Etienne Gaufrès

Nicolas Izard

Laurent Vivien

Saïd Kazaoui

Delphine Marris-Morini

Eric Cassan

Enhancement of semiconducting single-wall carbon nanotubes photoluminescence

Enhancement of semiconducting single-wall carbon nanotubes photoluminescence

Etienne Gaufrès 1 , Nicolas Izard 2+ , Laurent Vivien Single wall carbon nanotube (SWNT) is a unique monodimensional material, exhibiting unusual electronic and optical properties, making it an ideal candidate for future optoelectronic [START_REF] Avouris | Carbon-nanotube photonics and optoelectronics[END_REF] and all-optical devices [START_REF] Yamashita | Saturable absorbers incorporating carbon nanotubes directly synthesized onto substrates and fibers and their application to mode-locked fiber lasers[END_REF]. The nanotube structure, defined by the so-called (n,m) chiral index, fixes all the nanotubes properties, such as metallic or semiconducting behavior. Semiconducting nanotubes (s-SWNT) are of particular interest in photonics for their direct band-gap, allowing near-IR luminescence (PL) from electron-hole recombination. In the last few years, numerous impressive results have been obtained on the photoluminescence properties of SWNT, where luminescence arise from well-isolated s-SWNT, either in solution by wrapping with surfactant [START_REF] O'connel | Band Gap Fluorescence from Individual Single-Walled Carbon Nanotubes[END_REF][START_REF] Bachilo | Band Gap Photobleaching in Isolated Single-Walled Carbon Nanotubes[END_REF][5], or dispersion in gelatin matrix [START_REF] Wang | The Optical Resonances in Carbon Nanotubes Arise from Excitons[END_REF][START_REF] Berger | Temperature Dependence of Exciton Recombination in Semiconducting Single-Wall Carbon Nanotubes[END_REF]. Recently, photoluminescent composite gels have also been demonstrated [START_REF] Zamora-Ledezma | Photoluminescent single wall carbon nanotubesilica composite gels[END_REF]. Indeed, s-SWNT are extremely sensitive to their environment. An incomplete isolation, resulting in small bundles or metallic nanotubes (m-SWNT) in contact with s-SWNT, will lead to the creation of alternative non-radiative de-excitation paths and ultimately an effective quenching of the photoluminescence. It has been estimated that only around 10 % of the s-SWNT embedded in gelatin matrix effectively participate in the fluorescence, the remainder being quenched by left-over m-SWNT, bundles or impurities [START_REF] Berger | Temperature Dependence of Exciton Recombination in Semiconducting Single-Wall Carbon Nanotubes[END_REF]. The direct influence of m-SWNT removal on the nanotubes optical properties was never really studied. Several approaches have been explored in the past few years for optimizing the diameter and chirality selection of SWNTs, such as chemical functionalization, DNA, polymer wrapping and density gradient centrifugation techniques [START_REF] Ménard-Moyon | Separation of Semiconducting from Metallic Carbon Nanotubes by Selective Functionalization with Azomethine Ylides[END_REF][START_REF] Zhang | Assessment of Chemically Separated Carbon Nanotubes for Nanoelectronics[END_REF][START_REF] Arnold | Sorting carbon nanotubes by electronic structure using density differentiation[END_REF][START_REF] Nish | Highly selective dispersion of singlewalled carbon nanotubes using aromatic polymers[END_REF][START_REF] Chen | Toward the Extraction of Single Species of Single-Walled Carbon Nanotubes Using Fluorene-Based Polymers[END_REF][START_REF] Hwang | Polymer Structure and Solvent Effects on the Selective Dispersion of Single-Walled Carbon Nanotubes[END_REF]. The possibility to achieve chiral specific SWNT extraction is indeed very promising for future photonic applications. Recently, we demonstrated that the use of poly-9,9-di-n-octyl-fluorenyl-2,7-diyl (PFO) in toluene as an extracting agent followed by ultracentrifugation allows to obtain quasi metallic-free SWNT [START_REF] Izard | Semiconductor-enriched single wall carbon nanotube networks applied to field effect transistors[END_REF]. Using sample with a constant concentration in s-SWNT but differents concentrations in m-SWNT, we show in this paper that the presence of even a few metallic nanotubes quenche the strong intrinsic photoluminescence properties of s-SWNT.

Samples were prepared as previously described [START_REF] Izard | Semiconductor-enriched single wall carbon nanotube networks applied to field effect transistors[END_REF]. First, SWNT powder (as-prepared HiPco, Carbon Nanotechnologies Inc.), poly-9,9-di-n-octyl-fluorenyl-2,7-diyl (PFO) (Sigma-Aldrich) and toluene were mixed in the ratio of SWNT (5mg):PFO (20mg):toluene (30ml) and homogenized by sonication (for 1 h using a water-bath sonicator and 15 min using a tip sonicator). Then this mixture was centrifugated for 5-60 min using either a desktop centrifuge (angle rotor type, 10.000g) or an ultracentrifuge (swing rotor type, 150.000g), after which the upper 80 % of the supernatant solution was collected. We shall focus on two types of SWNT solution: solution centrifugated at 10.000g for 15 min (labeled L) and solution centrifugated at 150.000 g for 2 hours (labeled S). We also consider the non centrifugated solution (labeled R) for comparison purpose. At this stage, the concentration of solutions R and L were adjusted by dilution to match the most intense absorption peak around 1200 nm (1.03 eV) of sample S to assure the same amount of s-SWNTs in all off the solutions (cf. Figure 1a). Solutions were then spin coated several times onto glass substrate at 500 RPM for 60 s to achieve a layer thickness around 200 nm [START_REF]Layer thickness was estimated from ellipsometry spectroscopy and direct profilometer measurement[END_REF]. Lastly, samples were annealed at 150 o C for 15 min.

Absorption spectra of R, L and S samples in toluene are presented in Figure 1a. Sharp peaks in the range of 1050-1400 nm (1.18-0.89 eV, labeled E 11 ) and 600-900 nm (2.07-1.38 eV, E 22 ) are the optical absorption bands corresponding respectively to the first and second transitions between Van Hove singularities in the s-SWNT density of states. Peaks in the range of 500-600 nm (2.48-2.07 eV, M 11 ) correspond to the absorption bands of m-SWNTs [START_REF] Chiang | Purification and Characterization of Single-Wall Carbon Nanotubes (SWNTs) Obtained from the Gas-Phase Decomposition of CO (HiPco Process)[END_REF]. Peak intensities gradually diminished and eventually vanished when all the m-SWNTs were removed by ultracentrifugation in sample S. The absorption background also drastically reduced after the first low gravity (G) centrifugation. Indeed, the centrifugation step permits the effective removal of amorphous carbon and catalyst impurities initially present in the HiPCO SWNT powder. Raman spectra for samples L and S were recorded at 514.5 nm (2.41 eV) and the radial breathing mode (RBM) and tangential mode (TM) regions are reported in Figure 1 b andc, respectively. Raman spectroscopy is a resonant process for SWNT, and the RBM peak at 190 cm -1 is specific to s-SWNT while the RBM peak at 270 cm -1 is specific to m-SWNT [START_REF] Sauvajol | Phonons in single wall carbon nanotube bundles[END_REF]. We observed that m-SWNT peaks present in sample L completely vanished in sample S. This is coherent with the TM regions, where the broad asymmetric line shape between 1400 and 1600 cm -1 of sample L, characteristic of a metallic behavior, completely disappears in sample S. Further electric measurements presented in reference [START_REF] Izard | Semiconductor-enriched single wall carbon nanotube networks applied to field effect transistors[END_REF] confirm the absence of m-SWNT in sample S. According to absorption and Raman measurements, the first low G centrifugation step (sample L) effectively remove the initially present amorphous carbon and catalyst impurities, but also remove some m-SWNT. The high G centrigugation step (sample S) remove all the remaining m-SWNT. Thus, the composition of samples L and S differ only by the presence of m-SWNT in sample L.

The PL excitation-emission maps of each samples were obtained using a Titane Sapphire (Ti:Sa) laser pumped by a cw Ar laser. Ti:Sa laser delivers continuous light at a wavelength range from 700 nm to 840 nm (1.77 to 1.48 eV). Photoluminescence signal is then recorded at room temperature using a monochromator (JobinYvon 550) coupled with a cooled InGaAs detector with a 3 nm step. Emission intensity as a function of both excitation (from 700 to 840 nm / 1.77 to 1.48 eV) and emission wavelength (from 1000 to 1600 nm / 1.24 to 0.78 eV) was recorded. The experimental values were corrected by taking into account the system response. Results are presented in Figure 2a, b andc, each map being normalized by the maximal intensity. Figure 2d shows the chirality map for the photoluminescent nanotubes from raw nanotubes in sample R and PFO purified s-SWNT in sample S. We discovered, as previously observed by Chen et al. [START_REF] Chen | Toward the Extraction of Single Species of Single-Walled Carbon Nanotubes Using Fluorene-Based Polymers[END_REF], a chiral selectivity of PFO polymer. Indeed, chirality in the starting material were evenly distributed inside the measuring range, with chiral angle ranging from 3.96 • in (12,1) tube to 27.8 • in (8,7) tubes. In contrast, PFO-extracted nanotubes consist of only two tubes in the measuring range, 73 % of [START_REF] Zamora-Ledezma | Photoluminescent single wall carbon nanotubesilica composite gels[END_REF][START_REF] Wang | The Optical Resonances in Carbon Nanotubes Arise from Excitons[END_REF] and 27 % of (8,7) nanotubes [19]. Those two nanotubes possess near armchair structure with high chiral angles above 25 • . The exact mechanism of PFO's high chiral selectivity is still unknown, but the aromatic structure of the polymer may play a major role in the wrapping interaction with nanotubes [START_REF] Hwang | Polymer Structure and Solvent Effects on the Selective Dispersion of Single-Walled Carbon Nanotubes[END_REF].

In order to determine the influence of the purification process, SWNT photoluminescence intensity for an excitation energy of 1.7 eV is displayed in Figure 3. Particular care was given to obtain the three samples with roughly the same concentration of [START_REF] Zamora-Ledezma | Photoluminescent single wall carbon nanotubesilica composite gels[END_REF][START_REF] Wang | The Optical Resonances in Carbon Nanotubes Arise from Excitons[END_REF] s-SWNT as shown on Figure 1a, which permit a comparison of the photoluminescence intensity level. Further-more, photoluminescence intensity of S and L samples were corrected by the absorption peak intensity at 1.7 eV after background substraction. It was not possible to properly remove all background contribution in R sample; uncorrected photoluminescence intensities of all three samples are presented in the inset of Figure 3. The most striking feature is the impressive increase of [START_REF] Zamora-Ledezma | Photoluminescent single wall carbon nanotubesilica composite gels[END_REF][START_REF] Wang | The Optical Resonances in Carbon Nanotubes Arise from Excitons[END_REF] nanotube photoluminescence intensity with the degree of purification. Indeed, while sample R presents a rather low signal, samples L and S exhibit PL signal enhancement. An increase by a factor of 3 is observed in the PL intensity of [START_REF] Zamora-Ledezma | Photoluminescent single wall carbon nanotubesilica composite gels[END_REF][START_REF] Wang | The Optical Resonances in Carbon Nanotubes Arise from Excitons[END_REF] nanotube for sample S as compared to L. PL intensities of L and S samples increase by a factor of 2 and 6 respectively compared to sample R (unpurified). An analog increase was also observed for the intensity of the [START_REF] Zamora-Ledezma | Photoluminescent single wall carbon nanotubesilica composite gels[END_REF][START_REF] Berger | Temperature Dependence of Exciton Recombination in Semiconducting Single-Wall Carbon Nanotubes[END_REF] nanotube, but as the relative concentration in [START_REF] Zamora-Ledezma | Photoluminescent single wall carbon nanotubesilica composite gels[END_REF][START_REF] Berger | Temperature Dependence of Exciton Recombination in Semiconducting Single-Wall Carbon Nanotubes[END_REF] nanotube change between samples, the comparison is not as straightforward. It is remarkable that even if the concentration of (8,7) tube is higher in the starting material, the photoluminescence intensity is still 3 times higher in the PFO selectively extracted material. As the number of quenching centers constituted by m-SWNT decreases, most of the exciton recombination occurring in the [START_REF] Zamora-Ledezma | Photoluminescent single wall carbon nanotubesilica composite gels[END_REF][START_REF] Wang | The Optical Resonances in Carbon Nanotubes Arise from Excitons[END_REF] nanotubes likely contribute to the luminescence phenomenon, leading to an intensity increase of S sample by factor of 3 and 6 as compared to L and R samples, respectively.

In conclusion, we have studied the photoluminescence of pure semiconducting SWNT thin films, extracted by the PFO centrifugation technique. We found that the removal of m-SWNTs leads to an enhancement of the photoluminescence properties. Indeed, after removal of m-SWNT, the same quantity of (8,6) s-SWNT displays a 6-fold increase in photoluminescence intensity. The use of PFO as a matrix leads to the formation of well isolated nanotube thin films. These results are of major importance for the future use of s-SWNT thin film matrix in photonic applications.

N. Izard thanks the Japan Society for the Promotion of Science for financial support.

Fig. 1 .

 1 Fig. 1. (color online) (a) The optical absorption spectra in toluene, (b) and (c) the RBM and TM Raman spectra at 2.41 eV of R, L and S samples. Arrows indicates respectively position of the Raman laser probe (left) and where the concentration adjustment was performed (right).

Fig. 2 .Fig. 3 .

 23 Fig. 2. (color online) Photoluminescence map (emission wavelength vs excitation wavelength) of PFO-embedded SWNT thin film spin-coated onto glass substrate. The layer thickness is around 200 nm. (a) Sample R, (b) sample L and (c) sample S. (d) Chiral map showing the wrapping preference of PFO (in blue / dark gray) as compared to raw HiPCO SWNT sample (in yellow / light gray).

  1 * , Saïd Kazaoui 2 , Delphine Marris-Morini 1 and Eric Cassan 1

	1: Institut d'Electronique Fondamentale, CNRS-UMR 8622, Université Paris-Sud 11,
	91405 Orsay, France
	2: National Institute of Advanced Industrial Science and Technology (AIST), 1-1-1 Higashi,
	Tsukuba, Ibaraki 305-8565, Japan

+: Currently at Institut d'Electronique Fondamentale laurent.vivien@u-psud.fr Photoluminescence properties of semiconducting single wall carbon nanotubes (s-SWNT) thin films with different metallic single wall carbon nanotubes (m-SWNT) concentrations are reported. s-SWNT purified samples are obtained by polymer assisted selective extraction. We show that a few m-SWNT in the sample generates a drastic quenching of the emission. Therefore, highly purified s-SWNT films are a strongly luminescent material and a good candidate for future applications in photonics, such as near infrared emitters, modulators and detectors. c 2010 Optical Society of America OCIS codes:

160.2540, 160.4236, 160.4760, 260.2510, 300.6280


Michel Darmon

Pierre Calmon

Bertrand Bèle

Modelling of the ultrasonic response of inclusions in steels

teaching and research institutions in France or abroad, or from public or private research centers.

INTRODUCTION

Ultrasonic techniques are applied in the steel industry in the aim of controlling the quality of the produced steel. The inclusions are the defects which are searched for. The harmfulness of these flaws depends on several factors such as its composition, shape, size, position in the product. Therefore it is of great interest to ensure the ability of the methods to detect and characterize these flaws. That means firstly to distinguish inclusions responses from other ultrasonic diffraction sources such as porosity echoes or microstructural noise. In this context, modelling appears to be an efficient technique to reach this goal and to improve methods of inspection and data analysis.

This paper describes a modelling study performed in the framework of a collaboration between CEA (French Atomic Energy Commission) and IRSID (Research Center of ARCELOR company). The flaws under study here are alumina inclusions from 20 m to 1mm. The inspections are performed with ultrasonic focused probes functioning in pulse echo mode, the frequency being either 10 or 50 MHz depending on the size range of the defects which are searched for. From a review of the existing approaches, the two low-frequency approximations have been selected : the Born approximation and the extended quasistatic approximation 1. These approaches have been modified in order to improve their accuracy in the domain of application under consideration. The modification led to describe the wave propagation in the flaw by taking into account both density and wave velocity differences between the host medium and the inclusion. The accuracy of these approximated models, has been evaluated in the case of a spherical scatterer by comparison with the exact analytical solution developed by Ying and Truell [2].

The modified Born approximation and the Ying and Truell model have been both implemented in the CIVA software platform [START_REF] Paradis | Review of Progress in Quantitative Non Destructive Evaluation[END_REF]. This integration allows the connection of the inclusion-wave interaction model to the transducer field computation available in CIVA, and therefore enables the simulation of a real inspection by taking into account the probe characteristics.

In order to experimentally validate the models, artificial alumina inclusions have been manufactured by diffusion welding. We also compared simulated and experimental results obtained on real flaws detected by ultrasound and characterized by X-rays. In both cases, measurements are presented and compared with the simulation.

CIVA SIMULATION APPROACH

To model the whole configuration requires to take into account the interaction between the beam and the flaw, the radiation by the probe, the propagation and the reception of the ultrasonic wave. The CIVA software platform developed at the French Atomic Energy Commission enables to model all these effects : it includes two tools of ultrasonic modelling "Champ-Sons" and "Mephisto" [START_REF] Calmon | [END_REF][START_REF] Calmon | Nuclear Engineering and Design[END_REF].

First, Champ-Sons computes the transmitted field in the specimen after propagating in the coupling medium (here water as shown in Figure 1). In this study, we are only interested in plane specimen but henceforth complex blocks defined by CAD [START_REF] Mahaut | Review of Progress in QNDE[END_REF] or heterogeneous and anisotropic media [START_REF] Gengembre | Review of Progress in QNDE[END_REF] are able to be treated. The Champ-Sons model is based on an integral formulation of the field transmitted by the component which can be seen as an extension of the Rayleigh-Sommerfield expression established in the acoustic case. The field is obtained by summing all the contributions arising from infinitesimal surface elements distributed on the emitting surface. In this aim is applied a so-called pencil method and the acoustical version of the geometrical optics approximation. We can therefore express the velocity potential  radiated by the probe here in normal incidence : Then the Mephisto code is used to calculate the echoes backscattered by a flaw. We report here the main general assumptions applied to deal with the application under consideration: First the ultrasonic field radiated by the transducer and computed by Champ-Sons is approached by the product of a spatial function ) , , ( z q   describing the amplitude distribution in the beam and a time-dependent function describing the wave propagation. The transducer being a focused one, we assume that the wave fronts in the beam are plane. The incident field on the flaw can therefore be approximated as:

  ds R T t t C t M r transmitte e i tL         2 v * ) , ( ) , ( 0 , (1)
) ( ) , , ( ) , ( 0 t t z q t M        (2) in which ) , , ( z  
are the cylindrical co-ordinates of the point M in a beam-referenced system. That is,  measures the distance to the focal axis,  is the angle between the plane of incidence and the vector joining the point M and the focal axis, z is the distance along the focal axis and

L c z c H t 1   
, H being the coupling medium height (here water). The functions ) , , ( z q   and  0 ( ) t are model inputs extracted from the Champ-Sons calculation.

In the next section, we will point out in details the beam-inclusion models we applied. Let just say here that the flaw is assumed to scatter, at a point ' M in steel, a spherical wave which is characterized by a certain directivity ) , , , (

~t M    : r c r t M t M t M L diffr ) , ( ) , , , ( ) , ' ( 1 
       (3)     ) , , , ( ) , , , ( 1 
    M A TF t M   
is the inverse Fourier transform of the spectral scattering coefficient ) , , , (

   M A
and  represents the convolution in the time domain. Lastly, by assuming the transmission-reception reciprocity of the probe and in addition, by neglecting the variations of the directivity function ) , , , ( t M    on the surface of the receiver which is quite justified in a far field configuration, we can express the echo backscattered by the flaw by :

  ) , , ( ) 2 ( ) , ( ) ( 2 0 z q t t t M t t S           (4)

MODELLING OF THE BEAM-INCLUSION INTERACTION

The aim of the modelling study is to obtain the backscattering coefficient ) , (  M A for longitudinal wave of an alumina inclusion in steel which could be connected to the general approach described above. The requirement of intensive parametrical computations has led us to select analytical or approximated semi-analytical theories. We therefore reviewed such existing approaches and selected the Born and EQSA approximations. The Born approximation applied to elastodynamics [START_REF] Gubernatis | [END_REF]9] has been shown to be effective to model weak scattering [START_REF] Chiou | Review of Progress in QNDE[END_REF]. This low frequency (ka<1) model consists in approximating fields inside the scatterer by the incident fields. This assumes small variations of the structural parameters between the host medium and the inclusion (density and elastic stiffness tensor). The extended quasistatic approximation (EQSA) [1] is a second low frequency approximation which mixes some assumptions of the Born approximation and of the static limit (accurate at very low frequency). The strain field inside the defect is replaced by the static field, calculated with incident fields supposed to be permanent while the displacement field inside the defect is approached by the incident field. In theory, validity frequency range of EQSA is larger than Born's one : ka<1,5 [1].

An exact analytical model, the so-called Ying and Truell model, is also available in the literature [2]. The solution is limited to spherical voids and inclusions.

In Figure 2, we show a comparison between simulated A-Scans obtained with CIVA by applying, for the beam-inclusion interaction, the Born approximation and Ying and Truell model. We observe that Born approximation induces an erroneous time delay between the reflections from the front 1 and back 2 flaw surfaces. This effect is also observed with EQSA approximation. So as to correct this effect and also some amplitude discrepancies, we have modified both Born and EQSA approximations by adding in the initial solutions a new assumption (called D 2 W) that amounts to take into account wave velocity and density differences between the matrix and the inclusion. ). On Figure 3 are shown the comparisons between Ying and Truell, Born and Born + D 2 W results. In this example, ka = 3, a value inside our range of interest The initial approached models (EQSA and Born) give rise to two separate echoes arising from the front and back surfaces which are not observed in Ying and Truell result. On the contrary, by adding to these two approximate models the D 2 W method, both amplitude and wave form are in a good agreement with the exact solution. A wide range of computations have been carried out in the aim of comparing the previously reported modelling approaches. Have been added in our study the high frequency Kirchhoff approximation which is applied in Mephisto code for dealing with voids. In Figure 4, is presented the amplitude versus ka curves of echoes arising from spherical alumina inclusions and voids, and flat bottom holes (FBH) in steel. Regarding voids, the comparison of Ying and Truell results with both EQSA and the high frequency Kirchhoff approximation shows a good agreement in our study range (1< ka < 4). Regarding the alumina inclusions, we can confirm that the application of D 2 W quite improves the EQSA or Born approximation and the "exact" amplitudes (of the first echo when several) are well reproduced. These results have convinced us that the Born + D 2 W and EQSA + D 2 W reliability are quite equivalent in our domain of application. We therefore decided to adopt in Civa Born + D 2 W approximation because of its capability of dealing with various flaws geometries.

In Figure 5 and Figure 6 are presented some examples of Bscans simulated by applying the Born + D 2 W model illustrating the code capability. In Figure 5 are presented two results on a spherical inclusion showing the influence of the transmitted signal bandwidth on the possibility of distinguishing the reflections from the front and back surfaces. In Figure 6 are shown on the same result responses of inclusions of various geometries (cylindrical, ellipsoidal and spherical) : the cylinder gives rise to the most important echo owing to his plane face which is favourable to backscattering.

EXPERIMENTAL VALIDATION

Experimental validations of the Born + D 2 W model have been performed on both artificial and real inclusion-type defects. We have firstly created calibrated cylindrical inclusions by diffusion welding technique and spherical inclusions by hot isostatic pressing. We present, in Figure 7, results concerning two cylindrical inclusions with same diameter and length (2 mm and 5 mm) whose centre is located at a 17.5 mm depth. The axis of revolution of the flaws is the incidence axis of the transducer. The two echoes due to the reflection on the front and back surfaces of the cylinder are detected. Measurements have been carried out at different frequencies from 3.5 MHz to 15 MHz. On Figure 7 are plotted the amplitudes of the front surface echo measured at 3.5 MHz. The superimposition of these experimental data on the theoretical amplitude versus size curve obtaining by applying Born+D 2 W model shows the reliability of the model predictions. A similar agreement (better than 3dB) has been obtained for all the ka values from 3.7 to 26.7. It is noteworthy that the agreement is necessarily better for lower ka.

We present also on Figure 8 comparisons between experimental and simulated wave forms. The agreement is quite good both at low frequency (3.5 MHz : Figure 8.a) and high frequency (15 MHz : Figure 8.b). We can observe at 15 MHz the amplitude loss between the two echoes due to attenuation in alumina. This phenomenon is taken into account in the model. The inputted attenuation parameters have been deduced from measurements carried out on alumina samples. These samples were identical to the artificial inclusions and immersed in water.

Strongly laminated stainless steel products containing natural alumina inclusions have also been investigated in the aim of validation of the model. The sample acquisition were performed at 50 MHz. The results presented on Figure 9.a) concern an alignment of inclusions detected by ultrasonics and by X-rays measurements. The figure shows the superimposition of the ultrasonic Cscan and the corresponding X-rays view. From X-rays view have been deduced the size of inclusions assumed to be ellipsoidal which have been inputted in the model. We can see on Figure 9.b) the good agreement between the simulated and experimental Cscan; the amplitude discrepancy being less than 1 dB. These excellent results have to be confirmed by additional measurements on other defects.

CONCLUSIONS

To model the beam-inclusion interaction in Civa, we chose an approximated model to treat inclusions of different shapes (cylindrical, spherical, ellipsoidal) based on Born approximation. This model noted here Born+D 2 W accounts for wave velocity and density variations between the inclusion and host material.

This model has been extensively validated either theoretically by comparison with the Ying and Truell exact analytical solution available for spherical inclusions or experimentally.

The experiments have been performed on both artificial and real flaws and show a quite satisfactory agreement.

Work is now on progress for integrating to the theoretical predictions a modelling of the microstructural noise [START_REF] Chatillon | Review of Progress in QNDE[END_REF].

  component of the velocity on the surface of the probe, T is the time of flight between the source and M; the transmission coefficient at the liquid-solid interface and e R is the divergence factor.

FIGURE 1 .

 1 FIGURE 1. Ultrasonic testing inspection.

FIGURE 2 .

 2 FIGURE 2. a) A-Scans modelled by Born (dark curve) and by Ying et Truell (bright curve) in the case of a nickel inclusion in copper (

FIGURE 3 .

 3 FIGURE 3. A-Scans of a  560µm spherical alumina inclusion in steel inspected at 10 MHz (ka=3) with the exact model (Ying : bright curve) and the approached models (Born and Born with D 2 W).

  illustrate the improvements reached by applying the D 2 W assumption we have simulated the response of a spherical alumina inclusion in steel which is characterized by important variations of structural parameters (

FIGURE 4 .

 4 FIGURE 4. Echo amplitude versus ka for spherical alumina inclusions, flat bottom holes and spherical voids in steel.

FIGURE 5 .FIGURE 6 .

 56 FIGURE 5. Influence of the bandwidth : inspection of a  1,5 mm diameter spherical inclusion.

FIGURE 7 .

 7 FIGURE 7. Comparison of the first echo amplitude between model and experiment on the cylindrical inclusions using a transducer 3.5 MHz focused at the flaw centres.

Figure 8 .

 8 Figure 8. a) Comparison of the measured and simulated normalized wave forms for the  5 mm cylindrical inclusion at 3,5 MHz (ka=9.3) b) idem at 15 MHz (ka=40.1) .

FIGURE 9 .

 9 FIGURE 9. a) Superimposed experimental C-Scan and X radiography b) experimental C-Scan c) simulated C-Scan.


Hatice Akkoç
email: hakkoc@marmara.edu.tr

Tuğba Hangül

Exploring the TPACK of pre-service mathematics teachers in real classrooms: A techno-pedagogical integration matrix perspective

Keywords: SAMR model, mathematical quality of instruction, techno-pedagogical integration matrix, pre-service mathematics teachers, field experience

This study aims to investigate pre-service mathematics teachers' (PMTs') TPACK in the context of field experiences. We offer an alternative conceptualization of TPACK utilizing a revized version of the Techno-Pedagogical Integration Matrix (TPIM). We designed a case study and the participants were two senior PMTs. We analyzed two lesson videos based on the TPIM codes and prepared a matrix that summarized the findings. Data showed that PMTs used technology at the modification level of the SAMR model for linking different representations and mathematical explanations, and used the augmentation level for promoting mathematical sense-making. None of them used it at the levels of substitution and redefinition. Findings implied that the revized version of the TPIM could be a useful tool to design course modules that aim to improve PMTs' and mathematics teachers' TPACK.

Introduction

Mathematics education research has focused on using digital technologies for mathematics teaching and its effect on students' learning in the last four decades [START_REF] Clark-Wilson | Teaching with digital technology[END_REF]. However, effective use of technology in the classroom is still an issue, especially in terms of teacher knowledge. The nature of knowledge required to successfully integrate technology into the teaching of any subject matter and how this knowledge should be developed, have been among the important problems of teacher education research. This type of knowledge was defined as Technological Pedagogical Content Knowledge (TPCK) [START_REF] Pierson | Technology integration practice as a function of pedagogical expertise[END_REF][START_REF] Niess | Preparing teachers to teach science and mathematics with technology: Developing a technology pedagogical content knowledge[END_REF][START_REF] Mishra | Technological pedagogical content knowledge: A framework for teacher knowledge[END_REF] based on [START_REF] Shulman | Those who understand: Knowledge growth in teaching[END_REF] notion of pedagogical content knowledge (PCK). Later, [START_REF] Koehler | What is technological pedagogical content knowledge (TPACK)? Contemporary Issues in Technology and Teacher Education[END_REF] referred to the framework as TPACK. [START_REF] Pierson | Technology integration practice as a function of pedagogical expertise[END_REF], [START_REF] Niess | Preparing teachers to teach science and mathematics with technology: Developing a technology pedagogical content knowledge[END_REF], and [START_REF] Mishra | Technological pedagogical content knowledge: A framework for teacher knowledge[END_REF] conceptualized it as the intersection of three knowledge domains: content knowledge, pedagogical knowledge, and technological knowledge. Researchers investigated the intersections of these domains such as (TCK, TPK, or TPCK) and elaborated the framework [START_REF] Willermark | Technological pedagogical and content knowledge: A review of empirical studies published from 2011 to 2016[END_REF]. Later, there have been criticisms about the framework regarding ignorance of contextual aspects of the framework (defined as contextual knowledge i.e., XK by [START_REF] Mishra | Considering contextual knowledge: The TPACK diagram gets an upgrade[END_REF] or XTPACK by [START_REF] Brianza | Situating TPACK: A systematic literature review of context as a domain of knowledge[END_REF] or lack of a subject-domain oriented approach to TPACK [START_REF] Voogt | Technological pedagogical content knowledge-a review of the literature[END_REF].

This study focuses on TPACK as a domain of knowledge, that is the central domain, rather than the whole framework. We embrace TPACK as the knowledge required to successfully integrate technology into mathematics teaching. We propose an alternative conceptualization of TPACK that is specific to mathematics and its pedagogy rather than a generic approach that undermines the subject matter. Therefore, our conceptualization brings the subject (mathematics) into play by utilizing the Mathematical Quality of Instruction (MQI) framework. We aim to investigate two PMTs' TPACK from the lenses of our proposed conceptual framework as explained in detail below.

Conceptual framework

This study offers an alternative conceptualization of TPACK utilizing the TPIM matrix which has two dimensions. In the earlier version of the matrix, the x-axis represents instructional quality [START_REF] Backfisch | Professional knowledge or motivation? Investigating the role of teachers' expertise on the quality of technology-enhanced lesson plans[END_REF] and the y-axis represents the levels of technology integration using [START_REF] Puentedura | SAMR and TPCK: Intro to advanced practice[END_REF]2020) SAMR (Substitution -Augmentation -Modification -Redefinition) model [START_REF] Lin | Pedagogy* technology: A two-dimensional model for teachers' ICT integration[END_REF][START_REF] Şahin | An investigation of TPACK using the scenario technique: The case of a misconception related to the derivative-limit relationship[END_REF]. Later, [START_REF] Akkoç | An investigation of pre-service mathematics teachers' TPACK using the revised techno-pedagogical integration matrix[END_REF] proposed a revised version of the TPIM placing the richness of the mathematics dimension of the Mathematical Quality of Instruction (MQI) framework on the x-axis to focus on the pedagogy of mathematics. The y-axis of the TPIM represents the levels of the SAMR model. [START_REF] Akkoç | An investigation of pre-service mathematics teachers' TPACK using the revised techno-pedagogical integration matrix[END_REF] investigated the revised version of the TPIM by analyzing the micro-teaching practices of PMTs during which they taught technologyenhanced lessons to their peers. The highest level (redefinition) was not observed in none of the TPIM studies mentioned above. In the current study, we used the revised version of the TPIM in the context of real classroom settings during PMTs' field experiences. Below, we explain the x-axis and y-axis of the TPIM further.

Mathematical quality of instruction (MQI)

The MQI was developed to evaluate the teachers' work with students related to mathematical content (Hill, 2014a). It is an observational rubric that ensures a framework for analyzing mathematics instruction in five domains one of which is the richness of the mathematics (Hill, 2014a). This domain aims to catch the depth of the mathematics in the instruction within six sub-codes: mathematical sense-making, explanations, mathematical language, patterns and generalizations, linking between representations, and multiple procedures or solution methods [START_REF] Hill | Mathematical quality of instruction (MQI: 4-point version)[END_REF]. The code of mathematical sense-making seeks the connections between mathematical ideas [START_REF] Hill | Mathematical quality of instruction (MQI: 4-point version)[END_REF]. The code of explanations considers the question of "why". For instance, why a solution method is appropriate/inappropriate, why an answer is true/false, and why a procedure works/not [START_REF] Hill | Mathematical quality of instruction (MQI: 4-point version)[END_REF]. The code of mathematical language includes the fluency of the teacher/students' use of mathematical language [START_REF] Hill | Mathematical quality of instruction (MQI: 4-point version)[END_REF]. Patterns and generalizations code is about whether there is a development/working on mathematical generalizations [START_REF] Hill | Mathematical quality of instruction (MQI: 4-point version)[END_REF]. The code of linking between representations investigates teachers'/students' linking between different representations of a mathematical concept [START_REF] Hill | Mathematical quality of instruction (MQI: 4-point version)[END_REF]. Multiple procedures or solution methods code looks for multiple procedures of a problem type or multiple solution methods of a problem [START_REF] Hill | Mathematical quality of instruction (MQI: 4-point version)[END_REF].

SAMR model

SAMR is one of the models of technology integration [START_REF] Puentedura | SAMR and TPCK: Intro to advanced practice[END_REF]2020). It defines two main and four sub-levels: substitution, and augmentation (at the enhancement level); and modification, and redefinition (at the transformation level). The substitution level refers to the use of technology as a direct tool substitute with no functional change. The teacher conducts class activities that she can do without technology using technology. At the substitution level, digital technologies simply replace traditional tools without any fundamental changes in use and function e.g. using the smart board as a presentation tool. At the augmentation level, technology acts as a direct tool substitute with functional change. Technological tools replace previous tools while enhancing teaching practices and learning experiences e.g. drawing graphs with graphing tools. At the modification level, technology allows for significant task redesign e.g. use of the slide tool of GeoGebra to explore geometrical properties. At the highest level, redefinition, new tasks are created that were not possible without technology e.g. students model real-world geometric objects outside the classroom using augmented reality tools and share their findings with their peers. We used these four levels on the y-axis of the TPIM matrix. We also defined not using technology as another level when one decides not to use technology for specific purposes. Therefore, we also added this level on the y-axis.

Methodology

We designed a case study and the participants were two senior PMTs (one female and one male) enrolled in a four-year undergraduate mathematics teacher preparation program at a state university in Istanbul, Turkey. They took various courses such as Mathematics Teaching I and II, Technology Supported Mathematics Teaching, and Teaching Practicum I prior to the student teaching. We collected data in the context of the Teaching Practicum II course during which eight PMTs taught lessons in partnership schools in their senior year. We selected the only two PMTs who integrated technology into their lessons. Data consist of one lesson video of each participant. PMT1 (male) taught trigonometric functions using a modeling activity and PMT2 (female) taught a lesson on parabolas. Both of them used GeoGebra during their lessons. Students did not have their own devices to work with GeoGebra. We analyzed the transcripts of their lessons' video recordings based on the TPIM codes. First, we specified episodes of each lesson during which the PMTs used a specific technology-enhanced task (e.g. using a dynamic worksheet of GeoGebra) or did not prefer to use technology. We independently coded each episode according to the levels of the SAMR model. The most common disagreement between the two coders was about the distinction between the augmentation and modification levels. We negotiated the disagreements after discussing the codes to increase the validity of the study. After agreeing on the levels of the SAMR model, we coded each episode according to the sub-codes of the richness of the mathematics domain in the MQI framework to explore the purposes of using technology e.g. using the slide tool to link between representations. Episodes were also analyzed based on the indicators belonging to the sub-codes of the richness of the mathematics domain of the MQI framework (see Hill, 2014a). As a result, we obtained a point on the matrix e.g. linking between representations on the x-axis and modification on the y-axis.

Findings

We present the findings using the TPIM (See Figure 1). In the matrix, the x-axis represents the subcodes of the richness of the mathematics component of the MQI framework and the y-axis shows the technology usage levels of the SAMR model. Information in each cell of the matrix represents the PMTs and the frequencies of the coordinates. For example, PMT2*3 in the coordinate (Explanations, Modification) indicates that PMT2 used technology at the modification level to make explanations three times during her lesson.

As shown on the TPIM in Figure 1, PMTs preferred the modification (19) level the most. At this level, technology was used to allow students to make mathematical inferences by obtaining different variables with the slider tool and changing them in the GeoGebra software. In some cases, PMTs tried to facilitate this process by using the trace feature along with the slider. The second most observed SAMR level was augmentation (9). This level included drawing graphs with GeoGebra, including the check box button tool in GeoGebra. As can be seen in Figure 1, the redefinition and substitution levels were not observed in PMTs lessons. Below, we explore further the purposes of technology use at each level in terms of the sub-codes of the richness of the mathematics component of the MQI framework. 12). A balanced distribution has been observed for other purposes such as patterns and generalizations (6), explanations (5), and mathematical sense-making (5). We also analyzed parts of the lessons during which PMTs did not use technology (coded as not using technology level on the y-axis). These parts were also analyzed based on MQI sub-codes. This analysis revealed three sub-codes: patterns and generalizations (4), linking between representations (3), and mathematical language (3).

In the lesson videos, the linking between representations sub-code (on the x-axis) was identified fifteen times, eight of which were at the modification level (on the y-axis). For example, PMT2 aimed at helping students make inferences about the relationship between the coefficients of a parabola and its graph using the slider tool of GeoGebra. Obtaining different parabolas in a short time, she tried to help students discover how the change in coefficients affected the graph.

The patterns and generalizations sub-code were identified ten times during the lessons of PMTs. Four of them were coded as the modification level (PMT2). In other words, PMT2 used technology at the modification level the most to arrive at a mathematical generalization. For example, PMT2 aimed to investigate the trajectory of the vertex of the parabola 𝑓𝑓(𝑥𝑥) = 𝑎𝑎𝑥𝑥 2 + 𝑏𝑏𝑥𝑥 + 𝑐𝑐 first by using paper and pencil and then GoeGebra. She asked students to find out its equation when each coefficient of the parabola changes (Click here for the GeoGebra link). She used the slider tool of GeoGebra to find the relationship between the coefficient 𝑐𝑐 and the vertex of the new parabola. By defining the coefficient 𝑐𝑐 with a slider, she stated that the vertex of the parabola moves up and down. She led the students to conclude that the vertex was "moves along the line 𝑥𝑥 = 𝑟𝑟" where 𝑟𝑟 is the apsis of the vertex. The following excerpt and GeoGebra screenshot (see Figure 2) explain the codes, patterns and generalization on the x-axis and modification on the y-axis.

PMT2:

[…] when you start increasing 𝑐𝑐, where will the parabola move on that line? […] it will move up, and when you decrease it, it will go down. We've already found this without the software. Well, what do we call the trajectory? "𝑥𝑥 = 𝑟𝑟", right? As can be seen from the excerpt above and Figure 2, PMT2 used the technology at the modification level to make a generalization.

A total of seven explanations sub-codes was identified in the lessons of PMTs. During five of these instances, PMTs used the technology at the modification level. For example, PMT1 discussed the effects of the changes in the coefficients of the functions 𝑓𝑓(𝑥𝑥) = 𝑎𝑎 𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠 (𝑏𝑏𝑥𝑥 + 𝑐𝑐) + 𝑘𝑘 on their graphs. In this context, PMT1 made mathematical explanations for the changes in the graphs of the functions by connecting the coefficients of the functions to the slider tool in the GeoGebra software.

While doing this, he used the Ferris wheel problem. In this modeling problem from the real world, a boy called Ahmet is on a Ferris wheel (with a radius of 30 meters) and sees his house at an altitude of 55 metres (the altitude of the centre is 40 meters and the one round lasts for 60 minutes). The problem asks for how long Ahmet sees his house. There were different examples of the mathematical explanations that PMT1 made in his teaching. As one of them, he emphasized the relationship between the movement of a point on the circle with the period of the function (using the trace feature of GeoGebra). PMT1:

[…] Let's also look at the trace. This is a height versus time graph. As Ahmet travels, it draws the graph. Of course, this is for one turn. What difference will it make this time when Ahmet finishes two rounds? Two of these will be drawn. If he completes three rounds, three of these. It is a periodic function.

[…]

As the excerpt above shows, PMT1 used GeoGebra to explain the connection between the movement of the point on the circle and the periodic function. In this process, he stated that one full turn of the point on the circle corresponds to the period of the function (see the screenshot in Figure 3). As can be seen from Figure 3 and PMT1's statements, he used technology at the modification level to make mathematical explanations.

Six mathematical sense-making sub-codes were determined in PMTs' lessons. In the video sections where three of them were coded, PMTs used the technology at the augmentation level. For example, in the process of making sense of functions and graphs, PMT1 used GeoGebra to draw graphs quickly to explore how the graphs and periods changed as he changed the coefficients of 𝑓𝑓(𝑥𝑥) = 𝑐𝑐𝑐𝑐𝑠𝑠(𝑥𝑥) to 𝑓𝑓(𝑥𝑥) = 2𝑐𝑐𝑐𝑐𝑠𝑠(𝑥𝑥) + 1 and 𝑓𝑓(𝑥𝑥) = 2𝑐𝑐𝑐𝑐𝑠𝑠(2𝑥𝑥) + 1. In that sense, PMT1 aimed at making sense of functions and graphs. However, we think that this is not sufficient to be considered as the explanation sub-code of MQI because he merely focused on making sense of how the changes in the algebraic representations of the functions are reflected in the graphs. PMTs preferred to use technology mainly at the augmentation level in the mathematical sense-making process.

The parts of the lessons where PMTs did not use technology (as defined by the not using technology level on the y-axis) were also analysed according to the MQI sub-codes. As a result of this analysis, the mathematical language sub-code was identified three times. That is, PMTs used mathematical terminology/language either verbally or by writing relevant symbols and signs on the board. In other words, mathematical language was not an issue in the context of technology. The following excerpt below illustrates the mathematical language code: PMT1: [...] Yes, finally I specified the height function depending on time. (Writing the formula ℎ = 𝑎𝑎 + 𝑏𝑏𝑠𝑠𝑠𝑠𝑠𝑠(𝑐𝑐𝑥𝑥 + 𝑑𝑑) on the board) We said h. Now let's name the center. What is the height of the center from the ground? Let's say 𝑎𝑎. I got the equation ℎ = 𝑎𝑎 + 𝑏𝑏𝑠𝑠𝑠𝑠𝑠𝑠(𝑐𝑐𝑥𝑥 + 𝑑𝑑), right? It is the equation we found at the beginning of the lesson. This is the height of the center from the ground (𝑎𝑎), the radius (𝑏𝑏), and the coefficient that determines the period (𝑐𝑐), which is our delay (𝑑𝑑) if there is a delay.

PMT1 drew the geometric model on the board which was also on the GeoGebra screen (see Figure 3 above) and used mathematical language appropriately in this process as the excerpt illustrates.

Discussion

This study offered an alternative conceptualization of TPACK utilizing a revised version of the Techno-Pedagogical Integration Matrix (TPIM) and explored PMTs' technology integration practices in real classroom settings. Findings showed that PMTs used technology predominantly at the modification level (y-axis) to link different representations (x-axis). The highest level (redefinition) was not observed in this study, probably because students did not have their computers or tablets to engage in tasks to make constructions using GeoGebra software.

In both this current study and the previous study [START_REF] Akkoç | An investigation of pre-service mathematics teachers' TPACK using the revised techno-pedagogical integration matrix[END_REF], PMTs targeted more mathematical sense-making at the augmentation level. The aim was rather to make mathematical explanations at the modification level. This finding indicates that some components of the MQI framework are compatible with some levels of the SAMR model. In other words, the MQI perspective shed light on the levels of the SAMR model and makes them distinguishable. We recommend that future research explore this situation in the context of different mathematical topics and technological tools conducted with more teachers or PMTs so that one could explore the relationship between the MQI sub-codes and the levels of the SAMR model especially substitution and redefinition which were not observed in this study.

This study used only the richness of the mathematics domain of the MQI model, which constitutes the x-axis of the TPIM. Future empirical studies can be conducted to investigate which other domains of MQI can be useful for this purpose. The MQI-integrated TPIM framework can also be used to monitor the progress of participants in teacher preparation courses or professional development programs in intervention studies.

Considering the findings, we suggest mathematics teacher educators design modules that aim to train PMTs or mathematics teachers about the SAMR levels. MQI codes could also be included in these modules to help them be more critical about the purposes of using technology at each level. In other words, our revised version of the TPIM could be a useful tool to design course modules that aim to improve PMTs and mathematics teachers' TPACK. Future studies could be conducted with more participants using the matrix to enter the frequencies of the participants in each cell.

Figure 1 :

 1 Figure 1: Analysis of lessons using the revised version of the TPIM An examination of the pedagogical purposes of PMTs' technology use in their lessons based on MQI sub-codes shows that technology was mostly used for linking different representations (12). A balanced distribution has been observed for other purposes such as patterns and generalizations (6), explanations (5), and mathematical sense-making (5). We also analyzed parts of the lessons during which PMTs did not use technology (coded as not using technology level on the y-axis). These parts were also analyzed based on MQI sub-codes. This analysis revealed three sub-codes: patterns and generalizations (4), linking between representations (3), and mathematical language (3).

  […] isn't it a straight line passing through here? Let's see if it's right. Yes right[…]

Figure 2 :

 2 Figure 2: A screenshot of GeoGebra activity used by PMT2

Figure 3 :

 3 Figure 3: A screenshot of GeoGebra activity used by PMT1