate u, respectively. Each lanthanide pyrochlore has similar slope but is different in the absolute values. Fig. 3 bond against lattice constant a . We can see that the chemical pressures do not give so much contribution to the W as external pressures do and increasing W with pressure, which are consistent results of experimental data 10 , implies pressure effects enhance double exchange.

  Geometrically frustrated pyrochlore materials R 2 Mo 2 O 7 (R: a rare earth ion, F m 3m; Z = 8) 1 have attracted a great attention because of their unusual magnetism behaviors. Experiments evidently show ferromagnetic metallic states (FMM) for R = Nd, Sm, Gd and spin glass insulating states (SGI) for R = Tb, Dy, Y 2-5 . In addition, Gd (1-x) Dy x Mo 2 O 7 system is demonstrated to be on the boundary of FMM-SGI transition 2 . From the magnetic phase diagram, FMM-SGI transition is thought to be related to the rare earth ionic radius and consequently, structural parameters.The pyrochlore is composed of two parallel and half of lattice constant displaced sublattices of corner sharing chains of R and Mo. Mo is surrounded by 6 neighboring O forming a distorted octahedron that changes its shape accompanied by oxygen coordinate u and Mo-O-Mo bond angle θ. There have been several attempts to understand the mechanism of magnetic phase transition in R 2 Mo 2 O 7 . For example, it has been suggested that superexchange interaction should be controlled by the ratio of on-site Coulomb energy U ef f and Hund exchange coupling J H in molybdates6 . In addition, Taguchi et al. 7 observed ferromagnetic characteristics due to the spin chirality and the associated Berry phase of Mo 4d spin. It is, thus, particularly interesting to understand the mechanism of magnetic phase transition arising from Mo 4d spin. From the synchrotron radiation x-ray powder experiments, it is found that Mo-O-Mo bond angle θ is wider for FMM state than SGI state 8 . The bond angle increases with ionic radius r R from θ = 129.7 • for Dy 2 Mo 2 O 7 (SGI, r R = 1.027 Å) to θ = 131.5 • for Nd 2 Mo 2 O 7 (FMM, r R = 1.109 Å). It is important to see that this is opposite to Kanamori-Goodenough rule 9 for antiferromagnetism.

A c c e p t e d m a n u s c r i p t

  Perdew and Wang (PW91)13 with exchange and correlation potential. We used 5×5×5 kpoints mesh in the Brillouin zone with Gamma point centered. The pyrochlore structure is characterized by the cubic space group F m3m. The unit cell contains 2 R 2 Mo 2 O 7 units

  Ishikawa et al.10 suggested that chemical pressure and external pressure give same lattice constant a versus oxygen coordinate u linear relation. Meanwhile, Fig.1shows that the linear dependences are different in both cases. In Fig.1, oxygen coordinate u linearly decreased with the sharp slope -0.027 Å-1 for chemical pressure compared with other external pressures which have the slope around -0.01 Å-1 on average. We can see that the chemical pressure effects on the relations between u and a are much more than external pressures' by 2.7 times. Moreover, concerning the external pressure, the slopes are slightly reduced with increasing ionic radius. For example, Nd 2 Mo 2 O 7 has the slope of -0.010(3) Å-1 and Dy 2 Mo 2 O 7 does -0.011(2) Å-1 , which means that du/da shows not only almost same behaviors on external pressure but also slight differences with respect to the change of ionic radius.

FIG. 1 :

 1 FIG. 1: Oxygen coordinate u versus lattice constant a for chemical pressure(red crosses and line) and external pressure on each pyrochlore. From the left upper side of red line each point corresponds to the R = Nd, Sm, Gd, Tb, Dy, respectively.

A c c e p t e d m a n u s c r i p t

  effects than external pressure on θ. Also, with respect to crystallography, Mo-O-Mo bond angle θ should be decided by only u. In the inset of Fig.2, we showed θ dependent on only u so that both lines of chemical and external pressure follow the exact same line θ(u) derived from crystallography. This relations means that θ is the function of two variables(a, ionic radius) and the positive slope in the Fig.2and negative slope in the inset of Fig.2are opposite to Kanamori-Goodenough rule in which wider bond angle prefers antiferromagnetism.

FIG. 2 :

 2 FIG. 2: The relations of Mo-O-Mo bond angle θ and lattice constant a are shown. The highest slope line(red) correspond to the chemical pressure. The inset shows u dependence of θ. The straight dashed line in the inset results from crystallographic data.

A c c e p t e d m a n u s c r i p t 10 FIG. 3 : 7 FIG. 4 :A c c e p t e d m a n u s c r i p t In Fig. 4 ,

 103744 FIG. 3: (a) Lattice constant and (b) oxygen coordinate against external pressure. The straight lines in (a) and (b) are the results of the least-square-fitting.

A c c e p t e d m a n u s c r i p t

  double exchange is governed by itinerant 4d electrons while localized Nd 4f electrons gives superexchange interaction via coupling to Mo 4d. Therefore, we may refer the magnetic ordering of R 2 Mo 2 O 7 is due to the competition between the enhanced ferromagnetic ordering of Mo 4d and antiferromagnetic ordering of lanthanide 4f . In summary, using ab initio methods, we systematically investigated the dependences of Mo-O-Mo bond angle, lattice constants, and oxygen coordinate of different lanthanide pyrochlores R 2 Mo 2 O 7 on both of chemical and external pressure. Our study shows that most relations between structural parameters follow linear lines but different with the ionic radius of lanthanide. One electron band width shows the hoping rate increases with pressure, with which we would address that double exchange is enhanced. The change of 4d Mo bands shapes also indicates the increasing ferromagnetism. Thus, SGI states in lanthanide pyrochlores could be explained in terms of much stronger superexchange than enhanced double exchange. This work is financially supported by the Swedish foundation for International Cooperation in Research and High Education (STINT).

Chemical Nd 2 Mo 2 O 7 Sm 2 Mo 2 O 7 Gd 2 Mo 2 O 7 Tb 2 Mo 2 O 7 Dy 2 Mo 2 O 7

		0.342
		0.34
	u [arb. units]	0.334 0.336 0.338
		0.332
		0.33
		10.1	10.2	10.3	10.4	10.5	10.6	10.7
					a [ °A]

Chemical Nd 2 Mo 2 O 7 Sm 2 Mo 2 O 7 Gd 2 Mo 2 O 7 Tb 2 Mo 2 O 7 Dy 2 Mo 2 O 7 126 129

  (a) shows the relations of lattice constant and applied pressure. da/dp are -0.014(2) Å/GPa for Nd 2 Mo 2 O 7 , -0.015(1) Å/GPa for Gd 2 Mo 2 O 7 , -0.015(1) Å/GPa for Tb 2 Mo 2 O 7 and -0.016(5) Å/GPa for Dy 2 Mo 2 O

		131
		130
	θ [degree]	128 129
		127
						0.33	0.335	0.34
		126						u [arb. units]
		10.1	10.2	10.3	10.4	10.5 a [ °A]	10.6	10.7	10.8	10.9	11

7

, respectively. The slope da/dp is getting steeper with decreasing ionic radius of lanthanide. Thus, external pressure changes lattice constant more on smaller ionic radius lanthanide. Fig. 3 (b) shows oxygen coordinates' linear behaviors against applied pressures of different kinds of lanthanide pyrochlores. The lines in the figure are almost parallel each other and have the pressure coefficient (= du/dp) are 1.24 × 10 -4 GPa -1 for Nd 2 Mo 2 O 7 , 1.55 ×

Nd 2 Mo 2 O 7 Sm 2 Mo 2 O 7 Gd 2 Mo 2 O 7 Tb 2 Mo 2 O 7 Dy 2 Mo 2 O 7 Dy 2 Mo 2 O 7

  FIG. 5: The spin-up density of states(DOS) of 4d Mo versus energy are shown. the black colored solid line corresponds to Dy 2 Mo 2 O 7 with external pressure(94.7 kB). At the Fermi level, We can see DOS become higher with pressure effect. The band width corresponding to lower t 2g bands also become wider. From Gd 2 Mo 2 O 7 , the lower t 2g gets to split into two peaks In Fig 5, we draw spin-up t 2 g bands of Mo 4d while Fermi level is set to zero energy.All of spin-down t 2g bands are empty and located at higher states than Fermi level. We can see that the DOS at the Fermi level grows gradually as pressure goes up. The width of occupied t 2g become wider with pressure. This indicate that conduction of Mo 4d electrons becomes easier when pressure is increased as is understood from usual thought in a metallic system. The increased DOS at the Fermi level explains more electrons can participates in conduction by the double exchange mechanism. Moreover, the variation of DOS shapes shows that weak Jahn-Teller distortion occurs.Although the double exchange is strengthened by pressures, experiments show that Gd 2 Mo 2 O 7 undergoes magnetic phase transition from FMM to SGI with chemical/ external pressure. This means that antiferromagnetic ordering by superexchange interaction is dominant even double exchange is enhanced with increased pressure 14 . Indeed, Neutron scattering measurements suggest an important role of superexchange interactions between lanthanide 4f electrons and Mo 4d electrons7 . They explain that

	3
	DOS [States/eV]
	0
	0	2
	Energy [eV]

12 G. Kresse and Furthmüller, Software VASP, Vienna, 2001.


Dorel Crişan
email: dcrisan@icf.ro

Nicolae Drȃgan

Maria Crişan

Mȃlina Rȃileanu

Ana Brȃileanu

Mihai Anastasescu

Adelina Ianculescu

Diana Mardare

Dumitru Luca

Virgil Marinescu

Antoniu Moldovan

Nicolae Drăgan

Mălina Răileanu

Ana Brăileanu

Crystallization study of sol-gel un-doped and pddoped TiO 2 materials

Keywords: A. Oxides, A. Thin films, B. Sol-gel process, C. X-ray diffraction, D. Crystal structure

Sol-gel nanostructured titania materials have been reported to have applications in areas ranging from optics via solar energy to gas sensors. In order to enhance the photocatalytic activity, there are many studies regarding the doping of TiO 2 material with either nonmetals (S, C, N, P) or metals (Ag, Pt, Nd, Fe). The present work has studied some un-doped and Pd-doped sol-gel TiO 2 materials (films and gels), with various surface morphologies and structures, obtained by simultaneous gelation of both precursors Ti(OEt) 4 and Pd(acac) 2. Their structural evaluation and crystallization behaviour with thermal treatment were followed by DTA/TG analysis, IR spectroscopy, FTIR, spectroellipsometry (SE), XRD and AFM. The influence of Pd on TiO 2 crystallization both for supported and un-supported materials was studied (lattice parameters, crystallite sizes, internal microstrains). The changes in the optical properties of the TiO 2 -based vitreous materials were correlated with the changes of the structure. The hydrophilic properties of the films were also connected with their structure, composition, and surface morphology.

Introduction

Among the non-conventional wet chemical processes, the sol-gel method based on hydrolysis and polycondensation of metal-organic precursors such as metal alkoxides may be considered an adequate procedure for high purity and homogeneous oxide materials preparation. In metal-organic derived vitreous and ceramic systems, the Vitreous coatings obtained by the sol-gel method, at low temperature processing, represent a promising application, and TiO 2 -based vitreous films were among the first ones in this respect [START_REF] Dislich | Amorphous and crystalline dip-coatings obtained from organometallic solutions: procedures, chemical processes and products[END_REF][START_REF] Dislich | Glassy and crystalline systems from gels, chemical basis and technical application[END_REF][START_REF] Yoldas | Antireflective coatings applied from metal-organic derived liquid precursors[END_REF].

Titanium dioxide thin films have applications in optics and electronics because of their excellent properties e.g. chemical and physical stability, high refractive index, good transmission in VIS and NIR regions, high dielectric constants, high electrical resistance and interesting catalytic properties. We have previously published results on vitreous reflecting TiO 2 films on aluminized commercial glass [START_REF] Zaharescu | Herstellung von Glasschichten durch die Sol-Gel Technik[END_REF], TiO 2 films containing transition metals [START_REF] Zaharescu | TiO 2 -based vitreous coatings obtained by sol-gel method[END_REF][START_REF] Crişan | Vanadium doped sol-gel TiO 2 coatings[END_REF][START_REF] Crişan | Structural and optical study of the Fe 3+ doped TiO 2 sol-gel coatings[END_REF][START_REF] Trapalis | TiO 2 (Fe 3+ ) nanostructured thin films with antibacterial properties[END_REF], TiO 2 films obtained from different Ti-alkoxides [START_REF] Zaharescu | TiO 2 -films obtained from different Ti-alkoxides[END_REF] and an AFM study of these [START_REF] Zaharescu | Atomic force microscopy study of TiO 2 films obtained by sol-gel method[END_REF].

Highly dispersed nanoparticles of noble metals, such as Pt, Pd, Rh, Ru and Au in mesoporous supports as titania, alumina, silica are widely used as catalysts in organic synthesis, petrochemistry, etc. In a study concerning the photocatalytic activity of noble-metal-loaded TiO 2 , Ranjit et al. [START_REF] Ranjit | Photocatalytic reduction of dinitrogen to ammonia over noble-metal-loaded TiO 2[END_REF] suggest that an ohmic contact is formed between the metal and semiconductor. Hence electrons can easily flow to the metal sites on TiO 2 under irradiation and the role of the metal is to act as an electron sink and thus to enhance the activity. However, it is difficult to introduce metal nanoparticles into mesopores by traditional impregnation methods, because they tend to deposit richly on outer surface of mesoporous materials and moreover it is difficult to control the loading amount by impregnation [START_REF] Yuan | Synthesis of Pd nanoparticles in La-doped mesoporous Titania with polycrystalline framework[END_REF].

A c c e p t e d m a n u s c r i p t

Sol-gel method easily allows the preparation of nanocomposite materials such as inorganic matrices in which a metal phase could be highly dispersed. Many studies deal with Pt dopant [START_REF] Facchin | Sol-Gel Synthesis and Characterisation of TiO 2 -Anatase Powders Containing Nanometric Platinum Particles Employed as Catalysts for 4-Nitrophenol Photodegradation[END_REF][START_REF] López | Effect of pH on the incorporation of platinum into the lattice of sol-gel titania phases[END_REF][START_REF] López | Phenylacetylene hydrogenation on Pt/TiO 2 sol-gel catalyst[END_REF]. Concerning Pd doping, generally, its dispersion by sol-gel method was made in SiO 2 [START_REF] López | Synthesis and spectroscopic characterization of Pt and Pd silica supported catalysts[END_REF][START_REF] López | Structure of Pd/SiO 2 Sol-Gel and Impregnated Catalysts[END_REF], Al 2 O 3 [START_REF] Othman | On the characteristics and hydrogen adsorption properties of a Pd/γ-Al 2 O 3 prepared by sol-gel method[END_REF][START_REF] Noh | Characteristics of the Pd-only three-way catalysts prepared by sol-gel method[END_REF] and in vitreous matrices [START_REF] Carturan | Phenylacetylene half-hydrogenation with Pd supported on vitreous materials having different chemical composition[END_REF]. There are few references so far regarding the use of Pd 2+ as a dopant in the case of a sol-gel prepared TiO 2 matrix.

This paper presents our results on processing and structural and optical characteristics of un-doped and Pd-doped sol-gel TiO 2 materials (films and gels). The influence of Pd on TiO 2 crystallization for both supported and un-supported materials was studied in terms of lattice parameters, crystallite sizes, internal microstrains. Films wettability properties are also discussed in correlation with their composition, structure and surface morphology.

Experimental

Both un-doped and Pd-doped TiO 2 films were synthesized using the alkoxide route of the sol-gel method. In all cases the tetraethyl orthotitanate 95% (Merck), Hamilton motorized syringe (10 mL), operated by a computer, allowed controlling drop volume with 0.5 nL accuracy. Drop volumes of 0.5 µL were chosen to avoid drop shape alteration due to gravitational force and to diminish the evaporation effects. The samples were irradiated for photoactivation using filtered UV light (100 mW/cm 2 at 366

A c c e p t e d m a n u s c r i p t

nm) from a 150 W high-pressure mercury lamp. The CA angle versus the UV irradiation dose was monitored until the surface photoactivation reached the saturation.

After that, the UV lamp was removed and the time dependence of the contact angle was registered for 64 h, with an 8h step, keeping the samples in a dark place between measurements.

Results and discussion

Continuous and homogeneous films with good adherence to the support were obtained in the experimental conditions presented above.

Optical properties of the films as derived from SE measurements are presented in Table 2 andFigures 1 and2 Table 2 Figure 1 (a,b)

The packing density (relative to anatase phase) of the films was calculated [START_REF] Patil | Optical properties and adhesion of air oxidized vacuum evaporated bismuth thin films[END_REF] using equation (1):

A c c e p t e d m a n u s c r i p t

( ) ( ) ( )( ) 1 ) ( 2 ) ( 2 ) ( 1 ) ( ) ( 2 2 2 2 - + + - = λ λ λ λ λ A S A S n n n n PD (1)
where n A (λ) represents the refractive index dispersion of the anatase phase of TiO 2 [START_REF] Borgogno | Automatic determination of the optical constants of inhomogeneous thin films[END_REF] while n s is the dispersion of the refractive index of the sample (Figure 2 a-d). As will be seen further, according to the XRD data, the anatase was found to be the main phase in the analyzed gels and, therefore, was used as reference in obtaining the packing density of the samples. Nevertheless, the rutile phase can be also used as reference, producing a slight shift of the results towards lower values of the Packing Density.

Figure 2 ( a-d)

From SE measurements one can observe the increase of the refractive index and consequently the decrease of the film thickness with temperature as a result of the film densification. The presence of Pd induces an absorbtion component k≠0 [START_REF] Johnson | Optical constants of transition metals: Ti[END_REF]. The 0.5% Pd-TiO 2 thermally treated bi-layers films have the best packing density.

FT-IR spectra of un-doped and Pd-doped TiO 2 films thermally treated at 300 and 500°C are presented in Figure 3 (a,b). respectively, due to water and alcohol removal, associated with significant mass losses on TG curves. In the temperature range 150 -400°C, thermo-oxidative degradation processes of the organic matter, more pronounced for Pd-doped TiO 2 as exothermic peaks at 186 and 352°C, can be observed for both gels, on DTA and TG curves. At 352°C, an overlapping of the amorphous to anatase phase transformation, is not excluded. For the un-doped TiO 2 gel (Fig. 4a), a supplementary exothermal effect at 517°C due to rutile formation is evidenced.

Since the films are very thin (thickness derived from SE measurements ranges between 400 and 510 Å for thermally treated films, one-layer film, and 710 to 950 Å for two-layers film), their diffraction spectra, no matter of the temperature of the thermal treatment, are covered by the diffraction spectra of the support, which is amorphous.

Therefore, the crystallization study was accomplished on un-supported gels resulted from the gelation of the starting solutions used for the films preparation. The incipient crystallization (Ti-O bond formation) was also detected by IR spectroscopy.

IR spectra of the thermally treated (300-500°C) un-doped and Pd-doped TiO 2 gels show the following vibration bands: 3440 cm -1 (ν OH ), 1640 cm -1 (δ HOH ), 670 cm -1 (ν Ti-O of isolated tetrahedral TiO 4 ), 540 cm -1 (ν Ti-O of condensed octahedral TiO 6 ) and

A c c e p t e d m a n u s c r i p t

With the increase of the temperature, the intensity of the vibration bands due to Ti-O bonds increases and they are better defined. At the same time, the vibration bands due to the presence of molecular water and structural OH -diminish.

XRD results obtained with XRAY5.0 program [START_REF] Drăgan | Modeling, Aproximations and Program for Estimating some Structural Factors with XRD[END_REF][START_REF] Drăgan | Fast Analysis of the XRD profile with the XRAY3.0 Program[END_REF] for the same gels are presented in Table 3 (the identified phases, the lattice constants and some microstructural factors).

Table 3 XRD results presented in Table 3 evidence the presence of anatase as single crystalline phase in all samples thermally treated at 300°C, respectively un-doped and Pd-doped TiO 2 , no matter of the dopant concentration. At 500°C beside anatase, rutile phase could be observed, its presence depending strongly on the dopant concentration.

With the Pd concentration increase, the rutile crystallinity decreases, vanishing at 1%

Pd.

The profile analysis of the XRD peaks shows that, for 0.5% Pd concentration in the material annealed at 500°C, the anatase crystallite size has the smallest value (307 Å), and the local strain is the highest (1.9 x 10 -3 ). Our calculations show a slight decrease (~5%) of the interplanar spacing along the [200] and [004] crystallographic axes. This involves both a local densification of the anatase lattice (in agreement with SE results) and a decrease of the rutile cristallinity compared to the un-doped sample.

The further increase of Pd concentration to 1% leads to a slight relaxation of the lattice and, consequently to the increase of the crystallites size.

In spite of different mechanisms, the induced photo-catalytic activity and hydrophilicity of the surface are two phenomena strongly inter-related [START_REF] Fujishima | Titanium dioxide photocatalysis[END_REF]. Therefore,

A c c e p t e d m a n u s c r i p t

CA measurements are often used as an (indirect) indication of the photocatalytic activity of the photocatalytic thin films.

The hydrophilicity results are presented in Figures 5 and6.

Figure 5 Figure 6 The films are considered super-hydrophilic when CA values of water with their surface are below 10 degrees [START_REF] Luca | Increasing surface hydrophilicity of titania thin films by doping[END_REF]. For our non-irradiated film samples, the CA values range between 50 and 72 degrees. As shown in Fig. 5, a sharp decrease of the contact angle towards values lower than 10 degrees, for UV doses ranging between 6 and 10 J/cm 2 , can be observed. An exception occurs for the 1% Pd-doped TiO 2 film thermally treated at 500°C which presents a slower decrease, reaching only 18 degrees for an UV irradiation dose of 36 J/cm 2 . With this exception we may consider that the studied films become super-hydrophilic by UV irradiation. An inset was added to Fig. 5, where the saturation values of the CA were plotted with respect to Pd content (0-1%), for the thermally treated films. The thermal treatment of the samples at 300°C leads to an increased super-hydrophilic performance of the films, especially for the sample with the highest Pd concentration. At 500°C, the annealing of the films results in a marked degradation of the hydrophylicity of the surface.

Figure 6 shows the CA recovery vs. time, when keeping the samples in darkness, following the initial saturation photo-activation. As a general rule, the decay of the surface photo-activation is achieved after more than 40 hours, the CA values reaching the prior-to-irradiation values. As seen in Figs. 5 and6, the activation-decay processes are reversible to a high extent.

A c c e p t e d m a n u s c r i p t

To explain the observed differences in the UV light-induced hydrophilicity in the samples, we have to take into account the combined effects of the surface roughness, the amount of anatase phase, and the effect of the dopant. Additionally, the increased surface roughness results in decreased values of the CA of the surface with water [START_REF] Luca | Increasing surface hydrophilicity of titania thin films by doping[END_REF],

thus adding its effect to the photo-enhanced hydrophilicity.

There is a good agreement between the surface roughness values obtained by AFM and the initial values of the CA (for the non-irradiated samples) for all the films. Although the samples are very smooth with RMS values lower than 10 nm (Figure 7) they present a super-hydrophilicity. The slower decrease of the CA values versus UV irradiation dose and its high final value (18 deg at 36 J/cm 2 ) for 1% Pd concentration can be explained by a slight relaxation of anatase lattice and consequently by the increase of the crystallites size.

Conclusions

A crystallization study of sol-gel un-doped and Pd-doped TiO 2 materials (films and gels) was performed.

The influence of Pd on TiO 2 crystallization both for supported and un-supported materials was studied (lattice parameters, crystallite sizes, internal strains). The profile analysis of the XRD peaks, show that for 0.5% Pd concentration in the material annealed at 500°C, the anatase crystallite size has the smallest value (307 Å), and the local strain is the highest (1.9 x 10 -3 ). Our calculations show a slight decrease of the All the films present super-hydrophilicity properties which were connected with their structure, composition, and surface morphology.

A c c e p t e d m a n u s c r i p t

A c c e p t e d m a n u s c r i p t

  inorganic network structure is formed by mixing the starting components at the molecular level in solution. The solids (films, fibers, powders and bulk materials) with predetermined structure can be obtained by varying the experimental conditions during the gel formation stage, allowing an excellent compositional control of the product.

  . The experimental ellipsometric spectra were modeled with the multilayer and multicomponent Bruggemann's effective medium approximation (BEMA) model[START_REF] Bruggemann | Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen[END_REF]. The fitting parameters were: the thickness of the layers and the volume fraction of the components (amorphous TiO 2 , the anatase phase of TiO 2 , Pd, air-filled void). The optical constants of the films were obtained from the best fit. The dispersion of the refractive index (n) and of the absorption coefficient (k) versus wavelength (λ) for 0.5% Pd-TiO 2 bi-layers films is presented in Figure1 ((a) and (b)).

Figure 3 .

 3 Figure 3.

A c c e p t e d m a n u s c r i p tFigure 4

 4 Figure 4 (a, b) presents TGA/DTA results for un-doped and 1 % Pd-doped TiO 2

Figure 4 (

 4 Figure 4 (a, b)

Figure 7 (

 7 Figure 7 (a-c) presents AFM images of pure TiO 2 and Pd-doped TiO 2 films

Figure 7 .

 7 Figure 7.

A c c e p t e d m a n u s c r i p t

  interplanar spacing along the[200] and [004] crystallographic axes. This involves both a local densification of the anatase lattice (in agreement with SE results) and a decrease of the rutile cristallinity compared to the un-doped sample. The further increase of Pd concentration to 1% leads to a slight relaxation of the lattice and, consequently to the increase of the crystallites size.

Figure 1 .

 1 Figure 1.

Figure 2 .

 2 Figure 2. Packing density versus wavelength (λ) for Pd-doped TiO 2 films, as-deposited

Figure 3 .Figure 4 .Figure 5 .

 345 Figure 3. FT-IR spectra of un-doped and Pd-doped TiO 2 films thermally treated at

A c c e p t e d m a n u s c r i p tFigure 6 .Figure 7 .

 67 Figure 6. Time dependence of the contact angle, measured under surface de-activation

A c c e p t e d m a n u s c r i p t stirring

  . The nitric acid, HNO 3 65% Merck, was used as catalyst to obtain a final pH of the sol in the range 3-3.5. The experimental details are presented in Table1.

	Table 1
	Supported TiO 2 based vitreous films have been obtained by dip coating on microscopic
	slide substrates. The un-supported porous materials resulted from the gelling of the
	starting solutions at room temperature, were dried at 80°C and then thermally treated in
	air, at 300, 400, 500°C/1h according to DTA/TGA results. The densification of the films
	was realized at the same temperatures (Table 1). The second layer was deposited after
	the densification of the first one at 300°C.
	The following characterization methods for films and un-supported gels were used:
	spectroellipsometry (SE) with null-type Ellipsometer in 0.4-0.7 µm wavelength range at
	an angle of incidence of 70°, FT-IR with a Perkin Elmer Spectrum 100 spectrometer in
	the 550-4000 cm -1 range, XRD performed using a Bruker-AXS, D8 Advance type with
	copper radiation, AFM images with a Nomad microscope from Quesant, in the
	intermitent contact regim, hydrophilicity measurements, DTA/TGA using a Netzsch
	STA 409 PC Luxx thermal analyzer, and IR spectroscopy performed on a Carl Zeiss
	Jena Specord M-80 IR spectrophotometer with KBr pellets technique. A home-built
	2 samples goniometer allowed the study of TiO 2 films hydrophilicity, by measuring the contact
	containing 0.5% and 1 wt% Pd related to TiO 2 were prepared. absolute ethanol (Riedel angle (CA) between the deionized water and film surface. The CA measurements were
	de Haën), min.99.8 Vol % was used as solvent for both titania and palladium performed at room temperature under 65% environment humidity conditions. A
	precursors. The hydrolysis reaction took place in the presence of the water, added in
	substoichiometric quantity, in a closed system, in nitrogen atmosphere, under vigorous

Ti(OC 2 H 5 ) 4 , represented the TiO 2 source. The source of Pd was Pd acetylacetonate, 99%, (Merck), Pd(acac) 2 , C 10 H 14 O 4 Pd. The doped materials were obtained by simultaneous gelation of both precursors in the sol-gel process. Two Pd-TiO

Table 2 -

 2 The thickness (d) of the films, as-deposited and thermally treated, and the volume fraction of the components

			0.5 % Pd-doped TiO 2 mono layer films
	Sample	d(Ǻ)	TiO 2 (%)	Anatase(%)	Pd(%)	Air(%)	Error*10 -7
	I	630	65	0.05	0.45	34.50	1992
	300 C	350	67.21	14.63	0.54	17.62	1781
	400 C	330	66.74	14.75	0.54	17.97	1320
	500 C	460	65.93	14.57	0.54	18.95	1006
			0.5 % Pd-doped TiO 2 bi-layers films
	I	1010	83.32	0.07	0.57	16.04	129931
	300 C	710	70.92	14.87	0.40	13.81	2772
	400 C	730	70.97	14.84	0.40	13.78	5872
	500 C	720	70.91	14.87	0.40	13.81	2107
			1 % Pd-doped TiO 2 mono layer films
	I	990	61.58	0.05	0.81	37.57	2864
	300 C	470	67.30	5.97	0.91	25.82	1723
	400 C	510	65.17	5.78	0.88	28.16	1482
	500 C	400	69.14	5.96	0.91	23.99	1783
			1 % Pd-doped TiO 2 bi-layers films
	I	1150	74.09	0.04	0.71	25.16	98605
	300 C	950	70.94	6.87	1.05	21.14	7697
	400 C	760	70.37	6.81	1.04	21.77	3994
	500 C	770	72.51	7.02	1.07	19.40	2674

Table 3 .

 3 XRD data of the un-doped and Pd-doped TiO 2 gels

	Sample	Identified		Lattice constants	Microstructural
							factors
		phases	a	c	u.c.v.	<D>	10 +3 x<S>
		[%]	[Å]	[Å]	[Å 3 ]	[Å]
	TiO 2 /300°C	A	3.80	9.57	137.8	173	2.1
	TiO 2 +0.5%	A	3.79	9.51	136.8	185	6.6
	Pd/300°C
	TiO 2 +1% Pd/300°C	A	3.79	9.51	136.7	169	4.5
	TiO 2 /500°C	A -73.96	3.78	9.51	135.9	442	1.1
		R -26.04	4.59	2.96	62.3	405	1.2
	TiO 2 +0.5%	A -93.98	3.78	9.50	135.7	307	1.9
	Pd/500°C
		R -6.02	4.73	2.91	65.2	295	1.6
	TiO 2 +1% Pd/500°C	A	3.78	9.53	136.3	348	1.8
	A -Anatase; R -Rutile;
	a, c -lattice parameters; u.c.v. -unit cell volume; D -crystallite size; S -internal
	microstrains

Acknowledgements

This work was supported partially by the National Research Program CEEX, grants No. 37/2005-2007, 6113/2005-2008, 103 (12752)/2006-2008 and S5/23-2005.

A c c e p t e d m a n u s c r i p t


Marcin Przystalski

Paweł L Krajewski

Constrained estimators of treatment parameters in semiparametric models

Keywords: cubic smoothing splines, estimation, hypothesis testing, loess

Semiparametric models are generalizations of parametric regression models. We present a method of estimation of treatment effects in a semiparametric model with one smoothing term under additional conditions on their linear functions and its application to hypothesis testing.

A c c e p t e d m a n u s c r i p t 1 Introduction

Semiparametric models are one of the possible generalizations of parametric regression models. A model with one smooth term can be expressed in the form [START_REF] Durban | Approximate standard errors in semiparametric models[END_REF])

y = Xβ + f + e, (1)
where y = (y 1 , . . . , y n ) is a vector of observed values, X is an n × p design matrix, with 1, the column vector of 1' s, included in its column space, β is a p × 1 vector of treatment effects, f is a smooth term, that is, a function of a covariate z, e is a vector of i.i.d. errors with expected value 0 and covariance matrix σ 2 I n .

The term f in model ( 1) is called a smoother. Hastie and Tibshirani (1989) described several smoothers, e.g. cubic smoothing splines and locally weighted running line smoother (loess). [START_REF] Durban | Approximate standard errors in semiparametric models[END_REF] proposed a test for linear functions of treatment effects in semiparametric models with any number of smoothing terms based on approximate standard errors. To avoid using such approximations the following problem has to be considered: how to estimate the treatment effects in semiparametric models under some additional conditions on their linear functions, in particular under conditions forming a set of contrasts of treatment effects, and secondly how to apply these estimates in hypothesis testing.

The purpose of this paper is to give the answer to these two questions. In section 2 we introduce basic notions related to estimation in semiparametric models. In section 3 we discuss construction of the estimates under additional conditions. In section 4 we describe a testing procedure. Section 5 contains an example based on real data. A few concluding remarks are given in the final section.

A c c e p t e d m a n u s c r i p t 2 Estimation in semiparametric models

Usually, semiparametric models are fitted by the backfitting algorithm. For model (1) the estimates of β and f can be written explicitly as

β = Ay, (2)
f = S y -X β = S (I -XA) y,
where A = X (I -S) X

-1

X (I -S) and S is a smoother matrix which depends on smoothing parameter δ, provided that X (I -S) X is invertible. In this paper we do not consider the problem of choosing the smoothing parameter. Some methods of doing this may be found in [START_REF] Durban | The practical use of semiparametric models in field trials[END_REF], [START_REF] Eubank | Estimation in partially linear models[END_REF][START_REF] Schimek | Estimation and inference in partially linear models with smoothing splines[END_REF].

When, instead of the backfitting algorithm, we consider a penalized least squares criterion

y -Xβ -f 2 + f S -1 -I f , (3)
we derive the same estimates as (2) (see Buja et al., 1989, page 500). However this criterion can be used only for smoothers with a symmetric smoother matrix S.

If we consider approach proposed by [START_REF] Speckman | Kernel smoothing in partial linear models[END_REF], which minimizes

y -Xβ -S (y -Xβ) 2 , ( 4
)
where S is an arbitrary smoother matrix, we obtain a solution different from (2), namely

β = Ay, (5)
f = S y -X β , where A = X (I -S) (I -S) X -1
X (I -S) (I -S). The Speckman approach was considered by [START_REF] Buja | Linear smoothers and additive models[END_REF], [START_REF] Eubank | Estimation in partially linear models[END_REF], [START_REF] Schimek | Estimation and inference in partially linear models with smoothing splines[END_REF].

A c c e p t e d m a n u s c r i p t

For both discussed cases, the variance of the estimator of β is σ 2 AA . If H = S + (I -S) XA is the "hat" matrix for (1), then y = Hy = X β + f and, following [START_REF] Durban | Approximate standard errors in semiparametric models[END_REF], σ 2 can be estimated as

σ 2 = 1 n-trH n i=1 (y i -ŷi ) 2 .
3 Constrained estimators in semiparametric models

Let us assume that we have an additional condition for the vector of treatment parameters of the form

Rβ = r. (6)
We would like to find estimate b of β in model ( 1) such that condition ( 6) is satisfied. For this reason we consider two cases: when the smoother matrix S is symmetric and a second case when S is arbitrary. The following theorem gives the solution of this problem for the first case.

Theorem 3.1 If the smoother matrix S is symmetric then the estimate of β in model (1)

obtained by using the penalized least squares criterion (3), satisfying condition (6), is equal

to b = β + D -1 R RD -1 R -1 r -R β and its covariance matrix to Var (b) = σ 2 CC , where D = X (I -S) X and C = I -D -1 R RD -1 R -1 R D -1 X (I -S).
Proof: Let us assume that the smoother matrix S is symmetric and we want to find an estimate of β in model ( 1) satisfying condition (6) using the criterion (3). For this purpose we construct the function

F (β, f , λ) = y -Xβ -f 2 + f S -1 -I f -2λ (Rβ -r) .
A c c e p t e d m a n u s c r i p t

∂F (β, f , λ) ∂β = -2X (y -Xβ -f ) -2R λ = 0 ∂F (β, f , λ) ∂f = -2 (y -Xβ -f ) + 2 S -1 -I f = 0 ∂F (β, f , λ) ∂λ = Rβ -r = 0.
If we get f from the second equation as

f = S (y -Xβ)
and we put it into the first equation, we obtain

X [y -Xβ -S (y -Xβ)] + R λ = 0.
Solving this equation with respect to β we get that

β = X (I -S) X -1 X (I -S) y -R λ .
Such β we put into the third equation and we have

R X (I -S) X -1 X (I -S) y -R λ = r.
If we put D = X (I -S) X, and notice that X (I -S) X -1

X (I -S) y is equal to β in (2), we get the following equality

R β -RD -1 R λ = r.
Because RD -1 R > 0, we may write the optimal λ as

λ = RD -1 R -1 R β -r .
Finally, the estimate of β may be written as

b = β + D -1 R RD -1 R -1 r -R β .
= β + Z -1 R RZ -1 R -1 r -R β
and its covariance matrix to Var (b) = σ 2 BB , where Z = X (I -S) (I -S) X,

and

B = I -Z -1 R RZ -1 R -1 R Z -1 X (I -S) (I -S).
Proof: Transforming y and X in model ( 1), to y = (I -S) y and X = (I -S) X (see [START_REF] Speckman | Kernel smoothing in partial linear models[END_REF][START_REF] Eubank | Estimation in partially linear models[END_REF][START_REF] Schimek | Estimation and inference in partially linear models with smoothing splines[END_REF], applying RLSE for linear models and

(5) we get the thesis.

Hypothesis testing

Let us assume that we tested a hypothesis H 0 :

β = 0 against H 1 : β = 0. If we denote by y G i y, i = 1, 2, G 1 = (I -Q) (I -Q) and G 2 = (I -H) (I -H), the residual sum
of squares for model under H 0 and the non restricted model, respectively, then the test of null hypothesis against the alternative is based on the variance ratio

F = (y G 1 y -y G 2 y) /ν 1 y G 2 y/δ 1 = y (G 1 -G 2 ) y/ν 1 y G 2 y/δ 1 . (7)
According to Cleveland and Devlin (1988), this test statistic has an approximate F distribution with γ 1 = ν 2 1 /ν 2 and γ 2 = δ 2 1 /δ 2 degrees of freedom, where

ν 1 = tr (G 1 -G 2 ), ν 2 = tr (G 1 -G 2 ) 2 , δ 1 = tr (G 2 ) and δ 2 = tr G 2 2 .
If the general hypothesis H 0 : β = 0 is rejected, then we may be interested in testing hypotheses of the type H 0,R : Rβ = 0 against H 1,R : Rβ = 0, with an appropriate R.

From theorem 3.1 or proposition 3.2 with r = 0 we know how to obtain the estimate of β under this null hypothesis. Denoting again by y G i y, i = 1, 2, the residual sum of squares for the model with restriction and for the model without restriction, and using test statistic (7), we reject the null hypothesis when F > F α .

A c c e p t e d m a n u s c r i p t 5 Example

To illustrate the method we use the yield data from the spring barley experiment used by [START_REF] Durban | The practical use of semiparametric models in field trials[END_REF]. The data concern 272 varieties. Among them, first 12 were established as controls and 260 were tested varieties. The trial was sown in a row and column design with 2 replicates. In each replicate there were 8 rows and 34 columns. Plot yields were provided in tonnes per hectare.

We take as a smoother in our semiparametric model ( 1) the two dimensional loess smoother across rows and columns. As a smoothing parameter we take δ = 125/544 ≈ 0.2298. We compare the thirty-first variety with the controls. We can write the null hypothesis in the form

H 0 : Rβ = 0,
where the i-th row of R has -1 in th i-th position and 1 in the 31st. Using proposition 3.2 with r = 0 and ( 7) we get the values of the residual sums of squares and the test statistic as in Table 1. The degrees of freedom are γ 1 = 27.94 and γ 2 = 543.7

Table 1 Now we compare the thirty-first variety with each control variety separately, i.e. we want to test hypotheses

H 0 : r i β = 0 i = 1, . . . , 12, H 1 : r i β = 0,
where r i is the i-th row of R. Applying proposition 3.2 with r = 0 and (7) we get that the difference between thirty-first variety and first, second, third, fifth, and eighth is not significant. Contrasts between thirty-first variety and fourth, sixth, seventh, ninth, tenth, Table 2 If we apply the proposition of calculating standard errors of contrasts described by [START_REF] Durban | Approximate standard errors in semiparametric models[END_REF] to this example and use as the test statistic the ratio

F = r β 2 Var r β
with 1 and 272 degrees of freedom, we get that there are no significant differences between thirty first variety and first, second, third, fourth, fifth, eighth, tenth. The rest of the contrasts is significant.

Table 3 If for the same data set we consider the same null hypothesis, but as a smoother we use the cubic smoothing spline over row number and as δ we take 0.2, then using theorem 3.1 with r = 0 we get that the test statistic has γ 1 = 72.48 and γ 2 = 500.9 degrees of freedom, and we can not reject the null hypothesis as shown in Table 4.

  If S is an arbitrary smoother matrix then the estimate of β in model (1) under criterion (4) such that condition (6) is satisfied, is equal to b

Table 4 6

 4 DiscussionWhen the general hypothesis concerning treatment effects is rejected, usually more than one treatment comparison is of interest. The proposed method allows us to test a set of linear functions of treatment effects in a semiparametric model, not just only one function like in the method based on the approximate standard error. Both methods used in the example are approximate: ours because we use approximation of the F distribution, and the second because it uses approximation of standard errors described byDurban et al.

(1999)

.

Table 1 :

 1 Results of testing simultaneous comparison of variety no. 31 with the controls using loess smoother.

	Model	Residual SS Variance ratio P-value
	(a) without restriction	19.13	-	-
	(b) with restriction	22.91	4.260	< .001

Table 2 :

 2 Hypothesis testing to investigate the significance of the individual contrasts using loess smoother.

	Contrast Residual SS Degrees of freedom Variance ratio P-value
			γ 1	γ 2
	β 31 -β 1	19.14	4.253	543.7	0.1057	0.9840
	β 31 -β 2	19.20	4.543	543.7	0.9989	0.4135
	β 31 -β 3	19.25	4.434	543.7	1.593	0.1683
	β 31 -β 4	19.33	4.566	543.7	2.695	0.0241
	β 31 -β 5	19.24	4.606	543.7	1.549	0.1783
	β 31 -β 6	19.56	4.662	543.7	5.883	< 0.001
	β 31 -β 7	19.59	4.567	543.7	6.296	< 0.001
	β 31 -β 8	19.13	4.528	543.7	0.0004	1
	β 31 -β 9	20.05	4.407	543.7	12.67	< 0.001
	β 31 -β 10	19.44	4.409	543.7	4.255	< 0.001
	β 31 -β 11	19.51	4.596	543.7	5.190	< 0.001
	β 31 -β 12	19.40	4.600	543.7	3.655	0.0039

Table 3 :

 3 Hypothesis testing to investigate the significance of the contrasts using approximate standard errors.

	Contrast	r β	Standard error F statistic P-value
	β 31 -β 1	0.0590	0.2803	0.04	0.8334
	β 31 -β 2	0.3107	0.2839	1.20	0.2747
	β 31 -β 3	0.3913	0.2835	1.90	0.1687
	β 31 -β 4	0.4677	0.2839	2.71	0.1006
	β 31 -β 5	0.3959	0.2845	1.94	0.1652
	β 31 -β 6	0.7155	0.2861	6.30	0.0127
	β 31 -β 7	0.7347	0.2843	6.68	0.0103
	β 31 -β 8 -0.0028	0.2839	0.00	0.9922
	β 31 -β 9	0.9954	0.2817	12.49	< 0.001
	β 31 -β 10 -0.5281	0.2830	3.48	0.0631
	β 31 -β 11 0.6566	0.2837	5.36	0.0214
	β 31 -β 12 0.6063	0.2883	4.52	0.0345

Table 4 :

 4 Results of testing simultaneous comparison of variety no. 31 with the controls using cubic smoothing spline. The variance ratio is calculated with (a) as the denominator.

	Model	Residual SS Variance ratio P-value
	(a) without restriction	2.103	-	-
	(b) with restriction	3.248	0.3843	1

Acknowledgments

The authors would like to thank referees for careful reading and Christine Hackett for help in obtaining the barley data set. This work was partially supported by the Polish Ministry of Education and Science grant SPB/COST/KN/DWM 74/2005-2008.