L V B Diop

O Isnard

G Ballon

J P Itié

Investigation of the itinerant metamagnetic system Hf0.75Ta0.25Fe2 under extreme conditions of pressure or magnetic field

Keywords: neutron powder diffraction, spin reorientation, volume magnetostriction, spin fluctuations

The magnetic and structural properties of Hf0.75Ta0.25Fe2 have been investigated under the duplicate conditions, i.e., high magnetic fields and high pressures, by combining magnetization measurements up to 26 T, neutron powder diffraction experiments up to 14.5 T, and angle dispersive synchrotron x-ray diffraction up to 10 GPa. The intermetallic compound Hf0.75Ta0.25Fe2 exhibits an antiferromagnetic (AFM) ground state below the Néel temperature TN = 342 K. In the AFM structure, the Fe-6h magnetic moments (1.02  at 2 K) align along the c axis while Fe atoms at 2a sites are not magnetically ordered. We further show that the AFM ground state gets transformed into a ferromagnetic (FM) state via a field-induced metamagnetic phase transition. Magnetic-field dependent neutron diffractograms demonstrate that FM phase sets in with a change of the easy magnetization direction from axial to basalplane and an extremely anisotropic lattice expansion. The unit cell expands drastically in the basal plane upon increasing the applied magnetic field, while its dimension along c-axis shrinks slightly; resulting in a huge positive volume magnetostriction V/V = 0.78 %. The application of hydrostatic pressure leads to the continuous decrease of the cell parameters. The shrinkage of the hexagonal cell is anisotropic with a larger compression along the six-fold symmetry axis c. A bulk modulus of K0 = 247 GPa has been deduced from the pressure-volume relationship.

Introduction

The itinerant electron metamagnetism (IEM), i.e., field-induced first-order transition from a low magnetic moment state to a high magnetic moment state, is an essential physical phenomenon, describing magnetic properties of intermetallic compounds, demonstrating technologically important giant magnetocaloric, magnetoelastic and magnetoresistive effects [1][2][3][4]. The discovery of giant magnetocaloric effect in the itinerant metamagnetic system La(Fe,Si)13 [1], has opened new perspectives for room temperature magnetic refrigeration. This has triggered a spectacular upsurge of interest in magnetic cooling and the development of many researches from the viewpoints of both fundamental physics and technological applications.

One interesting example of IEM system is the Laves phase compounds Hf1-xTaxFe2. In the whole concentration range of x, the pseudo-binary system Hf1-xTaxFe2 adopts the hexagonal MgZn2-type structure (C-14 Laves phase, space group P63/mmc). There are two inequivalent positions for Fe atoms, 2a and 6h, and one position, 4f for Hf/Ta atoms in this structure. The magnetic ground state changes at a Ta concentration of about x = 0.25 from ferromagnetic (FM) for Hf rich to antiferromagnetic (AFM) for Ta rich side . In the intermediate concentration range 0.1 < x < 0.25, the alloys exhibit temperature-induced magnetic transitions from ferromagnetic (FM) to antiferromagnetic (AFM) order and then to the paramagnetic (PM) state upon heating [5][6][7][8][9][10][11][12][13]. In the FM ground state, the magnetism comes from Fe atoms on the two crystallographically inequivalent sites, 2a and 6h, both carrying a sizable magnetic moment.

The high-temperature AFM phase has a complex spin configuration with two independent Fe sublattices: only 3/4 of the Fe atoms are involved in the AFM arrangement, namely those located on 6h crystal sites, whereas Fe atoms occupying 2a Wyckoff positions carry no ordered magnetic moment [5,8,10,13,20]. The FM-AFM magnetic transition is of first order and is accompanied by an isomorphic (without change of symmetry) lattice collapse, large magnetocaloric [14][15][16], magnetoresistive [6,10] and magnetovolume effects [6,12,[17][18][19] as well as a giant negative thermal expansion [8,9,11]. Staircase-like transitions and anomalous thermomagnetic irreversibility arising due to kinetic arrest have also been observed in Hf1-xTaxFe2 compounds (x = 0.225, 0.230, and 0.235) [21][22][23]. Interestingly, peculiar and unconventional thermal effects across the FM-AFM first-order transition were most recently discovered in Hf0.82Ta0.18Fe2 [7]. Such thermal effects result in warming (cooling) when heat is extracted from (supplied to) the material, thus providing indisputable evidence of metastable superheated FM and supercooled AFM states [7]. Changes from ferromagnetism to antiferromagnetism can be induced by external pressure; for instance, ferromagnetic Hf0.825Ta0.175Fe2 becomes AFM at pressures beyond 0.75 GPa ( critical pressure) that corresponds to the decrease of volume of only 0.5% [6]. An investigation of the interatomic Fe-Fe exchange coupling coefficients and their lattice dependence demonstrated that the ferromagnetic interaction with the first shell is remarkably reduced upon shrinkage of the a cell parameter [19]. Another fascinating result of the theoretical calculations in Ref. [19] is that the Fe6h-Fe6h coupling is always positive meaning that these interactions will favour a ferromagnetic spin configuration within the atomic plane containing the Fe6h positions with Fe6h as near neighbours. The magnetic phase diagram of this pseudo-binary compounds Hf1-xTaxFe2 (0.1 < x < 0.25) has been qualitatively explained on the basis of the spin fluctuations theory of itinerant-electron system developed by Moriya and Usami (M-U) [24].

Up to now, the extensive investigations devoted to the Hf1-xTaxFe2 compounds have mainly focused on the intermediate composition range 0.1 < x < 0.25. It appears there have been no direct measurements of the magnetic structure of the compounds with x ≥ 0.25, that is, the precise magnetic structure of the AFM ground state remains an open question.

The present work aims at answering the open questions on the basis of an experimental study carried out on purposefully produced compound of composition Hf0.75Ta0.25Fe2. To determine the magnetic structures and understand their coupling with the lattice, we conducted a detailed neutron powder diffraction study as a function of both temperature and applied magnetic field. In addition, a careful look at the literature data led us to notice that despite the extensive investigations noted above, there appears to have been no experimental works addressing the high-pressure behaviour of the crystallographic sublattice in the series of compounds (Hf,Ta)Fe2. To contribute to the understanding of the role of volume in the pseudobinary system (Hf,Ta)Fe2, we have investigated the effect of hydrostatic pressure (up to 10 GPa) on the structural properties of Hf0.75Ta0.25Fe2 by means of high-resolution synchrotron x-ray diffraction.

Experimental methods

The polycrystalline alloy of composition Hf0.75Ta0.25Fe2 was synthetized by arc melting high-purity constituting metals (at least 99.95%) under a purified argon gas atmosphere. The sample was melted several times with the button flipped over after each melting, which is usually sufficient to achieve compositional homogeneity. The so-obtained ingot was wrapped in tantalum foil, sealed in an evacuated quartz tube and subsequently annealed in a resistive furnace as follows. It was heated up to 1273 K at a rate of 5 K/minute and kept at this temperature for 7 days. Then, the power was switched off and the furnace quickly cooled down to room temperature. The room temperature crystal structure and phase purity of the synthetized material was checked by standard x-ray powder diffraction using a Siemens D5000 diffractometer in reflection mode with the Bragg-Brentano geometry and Co-Kα radiation.

Magnetization curves were measured in steady magnetic fields up to 10.5 T at temperatures ranging from 2 K to 500 K using extraction-type magnetometers [25]. The magnetization measurements were extended up to 26 T using a non-destructive pulsed-field magnet at the French National High Magnetic Field Laboratory (LNCMI) in Toulouse. A single 1.25 MJ capacitor bank was employed. When fully charged, this could produce a maximum field of 53 T with a rise time of about 25 ms and a total pulse duration of 150 ms. In our measurements the capacitor module was charged to about one-half. A detailed description of the pulsed-field facility can be found in Ref. [26]. All pulsed-field data were calibrated against the magnetization recorded in steady magnetic fields.

Neutron powder diffraction (NPD) experiments were performed at the Institut Laue Langevin (ILL) in Grenoble, France. About 3 g of fine powder were introduced into a cylindrical vanadium container (D = 6 mm, H = 5 cm) and mounted on the stick of a He cryostat, whose contribution to the diffraction patterns was eliminated using a radial oscillating collimator. Several diffractograms were collected at selected temperatures ranging between 2 and 375 K on the high-intensity two-axis powder diffractometer D1B with a detector angular range coverage 5° ≤ 2θ ≤ 128° which is especially suited for magnetic structure determination.

The data were collected using a 3 He multicounter containing 1280 detection cells with a step of 0.1° between neighbouring cells. A neutron incident wavelength of 2.52 Å was selected by a (002) Bragg reflection of a pyrolytic graphite monochromator, the take-off angle being 44.2° in 2θ. All measurements were performed upon heating after a stabilization time of 3 minutes with typical acquisition times of 20 minutes per isotherm. Due to the high flux available on the instrument, a second set of diffraction patterns was recorded in situ every 3.5 K while ramping the temperature from 2 K to 300 K. Rietveld analysis of the collected data was done using the FullProf suite software package [27], which allows the simultaneous refinement of structural and magnetic profiles.

High-field neutron diffraction experiments were carried out on the Extreme Environment Diffractometer (EXED) at the BER II research reactor at Helmholtz-Zentrum Berlin, Germany. Neutron measurements under applied magnetic fields of up to 14.5 T (superconducting magnet) were performed on pelletized sample, to avoid reorientation of the crystallites. For neutron detection, the time-of-flight (TOF) instrument EXED is equipped with 1/2" diameter position-sensitive 3 He detector tubes (GE Reuter Stokes). These tubes are combined into 4 detector banks (48 tubes/bank) that can be positioned at different angles and distances around the sample environment [28][29]. Our diffraction experiments using EXED focused on the low angle (large d spacing) region with the aim to concentrate on the magnetic signal.

Angle-dispersive x-ray powder diffraction data under hydrostatic pressures of up to 10 GPa were obtained using gasketed diamond anvil cells (DAC) at the PSICHÉ beamline (monochromatic radiation λ = 0.3738 Å) of the French Synchrotron facility (SOLEIL), at room temperature. The powder sample is loaded into the gasket hole along with small chips of ruby.

A mixture of methanol and ethanol was employed as a pressure-transmitting medium. Accurate determination of sample pressure was made by the ruby fluorescence method. In order to improve the powder statistic, the sample was rotated ±5° during the exposure. The sample-todetector distance was calibrated using a polycrystalline powder CeO2 standard. Twodimensional diffraction images were recorded by means of a MAR detector and the resulting diffraction data were azimuthally integrated with the FIT2D program [30], yielding one dimensional intensity versus diffraction angle 2θ patterns. A precise determination of the unitcell parameters was obtained by a least-squares refinement of the diffraction patterns, including all the observed Bragg reflections. T. One remarks that, when the external magnetic field is reduced to zero, the system recovers the initial state; this indicates that the forced FM phase loses its stability when the applied field is brought back to zero. In the investigated temperature interval 1.5 -100 K, we observed a clear hysteresis against the field scan, which is characteristic of the first-order nature of the phase transition. The width of the magnetic hysteresis becomes smaller upon heating. The /f.u.)

Results and discussion

Magnetic susceptibility and magnetization

 0 H (T) 1.5 K 20 K 40 K 60 K 85 K 100 K
saturation magnetization Msat of the field-induced FM phase amounts to 3.02 B/f.u. at 1.5 K; leading to an average magnetic moment of 1.51 B per Fe atom. The saturation magnetization of the forced FM state of our sample Hf0.75Ta0.25Fe2 compares very well with the value of the spontaneous magnetization MS = 3.10 B/f.u. obtained at 4 K for the ferromagnetic parent compound HfFe2 [6].

 0 H up  0 H down  0 H cr (T) T (K)
compounds with the competing FM and AFM phases [24]. They proposed magnetic phase diagrams in which both FM and AFM components of spin fluctuations are present. The M-U theoretical model takes into account the equilibrium state of the free energy displaying a double minimum and assumes that the thermal evolution of 𝜒 Q 𝜒 0 ⁄ is monotonic, where 𝜒 0 and 𝜒 Q represent respectively the uniform and staggered magnetic susceptibilities. In the high temperature region, the behaviour of the critical field 0Hcr for Hf0.75Ta0.25Fe2 which exhibits an AFM ground state is similar to that for the compounds on the intermediate composition range 0.1 < x < 0.25 having a FM ground state (the increase of 0Hcr upon heating being correlated to the monotonic decrease in 𝜒 Q 𝜒 0 ⁄ ). By contrast, the critical field increases with decreasing temperature at low temperatures in Hf0.75Ta0.25Fe2 system against the M-U theoretical model.

This observed behaviour indicates that the thermal variation of 𝜒 Q 𝜒 0 ⁄ in the low temperature region is different from that of 𝜒 Q 𝜒 0 ⁄ at high temperatures. The M-U theory does not take into account interactions arising from magnetovolume coupling in the free energy expansion.

Consequently, we suggest that the thermal evolution of the critical field of the AFM-FM metamagnetic phase transition is explainable by considering the contribution from both the magnetic and elastic energies resulting from magnetovolume effects.

To further investigate the magnetic behaviour of Fe sublattice in the paramagnetic phase, magnetization isotherms have been measured at temperatures well above TN. The thermal evolution of the inverse susceptibility is shown in Figure 4. The magnetic susceptibility follows Curie-Weiss law but two distinct regions are observed involving different paramagnetic Curie temperatures p and Curie constants C. The determined values are summarized in Table 1. The paramagnetic susceptibility presents a break at T * = 400 K. The Curie constant above T * is found to be about 1.5 times smaller than that measured below. The effective moment eff derived from the Curie constant is 4.25 /Fe and 3.55 /Fe below and above T * , respectively.

This study of magnetic properties of Hf0.75Ta0.25Fe2 intermetallic in both paramagnetic and magnetically ordered states has led to remarkably different values of Fe magnetic moments in the ordered state and in the paramagnetic phase. We calculated the Rhodes-Wohlfarth ratio r for both paramagnetic domains, namely, below and above T * and the corresponding values are listed in Table 1. The determined r values are larger than 1 and point to the itinerant character of magnetism in this compound. Following Moriya and co-workers [31][32][START_REF] Kübler | Theory of Itinerant Electron Magnetism[END_REF][START_REF] Moriya | [END_REF], the observed bend at T * of the reciprocal magnetic susceptibility of Hf0.75Ta0.25Fe2 is due to the temperature variation of the local amplitude of spin fluctuations.

Neutron powder diffraction (NPD) A. Zero-field NPD measurements

In order to establish the magnetic structures of this complex system Hf0.75Ta0.25Fe2, we have performed NPD studies. So far, no member of the (Hf,Ta)Fe2 series of compounds with an AFM ground state has been investigated using neutron diffraction. Rietveld refinements have been performed on diffractograms recorded at selected representative temperatures in the PM and AFM states, in order to refine the crystal structure and magnetic ordering -Figures 5a and 2-Theta (degrees)

T = 2 K
The analysis of the NPD pattern obtained at 2 K shows that the crystallographic structure is retained at low temperature. The 2 K diffractogram presents the same Bragg peaks as the spectra collected in the PM phase. No additional magnetic reflections are detected, which implies that the corresponding AFM structure is defined by the magnetic propagation vector k = (0, 0, 0). In other words, the magnetic unit cell coincides with the crystallographic one. At 2 K, the best refinement is achieved for a magnetic arrangement in which Fe-6h spins align ferromagnetically within the same layer but opposite orientations on successive Fe-6h layers.

In the AFM structure, the Fe magnetic moments align along the six-fold symmetry axis c. The refined moment for Fe-6h atoms at 2 K is 1.02 ; the Fe-2a atoms are not magnetically ordered.

Since the 2a Wyckoff position constitutes an inversion centre for the 6h sublattice, the molecular field acting on the 2a crystal site from the Fe-6h magnetic moments is cancelled by symmetry in this AFM spin configuration. Therefore, Fe-2a atoms display null magnetic moment. The obtained Fe magnetic moment (1.02 ) is remarkably decreased in comparison with the pure Fe moment of 2.22 μB. This experimental result is consistent with the presence of non-magnetic Hf and Ta atoms as near neighbours. The AFM magnetic structure of Hf0.75Ta0.25Fe2 is similar to that established elsewhere for isostructural compound TiFe2 [20].

Not knowing a priori the orientation of the magnetic moments in the AFM ground state, Rietveld refinements of the 2 K pattern were also conducted considering an alignment of the moments in the basal-plane of the hexagonal lattice. The two principal basal-plane directions,

[100] and [120], were tested. The magnetic reliability factors (Rmag) obtained for the two directions are very close: 14.7% and 15.3% for [100] and [120] orientation, respectively. The orientation of the magnetic moments in the basal-plane leads to much worse fits compared to that along the c-axis [001] which leads to much smaller reliability factor 7.11%. Consequently, at 2 K, Rietveld refinement gives a significantly better fit when the magnetic moments are along the [001] direction giving confidence to such orientation of the magnetic moments. However, it should be noted that the magnitude of the magnetic moment for Fe-6h atoms differs only very slightly between the two orientationsc axis and basal-planeand Fe-2a atoms carry no ordered moment. Fitting the 2 K magnetic pattern with Fe-6h moments oriented along the caxis yields the results presented in table 2.

From now on, it is clear that the magnetic structure of the AFM ground state in Hf0.75Ta0.25Fe2 is different from the spin configuration of the high-temperature AFM phase found in compounds when 0.1 < x < 0.25, whose ground state is the FM state. It is recalled that in the high-temperature AFM the Fe magnetic moments are confined to the basal-plane of the hexagonal structure [13].

The thermal evolutions of the lattice parameters a and c, the c/a ratio, and the unit-cell volume V are plotted in Figures 6a,6b, and 6c, respectively. The results of the Rietveld analysis of the thermodiffraction data reveal that the lattice constants, a and c, decrease as the temperature is lowered, thus causing the overall reduction in the unit-cell volume upon cooling.

On the contrary, the c/a ratio decreases with increasing temperature. Below 25 K, the unit-cell parameters remain essentially constant; and then present a normal behaviour by increasing as a result of thermal expansion, with a rate of 1.03×10 -4 Å K -1 , 1.56×10 -5 Å K -1 , and 4.53×10 -3 Å 3 K -1 for a, c, and V, respectively. Hf0.75Ta0.25Fe2 intermetallic exhibit positive thermal expansion (PTE) over the investigated temperature interval. Based on the refined values of cell volume, we calculated the volumetric thermal expansion coefficient V = (1/V)(dV/dT) and a value of 26.9×10 -6 K -1 is obtained at room temperature. The observation of a metamagnetic phase transition by magnetization measurements prompted us to perform neutron experiments in applied magnetic field to determine the magnetic-field dependence of the structural and magnetic state of Hf0.75Ta0.25Fe2. For this NPD experiments we used the EXED instrument in the high-intensity mode, and focused on the large d-spacing lines, that is, maximizing the neutrons flux for peaks displaying large d-spacing which appear at low diffraction angles where the generalized increase of the magnetic signal in the Bragg positions is more evident. The normalized neutron diffraction patterns at 2 K and 20 K under various applied magnetic fields are depicted in Figures 7a-7b. Our choice of these (1 0 0)

T = 20 K
particular temperatures is due to the fact that we aimed to observe the field-induced phase transition (see Figure 2) within the maximum available magnetic field of 14.5 T in this experiment. All the magnetic intensities are observed on top of the nuclear Bragg peaks and the contribution of the magnetic order to the NPD diagrams is reflected in the intensities of the (100), ( 101) and (002) reflections. It can be clearly seen on the diffraction patterns in figures 7a and 7b that there is a large difference between the spectra at zero field and under applied magnetic fields. This is most reflected in the dramatic change of intensity of the (002) peak.

The intensity of the (002) line is much more susceptible to magnetic fields than that of the ( 100) and ( 101) lines. As expected the application of a magnetic field supresses the AFM phase and a FM state is induced, as can be observed in figure 7a looking especially at the diffractograms collected at 2 K at zero field and 14.5 T.

T = 2 K (0 0 2)
FM phase c/c (2 K, 14.5 T) = -0.059 %. Consequently, the forced volume magnetostriction is estimated to be V/V (2 K, 14.5 T) = 0.78 %. The AFM-FM metamagnetic transition is accompanied by large magnetovolume effects. Our results demonstrate that the effect of the applied field is to induce a magnetic phase transformation from a low-volume AFM state to a high-volume FM state through a first-order transition at a certain critical field. The obtained value for the field-induced volume magnetostriction compares very well with the volume change V/V = 0.75 % derived from dilatometric magnetostriction measurements for isostructural compound Hf0.825Ta0.175Fe2 [12].

We have found a different effect of the magnetic field in the intensities of the three peaks at low temperature. This is illustrated in Figure 9, where the integrated intensities of the (100), ( 101) and (002) reflections are displayed as a function of the magnetic field at 2 K. The difference in the influence of the applied magnetic field is remarkable. While the external applied magnetic field leads to a very fast rise of the intensity associated to the (002) peak, the intensities of the (100) and (101) lines become almost field independent below 9 T and then increase rapidly as the sample undergoes a metamagnetic transition. This phenomenon is in agreement with the macroscopic magnetic measurements, where the onset of significant change in the magnetization is found at 9.25 T. Upon application of an external magnetic field, a continuously increasing magnetic scattering is observed on (002) peak indicating that a ferromagnetic component is progressively induced in the material. The magnetic contribution to (002) reflection is proportional to the square of the in-plane magnetization component since neutrons are sensitive to the component of the magnetic moment perpendicular to the scattering vector. In the forced FM state, the presence of magnetic contribution on (002) peak reveals that the Fe moments are not aligned along the c-axis. One could conclude that Hf0.75Ta0.25Fe2 features a first-order field-induced spin-reorientation transition; the easy magnetization direction changes from axial to basal-plane upon application of magnetic field.

High-Pressure synchrotron x-ray diffraction (SXRD)

To probe the effect of external pressure on the crystal structure, angle dispersive synchrotron x-ray diffraction patterns have been recorded at room temperature for various applied pressures ranging from 0 to 10 GPa. In Figure 10 we compare selected SXRD patterns during compression of the sample. SXRD diffractograms collected at various pressures reveal that the hexagonal symmetry of the atomic arrangement remains unaltered within the investigated pressure range; no indication of a structural phase transition was found up to the highest applied pressure. All the observed Bragg reflections could be identified to arise from the original C-14 Laves phase structure (space group P63/mmc). The crystal symmetry is stable up to the highest applied pressure of 10 GPa. Analysis of the diffraction patterns shows that the diffraction peaks shift to higher angles due to the contraction of the lattice. The lattice constants (a and c), deduced from least-squares refinements, are plotted in Figure 11 as a function of applied pressure. Both cell parameters decrease continuously upon compression, and values of da/dP = -5.90×10 -3 Å GPa -1 and dc/dP = -9.51×10 -3 Å GPa -1 are obtained. The contraction of the hexagonal cell is anisotropic and more pronounced along the c direction. In the crystal structure, the arrangements of atoms and bonds perpendicular to c-axis direction are dense, while, along c-axis direction the atoms arrange loosely, which points out it has considerable relaxation space on compression. The pressure dependence of the unit-cell volume is illustrated in Figure 12. The experimental data were fitted to a third-order Birch-Murnaghan equation of state [35] where 𝐾 0

´ was fixed at 4. The fit yielded V0 = 168.56 Å 3 for the zero-pressure unitcell volume and K0 = 247 GPa for the bulk modulus. For comparison purposes we note that, TiFe2 and LaFe12B6 materials have a bulk modulus of 201 GPa [36] and 204 GPa [37],

respectively. The bulk modulus of Hf0.75Ta0.25Fe2 is ≈1.2 times larger than that of the antiferromagnetic itinerant-electron system LaFe12B6.

Conclusions

Competition between AFM and FM states in the itinerant-electron system Hf0.75Ta0.25Fe2 has been investigated by means of neutron powder diffraction and magnetizations measurements both performed under extreme conditions of low temperature and large magnetic fields. The studied compound Hf0.75Ta0.25Fe2 presents an antiferromagnetic order as a ground state below TN = 342 K. The AFM ordering is only due to the Fe atoms at 6h sites which carry a magnetic moment of 1.02  at 2 K. It is demonstrated that the AFM phase can be transformed into a FM phase via a field-induced metamagnetic transition accompanied with an in-plane lattice expansion and a change of the easy magnetization direction from axial to basal-plane. In the disordered phase, an unusual and intriguing transition from one paramagnetic state to another one involving two distinct Curie-Weiss behaviours has been observed at T * , a critical temperature at which the reciprocal magnetic susceptibility presents a bend. Thanks to synchrotron radiation, x-ray diffraction has been carried out upon application of a high external pressure. In the investigated pressure range, no change of crystal symmetry occurs. The unit cell shrinks continuously, but in an anisotropic way with a larger compression along the high-symmetry direction c.

Figure 1 .

 1 Figure 1. Temperature dependence of the magnetization for Hf0.75Ta0.25Fe2.

Figure 2 .

 2 Figure 2. Magnetization curves of Hf0.75Ta0.25Fe2 measured between 1.5 and 100 K in pulsed fields.

Figure 1

 1 Figure1presents the thermomagnetic curve, M(T), of the zero-field cooled (ZFC) sample, recorded upon heating in an applied magnetic field of 0.1 T. The peak at 342 K in the isofield magnetization curve of Hf0.75Ta0.25Fe2 corresponds to the Néel temperature TN reflecting the transition from an AFM state to a PM phase. Field-dependent magnetization curves M(H), at several fixed temperatures ranging from 1.5 to 100 K, are displayed in Figure2for the thermally demagnetized Hf0.75Ta0.25Fe2 sample. For the sake of clarity, just a selection of curves is shown in Figure2but all of the measured magnetization data were employed to plot the temperature dependence of the critical field illustrated in Figure3. The M(H) curves exhibit an interesting field-induced metamagnetic phase transition between the AFM and FM states. The metamagnetic phase transformation proceeds through a progressive conversion of the AFM phase into FM domains with increasing the applied magnetic field. At 1.5 K, the fieldinduced metamagnetic phase transition is remarkable in two ways: (i) by the huge magnetization change and (ii) by the presence of a particularly large magnetic hysteresis of 3

Figure 3 .

 3 Figure 3. Thermal evolution of the critical field 0Hcr upon increasing (up) or decreasing (down) the applied magnetic field for Hf0.75Ta0.25Fe2.

Figure 4 .

 4 Figure 4. Thermal variation of the reciprocal magnetic susceptibility for Hf0.75Ta0.25Fe2.

Figure 5 .

 5 Figure 5. Refinements of the neutron diffraction patterns recorded for Hf0.75Ta0.25Fe2 at 375 K (a) and 2 K (b). In figure (a) the top row of Bragg markers is for the Hf0.75Ta0.25Fe2 nuclear contributions, the second row corresponds to the position of the vanadium sample environment contribution. In figure (b) the top and second rows of Bragg markers are referring to the nuclear and magnetic contributions of the Hf0.75Ta0.25Fe2 phase. The third row is corresponding to the nuclear contribution from the vanadium sample environment.

Figure 6 .Figure 7 .

 67 Figure 6. Thermal dependence of the lattice parameters a and c (a), c/a ratio (b), and unit-cell volume (c) obtained from Rietveld refinement of NPD data measured upon heating for Hf0.75Ta0.25Fe2.

Figure 9 .

 9 Figure 9. Integrated intensities of (100), (101) and (002) reflections at 2 K as a function of the applied magnetic field.

Figure 10 .

 10 Figure 10. Synchrotron x-ray diffraction patterns collected at room temperature for Hf0.75Ta0.25Fe2 at various applied pressures (λ = 0.3738 Å).

Figure 11 .

 11 Figure 11. Lattice constants of Hf0.75Ta0.25Fe2 as a function of applied pressure. The solid lines correspond to linear fits.

Figure 12 .

 12 Figure 12. Unit-cell volume of Hf0.75Ta0.25Fe2 as a function of applied pressure. The continuous red line represents the third-order Birch-Murnaghan fit for the data.

Table 1 :

 1 TN or TC, magnetic ordering temperature; T

		TN/TC	s 1.5 K	T *	qs	C	eff	qc	p	r
	Compound	(K)	(/Fe)	(K)		( /f.u.T)	(/Fe)		(K)
						T<T * 8.08	4.25	3.37	103	2.23
	Hf0.75Ta0.75Fe2	342	1.51	400 1.51
						T>T * 5.64	3.55	2.69	193	1.78
	-Fe	1043	2.22	-2.22	2.26	3.18	2.33 1093 1.05

* temperature of change of the Curie Weiss behavior, qs, number of carriers deduced from saturation magnetization s; qc, number of carriers deduced from Curie-Weiss constant C; eff, paramagnetic effective moment; θp, paramagnetic Curie temperature and ratio r = qc / qs, for the Hf0.75Ta0.25Fe2 compared to that of the pure Fe.

Table 2 .

 2 Rietveld refinement results and reliability factors obtained from the analysis of the powder

	neutron diffraction patterns recorded at 2 K and 375 K for Hf0.75Ta0.25Fe2.
		AFM	PM
	T (K)	2 K	375
	a (Å)	4.904(1)	4.925(3)
	c (Å)	8.040(3)	8.045(2)
	Hf /Ta (4f)
	z	0.062(1)	0.066(3)
	Fe (6h)
	x	0.832(2)	0.834(3)
	y = 2x	1.664(4)	1.668(6)
	Fe 2a moment (µB)	-	-
	Fe 6h moment (µB)	1.02(6)	-
	Moment-c-axis angle (°)	0	-
	c 2	6.39	4.01
	RBragg (%)	7.51	4.34
	Rmag (%)	7.11	-
	Rwp (%)	11.8	11.1
	Rp (%)	12.6	11.1
	Rexp (%)	4.69	5.52

Figure 8 .

 8 Unit-cell parameters of Hf0.75Ta0.25Fe2 as a function of applied magnetic field.

	168.96
	168.52
	168.08
	167.64
	8.0410
	8.0393
	8.0376
	8.0359
		4.925
	(Å)	4.915 4.920
	a
		4.910
		4.905
		4.900
		0	2	4	6	8	10	12	14

Acknowledgments

The authors gratefully acknowledge the Institut Laue Langevin and Helmholtz-Zentrum Berlin for access to the neutron facilities. We acknowledge the French Synchrotron facility SOLEIL for the allocation of beam time. We also acknowledge the support of the French National High Magnetic Field Laboratory (LNCMI), member of the European Magnetic Field Laboratory (EMFL).


Quoc Thong Nguyen
email: quoc-thong.nguyen@univ-ubs.fr

Yves Delignon

Anh Thu Phan

Probabilistic Modelling of Printed Dots at the Microscopic Scale

Keywords: Probabilistic model, Bernoulli process, Metropolis Hastings within Gibbs, Microscopic printing, Markov chain

Microscopic analysis of paper printing shows regularly spaced dots whose random shape depends on the printing technology, the configuration of the printer as well as the paper properties. The modelling and identification of paper and ink interactions are required for qualifying the printing quality, for controlling the printing process and for application in authentication as well. This paper proposes an approach to identify the authentic printer source using micro-tags consisting of microscopic printed dots embedded in the documents. These random shape features are modelled and extracted as a signature for a particular printer. In the paper, we propose a probabilistic model consisting of vector parameters using a spatial interaction binary model with inhomogeneous Markov chain. These parameters determine the location and describe the diverse micro random structures of microscopic printed dots. A Markov chain Monte Carlo (MCMC) algorithm is thus developed to approximate the Minimum Mean Squared Error estimator. The performance is assessed through numerical simulations. The real printed dots from the common printing technologies (conventional offset, waterless offset, inkjet, laser) are used to assess the effectiveness of the model.

Introduction

Preventing and discouraging unauthorised printed materials is the expected achievement of the authentication methods. Papers dealing with this subject can be categorised in two strategies. The first one consists in embedding an extrinsic signature such as secure tags in the prints [START_REF] Picard | Towards fraud-proof id documents using multiple data hiding technologies and biometrics[END_REF][START_REF] Villán | Multilevel 2-d bar codes: toward high-capacity storage modules for multimedia security and management[END_REF]. While the second one, which is the motivation of this paper, consists in characterising the intrinsic features of printing and scanning process [START_REF] Chiang | Printer and scanner forensics[END_REF]. As examples of intrinsic features, the signature consisting on the banding artefact coming from fluctuation of the optical photoconductor angular velocity in case of laser printers [START_REF] Ali | Intrinsic and extrinsic signatures for information hiding and secure printing with electrophotographic devices[END_REF], dimple effect which is specific to inkjet printers or also texture features [START_REF] Ryu | Electrophotographic printer identification by halftone texture analysis[END_REF], or the analysis of quality signature of the unique print to differentiate one printer technology/supplier from another [START_REF] Oliver | Use of signature analysis to discriminate digital printing technologies[END_REF]. Authentication can be performed based on paper statistics, for example, the paper statistics is extracted by capturing microstructure images from different carton packages under different light conditions and with different cameras. These microstructure images are considered as digital fingerprinting on which the authentication frameworks are derived [START_REF] Voloshynovskiy | Towards reproducible results in authentication based on physical non-cloneable functions: The forensic authentication microstructure optical set (famos)[END_REF]. In [START_REF] Sharma | Paperspeckle: microscopic fingerprinting of paper[END_REF], the authors proposed the intrinsic feature of paper based on a texture speckle pattern, a random bright/dark region formation at the microscopic level when light falls on to the paper. The extracted signature of printer is a unique pattern of misplaced toner powder on each paper [START_REF] Zhu | Print signatures for document authentication[END_REF]. For electrophotographic printers, the intrinsic signals extracted from banding artefact are used to identify the device [START_REF] Mikkilineni | Signature-embedding in printed documents for security and forensic applications[END_REF]. Printer source identification is an important forensic tool.

The techniques using characters for identification of source scanner and printer have been proposed [START_REF] Khanna | Survey of Scanner and Printer Forensics at Purdue University BT -Computational Forensics: Second International Workshop[END_REF]. Source identification using printed character in different languages has been recently studied [START_REF] Tsai | Digital forensics of printed source identification for chinese characters[END_REF][START_REF] Tsai | Japanese character based printed source identification[END_REF][START_REF] Tsai | Digital forensics for printed character source identification[END_REF][START_REF] Tsai | Source identification for printed arabic characters[END_REF][START_REF] Tsai | Digital forensics of microscopic images for printed source identification[END_REF].

Besides the role in print authentication, depending on the purposes of the problems, there are a number of applications concerning the printing models. A specific model is considered in the scale of the experiments/applications. For instance, in the human vision scale, halftoning technique is a method that takes advantage of the optical illusion of human vision to display continuous tone grayscale image with only black or/and white dots [START_REF] Ulichney | Review of halftoning techniques[END_REF][START_REF] Velho | Digital halftoning with space filling curves[END_REF][START_REF] Pappas | Model-based halftoning[END_REF]. The scale of the model is the resolution of the printer. An image of print not only contains the properties of a printing process but also includes image degradation; Baird proposed a physics-based model in which the parameters defining imaging defects are described [START_REF] Baird | Document image defect models[END_REF]. Under the microscopic scale printing, a probabilistic model for the average coverage of the toner particles ink of electrophotographic printing process is introduced by Norris et al. [START_REF] Norris | Printer modeling for document imaging[END_REF]. However, the model has a limited validation, and the estimation procedure has not been considered. For authentication purpose, we propose a probabilistic modelling of microscopic printed dots.

The shapes of microscopic printed dots can be considered as the intrinsic feature of the printing process. At the microscopic scale, a dot is a random pattern whose shape depends on the technology, the setting of the printer, the ink quality and/or the paper properties. From a statistical point of view, the digital acquisition of these random dots can be modelled as a spatial interaction binary model based on an exponential power kernel that depends on location parameter and shape parameters. The accurate and consistent model can help to create an authentic signature to improve the authentication procedure. The model proposed in our previous works [START_REF] Nguyen | Printer technology authentication from micrometric scan of a single printed dot[END_REF][START_REF] Nguyen | Printer identification from micro-metric scale printing[END_REF] consists in a spatial distribution model that simulates the randomness of the shape of printed dots. The random shape is manipulated by the parameters in this probabilistic model. In [START_REF] Nguyen | Modélisation de points imprimés à l'échelle micro-métrique[END_REF], the numerical estimation procedure using MCMC algorithm is developed. However, in order to cover the diverse random shape of printed dots, the surrounding pixels or the conditional probabilities has to be taken into account.

In this paper, we propose a spatial interaction binary model that describes more precisely the random shape of the microscopic dots by taking into account the neighbourhood of the pixels. In particular, the Markov property between pixels in the image is considered. This property helps to describe more accurately the intrinsic features of the print, which can thus improve the discrimination power. An MCMC algorithm is developed for estimating the model parameters. Finally, we analyse the performance of the proposed unsupervised identification algorithm. The rest of this paper is organised as follows. Section 2 is devoted to visually describe the randomness of shape of microscopic dots. The formulation of spatial stochastic model is discussed in Section 3. The Metropolis-Hastings within Gibbs algorithm for the estimation is developed in Section 4. In Section 5, the Posterior Cramér -Rao bound used to evaluate the performance of the algorithm is derived. Section 6 is devoted to the performance of the new algorithm which are analysed through simulations, and the benefit of such a model and estimation algorithm are illustrated in the case of real microscopic scale printed dots. The basic authentication of printer identification using the model is also described in Section 7.

Concluding remarks are given in Section 8.

Printed dots at the microscopic scale

As aforementioned, we consider the shape of microscopic printed dots as an intrinsic feature to perform authentication on printer identification. This section is devoted to describing the random shape of microscopic printed dots. We then demonstrate intuitively that the shapes are different between different printers, which can benefit printers discrimination. Let us start by the observation of printed dots at the microscopic scale. A single printed dot is created by printing an image containing one black pixel. In our study, the microscopic printed dots are captured using an optical Zeiss Microscope with an AxioCam camera. The camera has a maximum solution of 2464 x 2056 (5Mp) with a sensor size of 8.5 mm x 7.1 mm and a pixel size of 3.45 µm x 3.45 µm. Since the focus of the study is on the physical shape of coverage area of the ink, the collected images are only in black and white.

The images on the left column in Fig. 1 illustrate the dots printed by offset, laser, inkjet printers respectively. Random shapes of these dots are observed under microscope scale. The physical size of the dots is very small. For example, size of a dot from the offset printer (1200 dpi) is theoretically 21.16µm. The printing resolution is an important element in this study. If the resolution is too high, the outcome will be similar to the dust noise from the printer. For instance in our study, the dots printed from a laser printer under resolution 1200 dpi are not distinguishable from the noise.

The resolution can neither be too low, since the shape of the dots are no more random except for the edge. Therefore, the resolution has to be chosen manually for a specific printer.

Visually, from the left column of Fig. 1, we can see that the shapes of printed dots from different printers are dissimilar. two different laser printers at the same resolution 600 dpi. On the right column in Fig. 1, each image illustrates the frequency of appearance of black pixels at each position (profile) from 100 captured samples. As observed, the probabilities of emergence of the black pixels are lower when the pixels are further from the centres. The shape of the profile contains the characteristics of the printed dots, which means the shape of the dots contains the identity of the source printer in some sense. The model in [START_REF] Nguyen | Printer technology authentication from micrometric scan of a single printed dot[END_REF][START_REF] Nguyen | Printer identification from micro-metric scale printing[END_REF] introduced the parameters controlling the width of the droplets, the density of the black pixels and the average number of black pixels. However, the interaction with the surrounding pixels was not taken into account. In particular, when a pixel is black, its neighbour is more likely to be blackened. In addition, when the pixels 100 are further from the centre of the dots, this influence of the black pixel(s) on their vicinity is also smaller. Including these properties could help the model to have a more precise simulation, thus describing in more details the identity of source printer. These properties are detailed in the next Section.

Spatial interaction binary model with Markov chain 105

Let S be a finite set corresponding to N pixels of a binary image U that displays K dots. We consider the random process U = (U s ) s∈S , where each U s is a Bernoulli random variable which takes its value from {0, 1}, respectively black state and white state.

Let V s be a neighbourhood of the pixel s, the geometric shape of V s is independent of s ∈ S. U is modelled by a Markov random field

P (U s = u s | (U t = u t ) t =s ) = P (U s = u s | (U t = u t ) t∈Vs ) , (1)
i.e. the probability that the pixel s takes the value u s conditional on all other pixels of the image equals to the probability of u s conditional on the values of the pixels in the neighbourhood V s . This probability also has to depend on the distance of the pixel s to the centre µ, which means P (U s = 0 | (U t = u t ) t =s ) decreases when the pixel s goes further from the centre, i.e. the Euclidean distance ||s -µ|| increases.

110

A two-dimensional image can be transformed into a one-dimensional chain by

Hilbert curve [START_REF] Hilbert | Ueber die stetige abbildung einer line auf ein flächenstück[END_REF] (see Fig. 3). Because the targeted interactions (statistical dependencies) between pixels within the microscopic dot are well preserved in the spatial structure by Hilbert curve, the Markov chain with Hilbert path is used in many image processing applications [START_REF] S.-I. Kamata | Interactive analysis of multi-spectral images using a hilbert curve[END_REF][START_REF] Fjortoft | Unsupervised classification of radar images using hidden markov chains and hidden markov random fields[END_REF][START_REF] Dridi | Em-based joint symbol and blur estimation for 2d barcode[END_REF]. Moreover, Pseudo-Hilbert Scan algorithm can be applied to arbitrarily-sized image [START_REF] Zhang | A pseudo-hilbert scan algorithm for arbitrarily-sized rectangle region[END_REF]. In convenient index, let U = (U 1 , U 2 , ..., U N ) be a vector of random variables ordered according to the Hilbert curve transformation of the image. In case the neighbourhood V s is restricted to one pixel, the image U becomes a Markov chain The Markov chain (2) encompasses the parameters of the initial probability

P (U = u) = P (U 1 = u 1 ) N i=2 P (U i = u i | U i-1 = u i-1 ). (2)
P (U 1 = 0), (3)
and transition probabilities

P (U i = 0 | U i-1 = u), (4)
for 1 < i ≤ N . The parametric probabilities for (3), (4) are

P (U 1 = 0) = 1 - K k=1 (1 -p k (s 1 )) (5)
P (U i = 0 | U i-1 = u) = 1 - K k=1 (1 -p k (s i ))λ (1-u) . (6)
The function p k (.), named kernel, measures the ability to blacken the surrounding pixels of the k-th dot. The kernel p k (s) has to decrease with respect to the distance ||s -µ k ||, so that, as requisite property, the initial and transition probabilities also decrease with respect to the distance. The support of p k (.) is a subset of (0,1].

The positive parameter λ characterises the interaction between pixels when the neighbour pixel is black, U i-1 = 0. With λ = 1, the probabilities (6) only depend on the relative position to the centres, P (U i | U i-1 ) = P (U i ), which is the approach proposed in [START_REF] Nguyen | Printer technology authentication from micrometric scan of a single printed dot[END_REF][START_REF] Nguyen | Printer identification from micro-metric scale printing[END_REF]. In case λ < 1, the smaller λ leads to a higher chance of being black of current pixel U i , since P (U i = 0 | U i-1 = 0) is greater with smaller λ (see Eq. ( 6)). When the neighbour pixel is white, U i-1 = 1, the probabilities depend on the distances of the current pixel to the dots centres. With these properties, the dots are more compact, and the density of black pixels at the areas closer to the centres is higher. The new proposed parameter λ helps to describe a wider variety of shape of dots.

The transition probabilities depend on p k (s) which relates to the position of the pixels, this implies that U is a non-stationary Markov chain. The expected number of black pixels of the image, denoted n b , is calculated as

n b = N i=1 E (δ(U i )) = N i=1 P (U i = 0), (7)
with δ(x) = 1 if x = 0, δ(x) = 0 elsewhere. The probability P (U 1 = 0) is computed by [START_REF] Ryu | Electrophotographic printer identification by halftone texture analysis[END_REF], while the other terms are calculated by [START_REF] Oliver | Use of signature analysis to discriminate digital printing technologies[END_REF] with the recurrence relation and the law of total probability

P (U i = 0) = P (U i = 0 | U i-1 = 0)P (U i-1 = 0) +P (U i = 0 | U i-1 = 1)P (U i-1 = 1), (8)
with 2 ≤ i ≤ N . Let us note that with smaller λ, P

(U i = 0 | U i-1 = 0) is greater,
that leads to the value of P (U i = 0) also becoming greater. This implies that smaller λ statistically increases the number of black pixels. In the following, the kernel p k is chosen as a Gaussian power kernel

p k (s) = η exp    - s -µ k 2 σ 2 β    . (9)
Fig. 4 illustrates the simulation of one dot with σ 2 , β equal to 70 and 5 respectively 130 and various values of λ and η. It can be seen that the dots with small λ are more continuous, in other words, the black pixels make their neighbourhood more likely to be black. The parameter η controls the amplitude of the kernel and the density of black pixels increases with η. When η is large, the density of black pixels near the centre is almost independent of λ.

θ(u) = θp(θ | u)dθ, (10)
where p(θ | u) is the posterior distribution. To approximate the integral [START_REF] Mikkilineni | Signature-embedding in printed documents for security and forensic applications[END_REF], an MCMC algorithm is applied [START_REF] Robert | Monte Carlo Statistical Methods[END_REF]. Since the vector of parameters is of high-dimension, it is more efficient to use Gibbs sampling procedure which consists in sampling successively each parameter from their conditional posterior distribution. Unfortunately, since no closed form expression is available for these distributions, we propose to use a Metropolis-Hastings within Gibbs algorithm [START_REF] Robert | Monte Carlo Statistical Methods[END_REF]. The chosen strategy is to run a random walk as proposal distribution:

θ * = θ (t) + t , ( 11
)
where t is a zero mean normal random vector with a diagonal covariance matrix.

We generate θ 2 = log(σ 2 ), θ 3 = log(β) instead of σ 2 , β to ensure their positivity; a transformation is also applied for η and λ to generate θ 4 , θ 5 respectively, i.e. Algorithm 1 Sampling algorithm with random walk

1: Initialise θ (0) = θ (0) 1 , θ (0)
2 , θ Generate θ b ∼ N (θ

(0) 3 , θ (0) 4 , θ
(i-1) b , σ 2 b ) 5:
Compute the acceptance ratio α(θ

(i-1) b , θ b ) = min    p θ b | θ (i-1) -b , u p θ (i-1) b | θ (i-1) -b , u , 1    6:
Decide

θ (i) b = θ b with probability α θ (i-1) b , θ b θ (i-1) b otherwise 7:
end for 8: end for Where θ

(i-1) -b = θ (i) 1 , . . . , θ (i)
b-1 , θ

(i-1) b+1 . . . , θ (i-1) 5
.

The initialisation of θ proceeds as following. The centres are initialised by the careful seeding [START_REF] Arthur | k-means++: The advantages of careful seeding[END_REF]. The other parameters are respectively initialised as σ 2 = 0.6 × N πK , β = 2. Since η and λ are in (0, 1], intuitively, we may initialise both of them equal to 0.5. From Bayes' law, we have:

p (θ b | θ -b , u) = p (u | θ b , θ -b ) p (θ b , θ -b ) p (θ -b , u) , (12)
where p (u | θ b , θ -b ) is the likelihood and p (θ b , θ -b ) is the prior. Since all parameters are independent, the acceptance ratio in Alg. 1 is rewritten as

α(θ (i-1) b , θ b ) = min    p (u | θ b , θ -b ) p b (θ b ) p u | θ (i-1) b , θ -b p b θ (i-1) b , 1    , (13)
with p b the prior of parameter θ b . The prior distributions are chosen with fixed pa-rameters by experience. Each centre {µ k } has multivariate normal distribution, σ 2 is log-normal distributed, β follows gamma law, while both η and λ are beta-distributed.

Let us note that when the priors are closed to uniform, the acceptance ratio mainly depends on the likelihood ratio. The choice of the variances of the priors depends on the confidence of the initialisation and the experience.

Posterior Cramér -Rao bound

Let u be the observation vector and θ be the r-dimensional vector of random parameters (with r = 2k + 4). The joint probability density of the pair (U, θ) is p(U, θ).

Let θ(u) denote an estimate of θ. The Posterior Cramér-Rao Bound (PCRB) states that [START_REF] Tichavskỳ | Posterior cramér-rao bounds for discrete-time nonlinear filtering[END_REF][START_REF] Van Trees | Detection, estimation, and modulation theory[END_REF]:

E[( θ(U) -θ)( θ(U) -θ) T ] ≥ I -1 , (14)
where I is r × r Fisher information matrix (FIM)

I = E ∇ θ log p(U, θ)∇ T θ log p(U, θ) = -E ∆ θ θ log p(U, θ) , (15)
where ∆ θ θ := ∇ θ ∇ T θ is the second derivative operator and ∇ θ is the gradient operator with respect to θ.

Using the fact that p(U, θ) = p(U | θ)p(θ), the expression of the FIM in (15) can be expressed as:

I = -E ∆ θ θ log p(U | θ) -E ∆ θ θ log p(θ) = I d + I p , (16)
where I p stands for the a priori information matrix and

I d is the "standard" FIM (I d (θ)
is used in the derivation of the CRB) averaged over the a priori distribution of θ:

I d = Θ I d (θ)p(θ)dθ. (17)
In many special cases, when only the i-th component of θ, say θ i , is of interest, the PCRB is expressed as

E[( θi (U) -θ i ) 2 ] ≥ [I -1 ] ii (18)

Experimental results and performances

In this section, we aim at assessing both the performance of the estimators and the adequacy of the model. The Metropolis-Hasting within Gibbs algorithm is analysed, the performance of the algorithm is evaluated from the images generated by the parametric model. The experiment on the real captured dots is also conducted.

Synthetic Data

The simulated images are generated with a size of 64 × 64. As mentioned previously, the priors are fixed, the expected values of the prior are set by the initial values, and the variance are chosen large in order to consider a highly uncertain prior knowledge. The estimation is implemented with 20000 MCMC iterations and a burnin period of 5000, which means that 15000 last samples are used to approximate the parameters over 20000 generated samples.

Analysis of the estimation with respect to β

In this analysis, only β is varying. The simulated images of single dot are generated. The true centres µ of the dot is [START_REF] Tichavskỳ | Posterior cramér-rao bounds for discrete-time nonlinear filtering[END_REF][START_REF] Tichavskỳ | Posterior cramér-rao bounds for discrete-time nonlinear filtering[END_REF]. With σ 2 = 50, η = 0.8, λ = 0.6, along with three values of β, β ∈ {0.8, 2, 5}, Tab. 1 shows the mean squared errors of the estimators. Larger value of β leads to smaller errors for σ2 and η. In contrast, the errors of β and λ are larger with large β. With a large value of β, the simulated dots are different mainly near the edges. It means that the information of λ is less significant, which makes the estimation of λ less consistent. While with smaller β, the contribution of λ to the connected black pixels is more visible (see Fig. 5). On the other hand, the simulated dots look similar for large values of β (see Fig. 6), which also makes the estimation of β less consistent. The mean squared errors of the estimators with various values of η, σ 2 = 50, β = 2, λ = 0.6.

Analysis of the estimation with respect to λ

The simulations of one dot were conducted with β = 2, σ 2 = 50, η = 0.8, along with three values of λ, λ ∈ {0.2, 0.6, 0.9}. Tab. 3 shows the MSE of the estimators.

The estimators of σ 2 , β, η have decreasing MSE when λ increases. Since the number of black pixels decreases when the value of λ increases, the information for the estimation of σ 2 , β, η exposes clearer on the realisations. On the other hand, the estimation of λ is better for small values, because the connection between black pixels strongly emerges.

σ 2 β η λ 0.2 66.06 0.247 0.026 0.0017 0.8 45.12 0.23 0.011 0.0068 0.9 36.57 0.199 0.0113 0.0181 Table 3: The mean squared errors of the estimators with various values of λ, β = 2, σ 2 = 50, η = 0.8.

Posterior Cramér-Rao bound for single dot case

The PCRB is used to assess the precision of the estimators. Since the posterior information matrix I in ( 16) cannot be calculated analytically, it is approximated by Monte Carlo simulation. The random values of vector of parameters θ generated N times by the priors. For each generated vector of parameter, M images are simulated to approximate the standard Fisher information matrix I d (θ) in [START_REF] Velho | Digital halftoning with space filling curves[END_REF]. In Tab. 4, we demonstrate the approximated values of the Posterior Cramér-Rao Bound in case of single dot with N = 100 generated random values of vector θ. The values of priors are as follows: centre (µ x , µ y ) has multivariate normal distribution with mean [START_REF] Tichavskỳ | Posterior cramér-rao bounds for discrete-time nonlinear filtering[END_REF][START_REF] Tichavskỳ | Posterior cramér-rao bounds for discrete-time nonlinear filtering[END_REF] and covariance matrix diag(5, 5); σ 2 is log-normal distributed with mean 50 and variance 100; β follows gamma law with mean 2 and variance 1.5; η and λ are beta-distributed B(3, 1.5). For each value of θ, M = 40 images are generated to approximate matrix I d (θ). Specifically, the size of each image is 64 × 64. With each generated image, the parameters are estimated using Algo. 1. In total, 40 × 100 images are used to approximate the Mean Squared Error. As we observe in this example, σ 2 is estimated least accurate.

The performance with many dots

The impact of the number of dots, K, is analysed. Instead of one dot on each image as the analyses above, each image contains multiple simulated dots. The experiment is set up with σ 2 = 50, β = 2, η = 0.8, λ = 0.6, and all generated dots are completely separated from each other. As demonstrated in Tab. 5, the estimators are clearly more consistent for the images with more dots. The accuracy and consistency of β, η, λ is crucial in authentication. Therefore, analysing many dots is a good mean to have a more consistent estimation of the parameters. Since more dots give more information about the shape, which are determined by the parameters of the model, the estimators are more consistent. The standard deviation of the estimators with different number of cluster-dots K, σ 2 = 50, β = 2, η = 0.8, λ = 0.6.

Estimation results from printed dots

The dots from four printers of four popular technologies are taken: conventional offset, waterless offset, laser and inkjet printings. The resolutions of each printing process for each sample are respectively 1200 dpi with offset printers, 600 dpi with laser printer, and 720 dpi with inkjet one. For each printer, 10 images of size 64x64 of one dot are collected for the experiment. Fig. 7 represents the images of the real dots and the realisation generated from the estimated parameters obtained from the developed MCMC method. Metropolis-Hasting within Gibbs algorithm is conducted with 30000 MCMC iterations with 10000 as burn-in period. The realisation from the Spatial binary model [START_REF] Nguyen | Printer identification from micro-metric scale printing[END_REF] is illustrated to compare the improvement as well. Tab. 6 gives the numerical values of the estimators by MCMC estimation method as well as the standard deviation values. Fig. 7 visually compares the printed dots, the realisations from the proposed Markov spatial model and from the spatial binary model that does not take into account the neighbour dependence developed in [START_REF] Nguyen | Printer technology authentication from micrometric scan of a single printed dot[END_REF][START_REF] Nguyen | Printer identification from micro-metric scale printing[END_REF]. The results are very promising when the interaction is taken into account. The realisation from Markov spatial model is more realistic to the real dot than the one from spatial binary model. With the offset and inkjet images, the fragment near the border is resembled better, which makes them more similar to the real dots. The shape of the dot from inkjet printer is more complicated to simulate due to the concave parts.

With the print from laser printer, we can see the substantial improvement using the Markov spatial model w.r.t. the spatial binary model. The simulated dot is much more realistic and continuous.

In Fig. 8 and Tab. 7, we give other estimation of four printed dots with the resolution 600 dpi from HP-600 M620 laser printer. We can again have a visual comparison to show the improved simulation using the Markov spatial model. The capability of the proposed model to capture more details can clearly increase the power of discrimination between printers. Moreover, since the characteristics of printed dots are represented by the parameters, the proposed model may significantly reduce training data than using machine learning generative models to characterise the printed dots.

Application in printer source identification

The capability of simulating the random shape of printed dots of the model and the performance of the estimation are demonstrated in previous Section. In this part, we use the typical random shape which is characterised by the parameters as a "signature" to identify the printer source. The vector of three parameters (β, η, λ) can be used as a characteristic of a printer with a specific setting. These three parameters are dimensionless, which means that the size of image of dots does not affect the value of these parameters. With the model, we are able to set up an authentic tag from an exact printer by setting specific values for the parameters of the printer (ink toner, paper...). Under a specific setting, a particular printer statistically creates the same shape of microscopic dots. In this experiment, we applied the maximum likelihood classification method to discriminate the printed dots from different sources [START_REF] Foody | Derivation and applications of probabilistic measures of class membership from the maximum-likelihood classification[END_REF][START_REF] Asmala | Analysis of maximum likelihood classification on multispectral data[END_REF]. Fig. 9 summarises the procedure of the proposed printer identification method.

The estimations of the shape parameters are displayed in Fig. 10 that shows respectively β vs. λ, β vs. η and λ vs. η. Each printer is 3D Gaussian cluster in the space of parameters. Their location and shape are respectively given by their means (Tab. 6)

and their covariance matrices (Tab. 8). For each printer (conventional offset, waterless offset, laser and inkjet), we take 10 other images for the discrimination experiment. In total, 20 images for each printer with 10 images are used to approximate the parameters of 3D Gaussian cluster. In this specific experiment, all the dots are printed on uncoated paper, and other conditions are identical during the experiments. In this case, the normal distributions of the shape parameters enable to use the maximum likelihood classification method. All the samples are correctly classified (100%) with respect to their printer sources. The confusion matrix is presented in Tab. 9 Other experiment is also conducted on two laser printers, a HP laser printer and a Dell laser printer. The resolution of 600 dpi is set up for both printers, and the samples are printed on uncoated paper. The relation between estimated parameters of each printer is given in Fig. 11 which clearly separates two printers. Their location and shape of the 3D Gaussian clusters of each printer are respectively given by the means in Tab. 10 and the covariance matrices in Tab. 11. Similar to the previous experiment, 20 images are captured from each printer with 10 images are used to approximate the parameters of 3D Gaussian cluster. The maximum likelihood classification method also gives a completely accurate classified result (Tab. 12). These results point out the relevance of the model for printed dots at the microscopic scale.

As demonstrated in Fig. 8, Tab. 7 and the analysis in the simulation experiment section, the estimation of parameters is more consistent with many dots. Therefore, the analysis of many dots will help to improve the accuracy of discrimination an exact printer source of a document from the other printers. In practice, a micro-tag consisting in printed dots can be embedded in an authentic document.

Concluding remarks

In this paper, we have proposed a novel statistical model that considers the spatial interaction of printed dots at microscopic scale. This model allows to describe more accurately the formation of the shape of the binary dot. The spreading of the ink on the surface was described as the emergence of the black pixels and the interaction between pixels that depends on the distance between the centres of the dots. All these features are modelled by non-stationary Markov chain.

An estimation method for the parameters based on the Metropolis-Hasting within Gibbs algorithm is proposed. The performance of this approach is assessed, not only

Fig. 2 Figure 1 :

 21 Figure 1: Binary image of 4 single dots printed by an offset printer (a), the empirical probabilities of appearance of black pixels in the given position (profile) of 100 samples of one printed dot with the resolution 1200 dpi (b). The dots from a laser printer with the resolution 600 dpi (c), and the profile (d). The dots from an inkjet printer at 720 dpi (c), and the profile (f).

Figure 2 :

 2 Figure 2: The dots printed by two laser printers, Dell laser printer (a) and HP laser printer (b). The conditions are identical, resolution is 600 dpi.

Figure 3 :

 3 Figure 3: Scanning of 8 × 8 image

A

  Bayesian framework is employed to estimate the model parameters. Let us recall our problem, the parameters of interest in (5), (6) are θ U = {µ k } , σ 2 , β, η, λ which belong to a 2K + 4 dimensional space. The Minimum Mean Squared Error (MMSE) estimator θ of θ U given the observed data u is

Figure 4 :

 4 Figure 4: The simulation of the model (5), (6) with Gaussian power kernel with σ 2 = 70, β = 5 and different values of λ, η.

η = 1 1+e θ 4 , λ = 1 1+e θ 5 .

 45 For convenient notation, we rewrite the vector of interested 140 parameters as θ = [θ 1 , θ 2 , θ 3 , θ 4 , θ 5 ]. The Metropolis-Hastings within Gibbs using random walk is summarised in Algo. 1.

2 : 3 :

 23 for i = 1 to N iter do for b = 1 to 5 do 4:

(a) λ = 1 ( 2 Figure 5 :

 125 Figure 5: The role of λ with small β = 0.8.

(a) β = 7

 7 (b) β = 7.5 (c) β = 8

Figure 6 :

 6 Figure 6: Realization with three different values of β.

Figure 7 :

 7 Figure 7: The estimation from four printing processes and the realisations from the models.

Figure 8 :Figure 11 :

 811 Figure 8: The real dots vs. the regenerated image, the approximated centres are demonstrated.

Table 1 :

 1 The mean squared errors of the estimators with various values of β, σ 2 = 50, η = 0.8, λ = 0.6.

	σ 2	β	η	λ
	0.5 100.9 0.2041 0.0065 0.0058
	0.8 45.12	0.23	0.011 0.0068
	0.97 49.85 0.1845 0.032	0.01

Table 2 :

 2

Table 4 :

 4 The approximated PCRB and the Mean Squared Error (MSE) of each parameter in case of single dot.

Table 5 :

 5

		1 dot 2 dots 4 dots 7 dots
	σ 2 6.03	3.08	2.42	1.92
	β	0.25	0.19	0.18	0.15
	η	0.08	0.04	0.04	0.03
	λ	0.08	0.06	0.05	0.03

Table 6 :

 6 The estimators of the real dots by MCMC.

	Conventional Offset Waterless offset
	Avg.	Std.	Avg.	Std.
	β 3.71	1.4	6.55	1.74
	η 0.96	0.05	0.99	0.0004
	λ 0.67	0.15	0.52	0.13
		Laser	Inkjet
		Avg. Std. Avg. Std.
	β 1.21 0.92 2.28 0.51
	η 0.59 0.32 0.99 0.01
	λ 0.30 0.06 0.20 0.04

Table 7 :

 7 The estimators of σ 2 , β, η, and λ from four printed dots.

Table 8 :

 8 The covariance matrices calculated from the estimators of four printers.

	Actual/Predicted Con. offset Wat. offset Laser Inkjet
	Con. offset	20	0	0	0
	Wat. offset	0	20	0	0
	Laser	0	0	20	0
	Inkjet	0	0	0	20

Table 9 :

 9 Confusion matrix for ML classification of four different types of printer: conventional offset, waterless offset, laser and inkjet.

Table 10 :

 10 The mean values of the estimators of HP laser printer and Dell laser printer.

	0.0147	0.001 -0.0003	0.243 0.054 -0.003
	0.001	0.009	0.0009	0.054 0.028 0.001
	-0.0003 0.0009 0.0007	-0.003 0.001 0.0007
		(a) HP		(b) Dell

Table 11 :

 11 The covariance matrices calculated from the estimators of HP and Dell laser printers.

	Actual/Predicted HP laser Dell laser
	HP laser	20	0
	Dell laser	0	20

Table 12 :

 12 Confusion matrix for ML classification of two laser printers.

Acknowledgments Q-T. Nguyen wants to dedicate this work to Prof. Y. Delignon, 2nd author of the paper who passed away before this work can be published.


Samia Moumade
email: samia.moumade@iae-aix.com

Aurélie Hemonnet-Goujot

Pierre Valette-Florence

THE ERA OF "WOKE-WASHING": HOW CONSUMERS PERCEIVE BRAND ACTIVISM?

HAL is

PERÇOIVENT L'ACTIVISME DE MARQUE ?

Résumé : Alors que la recherche consacrée à l'activisme de marque (AM) se développe, nous ne savons pas comment les consommateurs perçoivent l'AM et son influence sur la relation avec les marques. Cette recherche exploratoire vise à comprendre la perception et les réactions des consommateurs face aux campagnes d'AM à travers une double perspectivecognitive et affective. Premièrement, il contribue à mieux délimiter la définition d'AM de marque à partir de construits étroitement liées. Deuxièmement, basée sur une technique projective AOL, d'un côté affectif, cette étude révèle le rôle potentiel de la distance psychologique dans la perception par le consommateur des efforts de la marque soutenant une cause sociale et souligne la légitimité de la marque acquise en s'engageant dans l'AM. Du côté cognitif, cette étude montre les incitations managériales derrière les marques à s'engager dans l'AM, tout en offrant aux consommateurs un moyen d'exprimer leur identité et leurs valeurs.

Mots clés : Activisme de marque; Relation consommateur-marque; Qualité de la relation avec la marque

THE ERA OF "WOKE-WASHING": HOW CONSUMERS PERCEIVE BRAND ACTIVISM?

Abstract: While research dedicated to brand activism is growing, we still do not know how consumers perceive brand activism (BA) and how it influences their relationship with brands. This exploratory research aims at gaining a better understanding of the perception and reactions of consumers towards BA campaigns through a dual perspective -cognitive and affective. First, it contributes to better delineate the construct of BA from closely related constructs. Second, based on an AOL projective technique, from an affective perspective, this study reveals the potential role of psychological distance in consumer's perception of the brand's efforts in supporting a social cause. It underlines the acquired brand legitimacy through engaging in brand activism. From a cognitive perspective, this study shows the managerial incentives behind brands in engaging in brand activism, while also providing a way for consumers to express their identity and values. Keywords: Brand Activism; Consumer-Brand Relationship; Brand Relationship Quality