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Experimental Procedure

  • Ece UykurEmail author
Chapter
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Part of the Springer Theses book series (Springer Theses)

Abstract

In this thesis study, the temperature dependent reflectivity measurements have been performed on Zn-substituted YBa\(_2\)Cu\(_3\)O\(_y\) single crystals. In this chapter, information about the samples, sample preparation for measurements has been given. Moreover, the experimental setup, equipments used in measurements has also been introduced. Finally, the basics of the data analysis (fittings, error calculations, etc.) has been discussed.

Keywords

Zn-YBCO FTIR spectrscopy Reflectivity measurments Dc-resistivity 

3.1 Samples

To obtain large homogeneous single crystals with a good quality surface along c-axis is relatively difficult process, compared to the ab-plane surface. Samples used in this study were grown with a pulling technique [1]. Sample sizes were approximately 2.5 mm\(^2\) in the ac-plane, and the Zn concentrations in the samples were homogeneous, which can be seen with a sharp superconducting transition (Fig. 3.1). Superconducting transition temperatures of the samples are determined with dc susceptibility measurements. Susceptibility measurements are performed on samples, that the FTIR measurements are performed. The magnetic susceptibilities of the samples are obtained with the Magnetic Property Measurement System Superconducting Quantum Interference Device (MPMS SQUID) with applied 10 Oe magnetic field and with the zero field cooling (ZFC) method. In ZFC configuration, the sample is cooled down below the superconducting transition temperature without applying any magnetic field and at low temperature a magnetic field is applied. Then the magnetization of the sample will be measured with a temperature sweep under a constant magnetic field. T \(_c\) values are determined as the middle of the transition curve, and the width \(\varDelta \) T \(_c\) of the superconducting transition was estimated by the 10–90 % criterion of the transition curve.
Fig. 3.1

Magnetic susceptibility of the samples used in measurements. Please note that, the FTIR spectroscopy of the p = 0.16 of x = 0.012 and 0.04 had not been measured. These data has been given for comparison

All the samples are placed in a tube furnace with an alumina boat and the tube is sealed with glass fibers to keep the temperature stable inside the furnace. Samples are annealed under 100 % oxygen flow at specific temperatures for each doping level. Temperature is raised to the desired temperature quickly (within 15–20 min.) and kept at that temperature for 2–3 weeks that is depend on the doping level. The minimum annealing time depends on the sample size and the doping level that we want to achieve. At higher doping region, samples are annealed at low temperatures; therefore, it is necessary to anneal the samples longer time. For the low doping samples, we use high temperature annealing, where the oxygen diffusion becomes easier, hence the annealing time is decreasing. Annealing temperatures are determined from the annealing temperature versus oxygen concentration curve that is given by the Jorgensen et al. [2]. Zn-substitution to the Cu site does not alter the oxygen concentration; therefore, the annealing conditions to obtain the same doping level were the same as the Zn-free samples. We annealed the Zn-free and Zn-substituted samples together to keep the annealing condition in the maximum accuracy. After the annealing is finished, samples are quenched into liquid nitrogen. For small samples a copper plate can be used as the quenching environment, as well. However, in our case the samples are quite big, therefore, it is difficult to cool them down to the room temperature quickly with the copper plates, hence, liquid nitrogen has been chosen. If the cooling rate is low, the oxygen concentration in the system changes very quickly, and it becomes difficult to obtain the wanted doping level.
Fig. 3.2

Doping level versus T \(_c\) curve of YBa\(_2\)Cu\(_3\)O\(_y\). Solid symbols are experimental results, while the dashed curve is the empirical formula for YBa\(_2\)Cu\(_3\)O\(_y\). Reprinted with permission from Ref. [3]. Copyright (2006) by the American Physical Society

Previously an empirical formula has been proposed to determine the doping level of the cuprates with a known T \(_c\), as is given in Eq. 3.1. However, this empirical formula does not work for all the cuprate systems. For instance YBa\(_2\)Cu\(_3\)O\(_y\), used in this study, shows a plateau region around 1/8 doping level. Therefore, the T \(_c\) versus doping level p curve significantly deviates from the empirical formula as can be seen in Fig. 3.2. Instead of using the empirical formula we determined the doping levels of the samples by the literature values [3], which has been evaluated from the relation between the c-axis lattice constant versus oxygen concentration. Since the doping levels do not change with Zn-substitution, we determined the doping level of the Zn-substituted samples based on the Zn-free ones. This determination works well, since we can see a specific behavior in the optical conductivity, that will be explained in later chapters. The annealing conditions, T \(_c\) values with the transition width \(\varDelta \) T \(_c\) and the corresponding doping levels are given in Table 3.1.
$$\begin{aligned} \frac{{T}_c}{{T}_{c,max}}=1-82.6\times ({p}-0.16)^2 \end{aligned}$$
(3.1)
The Zn-contents, x, of the samples are determined from the work of the Masui et al. [4]. Zn content versus maximum T \(_c\) curve is prepared by using their data, where the amount of Zn is determined with the inductively coupled plasma (ICP) analysis (Fig. 3.3). And then from the linear fitting of the data, the Zn-contents have been decided for the samples that are used in this study.
Table 3.1

Zn-content (x), doping level (p), T \(_{c}\), transition width \(\varDelta \) T \(_{c}\) and annealing condition

Zn-content (x)

Doping level (p)

T \(_{c}\) (K)

\(\varDelta \) T \(_{c}\) (K)

Annealing condition

0

0.17

92

0.5

\(450\,{^\circ }\mathrm{C}\), 5 weeks

0

0.16

93.5

0.5

\(500\,{^\circ }\mathrm{C}\), 3 weeks

0

0.145

89

0.5

\(540\,{^\circ }\mathrm{C}\), 3 weeks

0

0.137

81

2

\(580\,{^\circ }\mathrm{C}\), 3 weeks

0

0.13

71

3

\(625\,{^\circ }\mathrm{C}\), 2 weeks

0

0.11

61

3

\(675\,{^\circ }\mathrm{C}\), 2 weeks

0

0.06

15

7

\(740\,{^\circ }\mathrm{C}\), 2 weeks (in nitrogen flow)

0.007

0.16

82

1

\(500\,{^\circ }\mathrm{C}\), 3 weeks

0.007

0.137

64

4

\(580\,{^\circ }\mathrm{C}\), 3 weeks

0.007

0.13

53

6

\(625\,{^\circ }\mathrm{C}\), 2 weeks

0.007

0.11

43

7

\(675\,{^\circ }\mathrm{C}\), 2 weeks

0.012

0.137

53

5

\(580\,{^\circ }\mathrm{C}\), 3 weeks

0.012

0.13

37

5

\(625\,{^\circ }\mathrm{C}\), 2 weeks

0.012

0.11

29

7

\(675\,{^\circ }\mathrm{C}\), 2 weeks

0.04

0.17

40

2

\(450\,{^\circ }\mathrm{C}\), 5 weeks

0.04

0.13

Non superconducting

\(625\,{^\circ }\mathrm{C}\), 2 weeks

Fig. 3.3

Zn content versus maximum T \(_c\) curve of YBa\(_2\)Cu\(_3\)O\(_y\). Open symbols are from [4], close symbols are the samples that are used in this study

Flat, mirror-like surfaces are necessary for the infrared measurements. The samples are polished with the Al\(_2\)O\(_3\) powders gradually as fine as \(0.3\,\upmu \mathrm{{m}}\) prior to the annealing. After annealing, before the reflectivity measurements, a final polishing is performed with \(0.3\,\upmu \mathrm{{m}}\) Al\(_2\)O\(_3\) powders to obtain clean surface before the measurements. The T \(_c\) of the samples are checked after the final polishing performed and the sample is placed on the sample holder by using silver paste.

3.2 Principle of the Fourier Transform Infrared (FTIR) Spectroscopy

Infrared radiation is the region that lies between the microwave and the visible region of the electromagnetic spectrum. Electromagnetic radiation interacts with material in different ways and different phenomena can be observed for the light propagating the material such as refraction, scattering, interference, and diffraction. Infrared spectra are based on the transitions between quantized vibrational energy states. Samples in all phases of matter can be studied with infrared spectrometry.

FTIR (Fourier Transform InfraRed) spectrometers have a dominant position for the measurements of infrared spectra. In FTIR spectrometry, all the wavelengths are measured at all times during the measurement (the multiplex or Fellgetts advantage). Moreover, more radiation can be passed between the source and the detector for each resolution element (the throughput or Jacquinots advantage). Due to these advantages, transmission, reflection and emission spectra can be measured significantly faster and with higher sensitivity with FTIR, compared to prism or grating monochromator.

The design of many interferometers used for infrared spectrometers today is based on the two-beam interferometer originally designed by Michelson in 1891. Moreover, some other two- beam interferometers also have similar basics. A schematic of this interferometer is shown in Fig. 3.4. The Michelson interferometer is a device that can divide a beam of radiation into two paths and then recombine them after a path difference is introduced. The variation of intensity of the beam emerging from the interferometer is measured as a function of path difference by a detector.
Fig. 3.4

Schematic of a Michelson interferometer

Fig. 3.5

Schematic of an interferogram of monochromatic light

Fig. 3.6

Upper panel interferogram for a light source with two wavelength. Bottom panel interferogram for a polychromatic light

A detailed explanation regarding to the interferometers has been given in Ref. [5]. If we consider a monochromatic light, the obtained intensity will be given as a function of the displacement of the moving mirror. The split light reflected from the fixed and moving mirrors will produce a destructive interference when the wavelength of the light source is equal to \(m\lambda \), while they exhibit constructive interference when the wavelength of the light source is equal to \((m+\frac{1}{2})\lambda \). Therefore, we will observe an interferogram as presented in Fig. 3.5. Here \(m\) is an integer.

In FTIR spectrometer, a polychromatic light source is used, hence we can obtain all the information of all wave numbers at the same time. In this case the interferometer can be shown as in Fig. 3.6 for a two-wavelength light source and a polychromatic light source. Please note that these are for an ideal light sources obtained for the infinite mirror displacement. In reality we can measure the interferogram only for a finite mirror displacement. Therefore, an apodization function will be used beyond the measured displacement.

After an interferometer has been obtained by taking the Fourier transform of this interferometer, we can obtain the power spectrum. Comparing the power spectrum of the sample with that of background gives us the spectrum of the sample in the measured energy range.

3.3 Experimental Details

3.3.1 Fourier Transform Infrared (FTIR) System

In our laboratory, we perform spectroscopic measurements with a Bruker Vertex 80 V Fourier transform infrared spectrometer. By utilizing a series of different sources, beamsplitters, and detectors, we can measure an energy range from 20 to 20,000 cm\(^{-1}\). Above this energy region up to \(\sim \)320,000 cm\(^{-1}\), the spectra were measured in UVSOR (Ultraviolet Synchrotron Orbital Radiation) facility, Okazaki, JAPAN. In Fig. 3.7, the relative positions of the used light sources, beamsplitters, and detectors are illustrated with a general view to the FTIR system. The system control and measurements are done with OPUS software program.
Fig. 3.7

Schematic view of Bruker 80 V FTIR system

In this system, the light emitted from one of the sources passes through the aperture and the filter, then sent to the beamsplitter. After the path difference is introduced, it is redirected to the sample space, and finally reaches the detector. In our measurements we use the reflectivity configuration, therefore, the light should be reflected at the sample surface after the beamsplitter. In Fig. 3.8, we plot the schematic picture of the sample space for the reflectivity measurements. We use an external aperture before the sample, to adjust the spot size according to our sample size. Then the reflected light is polarized to perform axis dependent measurements (polarizer information has been given in Table 3.3). Moreover, by using the s-polarization, we eliminate the additional reflection component (in the present case in-plane component). A second aperture cut the window reflections of the cryostat, where the details are given later. In each temperature measurement, we measure a spectrum of a reference mirror, preferably reflective 100 % in the chosen energy region as a background to our sample. Then we measure the sample spectrum. During this process, the optical path should be identical for both of the reference mirror and the sample. Therefore, the positions and angles of these two surfaces are checked by a He-Ne laser during the measurement.
Fig. 3.8

Schematic view of sample space

We perform our measurements in five energy region, where the details are given in Table 3.2. The power spectra of each energy region are shown in Fig. 3.9. The overlapping energy regions in each measurement range allow us to obtain a continuous spectrum over the whole energy range.
Table 3.2

Light sources, detectors, and beamsplitters used in measurements

Region

Measurement range

Light source

Beamsplitter

Detector

Far infrared_1

20–120 cm\(^{-1}\)

Mercury lamp

Mylar \(50\,\upmu \mathrm{m}\)

1.7 K bolometer

Far infrared_2

50–600 cm\(^{-1}\)

Mercury lamp

Mylar \(6\,\upmu \mathrm{m}\)

4.2 K bolometer

Middle infrared

550–5,000 cm\(^{-1}\)

Globar lamp

Ge/KBr

DTGS

Near infrared

3,500–10,000 cm\(^{-1}\)

Tungsten lamp

Si/CaF\(_{2}\)

InGaAs diode

Visible

9,000–20,000 cm\(^{-1}\)

Tungsten lamp

UV/CaF\(_{2}\)

Silicon diode

Fig. 3.9

Power spectra of the different energy regions used in measurements. These curves were obtained with 1 cm\(^{-1}\) spectral resolution. a Far infrared region, b middle infrared region, c near infrared region, d visible region

3.3.2 Low Temperature Measurements

The low temperature measurements have been performed in a He-flow cryostat of which the schematic view is given in Fig. 3.10. To avoid the absorption of the environmental gasses, we need to perform our measurements in vacuum. Moreover, to achieve the low temperature measurements, we need a higher vacuum condition that requires to isolate the sample space from the rest of the system. Therefore, we close the system with inner and outer windows. We give the relevant information in Table 3.3. Appropriate windows have been chosen to the measurement range, since the specific windows are transparent only for the specific energy ranges. Some of the windows, like the quartz ones, are crystal quartz window. Therefore, even though they transmit the light with high efficiency, they also reflect some portion. Moreover, we also observe multiple reflections between samples and the windows, as well. These might affect the spectrum greatly in some conditions; therefore, we cut these reflections by using a second aperture.
Fig. 3.10

Schematic view of the cryostat

Table 3.3

Windows and polarizers used in measurements

Region

Inner windows

Outer windows

Polarizer

Far infrared_1

Polypropylene

Polypropylene

Polyethylene wire grid

Far infrared_2

Polypropylene

Polypropylene

Polyethylene wire grid

Middle infrared

Zn-Se

KBr

KRS-250 wire grid

Near infrared

Quartz

Quartz

Glan-Taylor prism

Visible

Quartz

Quartz

Glan-Taylor prism

3.3.3 Reflectivity Measurements

The reflectivity measurements are performed in five different energy regions from 20 to 20,000 cm\(^{-1}\). Measurements at each temperature are performed with 256–512 scans. Moreover, measurement at each temperature is repeated several times (3–5 times). The repetition of the measurements at each energy region allows us to specify an error bar in this energy range. Moreover, we can also define another error bar in the overlapped energy regions for different measurement ranges. The overall error in the reflectivity is chosen as the highest error calculated in the measurements (from the repetition of the same energy range or from the overlapped regions). The error bars are calculated with the same way for each temperature separately. As a result, the maximum error in our reflectivity measurements is better than 0.4 %. In Fig. 3.11, the spectra measured for each energy region and the overlapped part for the different energy regions are given for a chosen sample as an example. Errors in measurements are determined with the spectral weight calculations of the used range of the spectra. The regions for which the spectral weights are calculated were shown with the vertical lines and the calculated errors for each spectrum is given on the corresponding graph as percentage.
Fig. 3.11

The errors in each energy scales had been determined from the spectral weight difference of the repeated measurement in the same energy range (ae). The error of the connection of the different energy scales are also determined from the spectral weight difference between the different energy scales that is calculated for the overlapped energy region of the spectra (fi). The dotted lines are the limits where the spectral weight calculations had been done. These regions show the reliable part for the each energy region

The optical conductivity spectra are determined with the Kramers-Krönig transformations from the measured reflectivity spectra, where the formalizations are given in the next section. Errors in the optical conductivities (real and imaginary) and superfluid densities are determined with the recalculations from the reflectivity spectra after the errors are included. One of the recalculated spectra for the real optical conductivity, as well as the relative error is given in Fig. 3.12. In the next chapter, the temperature dependence of the superfluid densities is given with the error bars in Fig.  4.21. As can be seen from the figures, the errors on the calculated superfluid densities have no effect on the obtained temperature scales.
Fig. 3.12

Optical conductivity has been determined from the reflectivity measurements with Kramers-Kronig transformation. After the calculated error bar is added to the reflectivity, the optical conductivity recalculated with the same method. And the difference between two calculations are determined as the error on the optical conductivity

3.4 Optical Relations

The interaction of electromagnetic radiation with matter is based on the calculations that are derived from Maxwell’s equations. The interaction of the material with light can be defined with response functions. Complex dielectric function \(\hat{\varepsilon }\) and complex conductivity \(\hat{\sigma }\) can be taken into account as the prime response functions that describe the response to the applied electric field. These response functions contain both dissipation (real part) and phase (imaginary part) information. The general considerations that can be used to derive important relations between these real and imaginary parts of the response functions are given by Kramers and Krönig [6, 7]. These are called Kramers-Krönig relations (KK relations). It is possible to set series of relations between the real and the imaginary parts of the several response functions. For instance these relations for the complex dielectric function can be given as Eqs. 3.23.4:
$$\begin{aligned} \hat{\varepsilon }(\omega ) = \varepsilon _1(\omega ) + i\varepsilon _2(\omega ) \end{aligned}$$
(3.2)
$$\begin{aligned} \varepsilon _1(\omega )-1 = \frac{2}{\pi }\fancyscript{P}\int \limits _{0}^{\infty }{\frac{\omega ^\prime \varepsilon _2(\omega ^\prime )}{\omega ^{\prime 2}-\omega ^2}}d\omega ^\prime \end{aligned}$$
(3.3)
$$\begin{aligned} \varepsilon _2(\omega ) = -\frac{2\omega }{\pi }\fancyscript{P}\int \limits _{0}^{\infty }{\frac{\varepsilon _1(\omega ^\prime )}{\omega ^{\prime 2}-\omega ^2}}d\omega ^\prime \end{aligned}$$
(3.4)
Usually the normal-incidence reflectivity \(R\)(\(\omega \)) spectra are obtained experimentally and the real and the imaginary parts of the complex dielectric function are calculated from this reflectivity value. Both \(\varepsilon _1(\omega )\) and \(\varepsilon _2(\omega )\) depend to an unknown phase of the complex reflectivity \(\hat{r}(\omega )\) (Eq. 3.5).
$$\begin{aligned} \hat{r}(\omega ) = \frac{1-\sqrt{\hat{\varepsilon }(\omega )}}{1+\sqrt{\hat{\varepsilon }(\omega )}} = \sqrt{R(\omega )}exp(i\theta (\omega )) \end{aligned}$$
(3.5)
Here \(\theta (\omega )\) is the phase, in other words the imaginary part of the complex reflectivity, while \(R\)(\(\omega \)) is the real part of the complex reflectivity response function. \(R\)(\(\omega \)) is experimentally measured, and we can calculate the phase with KK relations as Eq. 3.6.
$$\begin{aligned} \theta (\omega ) = -\frac{2\omega }{\pi }\fancyscript{P}\int \limits _{0}^{\infty }{\frac{ln\sqrt{R(\omega ^\prime )}}{\omega ^{\prime 2}-\omega ^2}}d\omega ^\prime +\theta (0) \end{aligned}$$
(3.6)
Once we have the real and imaginary components of the complex reflectivity function, we can obtain the complex dielectric function and its components, as well, by using the Eq. 3.5.
Moreover, we can obtain the complex optical conductivity and its components, as well, by using the Eqs. 3.7 and 3.8.
$$\begin{aligned} \sigma _1(\omega ) = \frac{\omega \varepsilon _2(\omega )}{4\pi } \end{aligned}$$
(3.7)
$$\begin{aligned} \sigma _2(\omega ) = \frac{(1-\varepsilon _1(\omega ))\omega }{4\pi } \end{aligned}$$
(3.8)
Fig. 3.13

Low energy fitting of the reflectivity used for extrapolations

3.5 Fitting of the Reflectivity Spectra

As can be seen from the KK relations, these relations require data set from zero frequency up to infinity. Such kind of measurement is not possible in reality. Therefore, we need to consider some extrapolations beyond our measurement limits. These extrapolations can be chosen in different ways depending on the measurement sample. As mentioned previously, c-axis optical spectra of the cuprates are dominated by phonon modes. Therefore, in the normal state, Lorentz oscillators (for phonon modes) and a Drude component (for the weak electronic background) had been used to fit our experimental data (Dielectric function given in Eq. 3.9). In the superconducting state, instead of Drude, two-fluid fitting with Lorentz oscillators was chosen (Dielectric function given in Eq. 3.10). The fitting results were used as the low energy extrapolations (Below \(20\,\mathrm{cm}^{-1}\)). At high energy region (above 40 eV) free carrier approximation had been used as an extrapolation (\(R(\omega )\propto \omega ^{-4}\)). Low energy fittings are shown in Fig. 3.13.
$$\begin{aligned} \hat{\varepsilon }(\omega ) = \varepsilon _{\infty } - \frac{\omega _{p,Drude}^2}{\omega ^2 - i\omega \varGamma _{Drude}} + \sum _j{\frac{\omega _{p,j}^2}{\omega _{0,j}^2 - \omega ^2 - i\omega \varGamma _j}} \end{aligned}$$
(3.9)
Here \(\omega _p\) is the plasma frequency that can be defined as \(\omega _p^2\) = \(ne^2/\varepsilon _0m\). \(\varGamma \) is the damping term. \(\omega _{0,j}\) is the resonance frequency of the phonon oscillators.
$$\begin{aligned} \hat{\varepsilon }(\omega ) = \varepsilon _{\infty } - f_s\frac{\omega _{p}^2}{\omega ^2} - f_n\frac{\omega _{p}^2}{\omega ^2 + i\omega \varGamma _n} + \sum _j{\frac{\omega _{p,j}^2}{\omega _{0,j}^2 - \omega ^2 - i\omega \varGamma _j}} \end{aligned}$$
(3.10)
Here \(f_s\) and \(f_n\) are the superconducting and normal carrier volume fractions, respectively. \(f_s + f_n = 1\)
Fig. 3.14

Comparison of the dc resistivity data with the optical conductivity extrapolations to the zero frequency. Resistivity data are taken from [8]. There might be 2–3 K T \(_c\) differences between samples used in this study and the samples that the resistivity data belong to. Here U underdoped, O optimally doped, and Ov overdoped

The extrapolation values of the optical conductivity to the zero frequencies also confirm that our fittings are reasonable. dc conductivity values can be compared directly with the dc resistivity values (1/\(\rho _{dc}\) = \(\sigma _{dc}\)). In Fig. 3.14, the dc resistivity values obtained with the extrapolations of our \(\sigma _{1}\)(\(\omega \)) to the zero frequencies are given for the Zn-free samples (symbols) for several doping levels. The dc resistivity curves are also plotted for some of the samples. We chose the resistivity curves of the samples, which have the closest T \(_c\) to our samples; nonetheless, there might be 2–3 K T \(_c\) differences between the resistivity curves and our samples. The T \(_c\) of the optimally doped sample is the only one that exactly matches to our sample. In the lower doping regions, the physical properties of the system changes rapidly, but, be that as it may, our extrapolation values show a reasonable agreement with the dc resistivity measurements. Moreover, our extrapolation values show a gradual change with doping.
Fig. 3.15

Zn-substitution effect on in-plane (a) and c-axis (b) resistivity. c shows the absolute values of the resistivity at 273 K as a function of the Zn-content for in-plane and c-axis resistivity. Reprinted from Ref. [9] with permission from Institute of Physics Publishing. All rights reserved (2009)

Fig. 3.16

Doping and Zn-dependence of the c-axis resistivity obtained from the extrapolation of the optical conductivity

With Zn-substitution, although the in-plane resistivity and the residual resistivity are gradually increasing, it has been shown that the c-axis resistivity does not change significantly. In Fig. 3.15, a comparison of the in-plane and c-axis resistivity as a function of Zn-content is shown [9]. Other groups also report similar Zn-insensitive behavior along the c-axis direction for lower doping region [10]. In Fig. 3.16, we plotted the extrapolation values of the optical conductivity to the zero frequency for several doping levels for several doping levels.

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Copyright information

© Springer Japan 2015

Authors and Affiliations

  1. 1.Experimental Physics IIAugsburg UniversityAugsburgGermany

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