J. Phys. Chem. C 2007, 111, 5599-5604
5599
Microstructure Evolution and Grain Growth Kinetics in Annealed Nanocrystalline Chromium Grzegorz Chojnowski, Radosław Przeniosło,* and Izabela Sosnowska Institute of Experimental Physics, Warsaw UniVersity, Hoz˘ a 69, PL 00-681 Warsaw, Poland
Mirko Bukowski, Harald Natter, and Rolf Hempelmann Institute of Physical Chemistry, UniVersity of Saarbru¨cken, D-66123 Saarbru¨cken, Germany
Andrew Fitch and Volker Urban† European Synchrotron Radiation Laboratory, F-38043 Grenoble, France ReceiVed: October 25, 2006; In Final Form: February 13, 2007
The kinetics of thermal evolution of the microstructure of nanocrystalline chromium (nano-Cr) has been studied by time-resolved synchrotron radiation techniques: high-resolution powder diffraction and smallangle X-ray scattering (SAXS). The as-prepared electrodeposited nano-Cr with average grain size of 27 nm shows the same bcc structure as R-Cr. The nano-Cr cubic lattice parameter thermal expansion is the same as that of reference polycrystalline R-Cr. Annealing of nano-Cr at temperatures above 400 °C leads to a grain growth process with the final grain size not exceeding 125 nm even at a temperature of 700 °C. The single power-law behavior is observed by SAXS in as-prepared nano-Cr changes during annealing above 400 °C. In nano-Cr samples annealed at temperatures between 400 and 700 °C, the low-q part of the SAXS signal shows a Porod-type behavior while the high-q part shows a power-law Q-R with the exponent R < 4. This effect is probably due to changes of the grain surface roughness during grain growth.
I. Introduction The thermal behavior of nanomaterials is a very important phenomenon for their industrial applications, because the working conditions for these materials are often related to large changes of temperature. Our present study is devoted to electroplated chromium, which is widely used as a coating material. Chromium coating improves, for example, the properties of steel by reducing the friction coefficient1 or by improving its fatigue properties.2 It is known that the reduced grain size and grain boundary microstructure may influence the physical properties of the material, for instance its magnetic properties. Earlier investigations of the influence of the grain size on the magnetic properties of nano-Cr by neutron diffraction3-5 have shown that the magnetic ordering and magnetic phase transitions observed in nano-Cr are different from those of bulk polycrystalline chromium.6 Our previous studies have shown that electrodeposited chromium is a nanomaterial (nano-Cr) composed of small crystallites with an average diameter of 27 nm7 with high density of about 97% of the single-crystal Cr density. Small-angle neutron scattering (SANS) studies7 have shown that electroplated nano-Cr may have a surface fractal-type morphology of pore distribution. The electroplated amorphous chromium (am-Cr) prepared by a similar method also shows high relative density and fractal-type morphology.7 The preparation method influences the microstructure of nanometals (e.g., electrodeposited nano-Ni and nano-Co have a fractal-type morphology)8 while the inert gas-condensed and ball-milled nano-Ni and nano* To whom correspondence should be addressed. E-mail:
[email protected]. † Present address: Oak Ridge National Laboratory (ORNL), Oak Ridge, TN 37831.
Co show a SANS signal characteristic for nanoparticles with smooth boundaries.9,10 It is interesting to study the changes of the microstructure due to annealing (e.g., small-angle X-ray scattering (SAXS) studies of annealed amorphous metallic glass Mg62Cu25Y10Li314 have shown important changes of the fractaltype microstructure as a function of annealing time). The combination of both diffraction and SAXS provides important complementary information on the nanomaterial microstructure. This paper describes the studies of thermal evolution of the crystalline microstructure of electrodeposited nano-Cr by highresolution X-ray diffraction and SAXS at a synchrotron radiation (SR) source. By combining high-brilliance and high-angular resolution at SR beamlines, both diffraction and SAXS experiments could be performed on nano-Cr annealed in situ. The main motivation of this study is to obtain detailed information on the time evolution of the nano-Cr microstructure during the annealing process with special emphasis on crystallite grain growth and changes of the fractal pore morphology. Similar SR diffraction studies on annealed nano-Fe11,12 and nano-Ni13 provided important insight into the grain growth phenomena occurring in these materials. II. Materials and Methods A. Preparation of Samples. The electrodeposition of Cr was performed by the method described by Hoshino et al.15 and Tsai and Wu16, as already described in ref 7. The electrolyte solution contained 100 g/l of CrO3 and 5 g/l of H2SO4. The deposition temperature was 25 °C and the direct current density was 0.4 A/cm2. The anode was a Pb plate while the cathode was a thin Cu foil of dimensions 20 × 20 mm. After deposition, the Cu electrode was removed by dissolving in a 50% solution of nitric acid at room temperature.
10.1021/jp066993q CCC: $37.00 © 2007 American Chemical Society Published on Web 03/24/2007
5600 J. Phys. Chem. C, Vol. 111, No. 15, 2007 The obtained samples were thin foils of a surface area close to that of the electrode surface (i.e., 20 × 20 mm) and of a thickness varying from 10 to 50 µm. The densities of the nanoCr samples were measured by the Archimedes method. The mass of the samples was measured in air and in liquid diethylphthalate (density 1.1175 g cm-3) by using a Mettler MT5 FACT balance. The obtained density of nano-Cr was 6.88 g cm-3 (relative density 96.3%). This nano-Cr material obtained in a form of thin foil will be referred to as “as-prepared nano-Cr”. B. Experimental. 1. SR Diffraction. The experiment was performed on the high-resolution powder diffraction beamline ID3135 at ESRF, Grenoble. Monochromatic radiation of wavelength λ ) 0.25000(23) Å was used. The as-prepared nano-Cr samples were ground and sealed in quartz capillaries (L: 0.5-0.7 mm). The grinding was necessary because of the requirements of the high-resolution SR diffraction sample mounting. Nano-Cr samples are brittle so the grinding was done with small forces, and it has been verified by laboratory X-ray diffraction that the grinding process does not change the nanoCr microstructure within experimental errors. The temperature of the sample was controlled by using a hot-air blower located below the capillary. The power of the hot-air blower is 500 W and the maximum temperature is 950 °C. There is no thermocouple near the sample. The temperature difference between the sample and the thermocouple in the unit has been calibrated using the lattice parameter of silicon. SR powder diffraction patterns were measured in the angular range from 6.5° < 2θ < 25° corresponding to the scattering vector range 2.8 Å-1 < q < 10.8 Å-1, where q ) (4π/λ)sin θ and 2θ is the scattering angle. The time of the measurement of a single scan from 6.5° up to 25° 2θ varied from 1 min up to 20 min depending on the required statistical accuracy. The detector system is made of a bank of nine scintillation detectors with approximately two degrees between channels, so a short scan of 2.2 degrees can cover an acquisition range of up to 18 degrees. To reduce texture effects, the samples were rotated. The resolution function of the diffractometer was estimated by measuring the diffraction pattern of the LaB6 National Institute of Standards and Technology (NIST) standard sample. 2. SAXS. The SAXS experiment was performed on the beamline ID0217-20 at ESRF, Grenoble. Monochromatic radiation of wavelength λ ) 0.7280(2) Å and beamsize of approximately 0.3 × 0.3 mm was used. The as-prepared nano-Cr samples in the form of thin foils were mounted between two microscope glasses in a heating setup with controlled temperature ranging from room temperature (rt) up to a maximum of 500 °C. The measurement of a single SAXS pattern took about 0.1 s. The SAXS pattern was recorded on a position-sensitive detector. A scattering vector range of 0.03 < q < 0.9 nm-1 was covered using a sample-to-detector distance of 10 m. SAXS data treatment included intensity, flat-field and angular normalizations, background subtraction, masking, regrouping, and azimuthal averaging.18,19 3. Annealing Procedures. In the SR diffraction, the annealing was performed in situ by fast-heating from rt up to the required annealing temperature. The as-prepared and ground nano-Cr sample was mounted on the diffractometer and the hot-air blower was heated to the required temperature far away from the sample. The annealing process started at time t ) 0 (i.e. when the hot-air blower was placed below the sample). The first SR diffraction measurement started at t ) 0. Each SR diffraction pattern was measured during 75 s (60 s measurements and 15 s needed to move the detectors back), and the next pattern
Chojnowski et al. measurement started immediately after the preceding one. The observed Bragg peaks positions show that the nano-Cr sample reached the annealing temperature in about 1 min. After at least 1 h of consecutive SR diffraction measurements when the diffraction patterns did no longer change with time, the measurements were stopped and the sample was fast-cooled to rt (by moving the hot-air blower away). Next, a new as-prepared sample was mounted in a new capillary, and the whole procedure was repeated at a different annealing temperature. These SR diffraction measurements with in situ annealing were performed at nine different temperatures ranging from 300 to 700 °C in 50 °C steps. In SAXS, the in situ annealing procedure was similar. The heating of as-prepared nano-Cr samples from rt up to the annealing temperatures was much slower and amounted to 50 °C/min. When the annealing temperature was reached (at time t ) 0), SAXS patterns were measured sequentially in 30 s intervals. After at least 1 h of measurements when the observed SAXS patterns were no longer changing with time, the measurements were stopped and the sample was fast-cooled to rt. Next, a new as-prepared sample was mounted, and the annealing sequence was repeated. These SAXS measurements with in situ annealing were performed at 150, 200, 250, 300, 350, 400, and 500 °C. 4. Analysis Methods. The SR diffraction patterns were analyzed by the Rietveld method to determine the lattice parameters and to detect possible texture effects. The analysis was done by using the program FullProf.21 The microstructure of nano-Cr was studied by applying the Warren Averbach method22,23 to SR diffraction data according to the procedure given in ref 24. Each diffraction peak was fitted with a symmetric pseudo-Voigt function that was used in Fourier transformations. The correction for resolution effect was applied in reciprocal space according to the Stokes’ method25 by using the parameters obtained from the reference LaB6 standard sample measurements.26 The column length distribution and the microstrain content were calculated from the peak profiles of several orders taken from the same family of planes (e.g., (110) and (220)). By assuming a log-normal distribution function of spherical-shaped grains, the procedure gave values of the median D0, the variance σ0, the area-averaged crystallite diameter 〈D〉area, and the volume-averaged crystallite diameters 〈D〉vol as described in ref 24. The whole procedure was implemented into the computer program GADGET.27 The recorded nano-Cr SAXS patterns were isotropic with respect to the azimuthal angle. Every SAXS pattern was averaged with respect to the azimuthal angle and was reduced to the SAXS signal denoted as I(q) in which q is the length of the scattering vector. The SAXS signals I(q) were analyzed by fitting linear combinations of power-law functions. When the system is composed of scattering objects with homogeneous electron density and well-defined smooth surfaces, the asymptotic part of small-angle scattering intensity distribution should obey the Porod law28
I(q) ∝ q-4
(1)
The Porod law describes the scattering of an interface that is flat with respect to the length scale probed in the scattering experiment. One possible explanation of q-R behavior with R < 4 is due to the roughness of the grain surfaces corresponding in specific cases to a surface fractal model of the microstructure.36 Surface fractals with fractal dimension 2 e dS e 3 contribute to small-angle scattering with exponents
Evolution and Kinetics in Annealed Nanocrystalline Cr
Figure 1. Rietveld analysis of the SR powder diffraction (λ ) 0.25 Å beamline ID31) pattern of as-prepared nano-Cr at room temperature. The experimental data are shown as black dots, while the intensities calculated by using FullProf21 are shown as a line. The line below indicates the difference between the observed and the calculated intensities. Ticks show the predicted positions of the Bragg peaks. The insert shows the enlarged large-angle part of the same diffraction pattern.
J. Phys. Chem. C, Vol. 111, No. 15, 2007 5601
Figure 3. Temperature dependence of the cubic lattice constant, normalized to the lattice constant at rt, obtained for nano-Cr after 60 min annealing at various temperatures (crosses) compared with reference polycrystalline Cr data31 (squares).
Figure 4. The volume-averaged grain size 〈D〉vol in nano-Cr as a function of annealing temperature. 〈D〉vol was determined from SR diffraction data after 60 min annealing at each temperature. Figure 2. Radial SAXS intensity distribution measured for as-prepared nano-Cr at room temperature (open circles). The line corresponds to the fitted power-law dependence with a flat background.
3 e R ) 6 - dS e 4.29 An alternative explanation of a q-R behavior with R < 4 may be a specific pore distribution as discussed in refs 37 and 38. III. Results A. Room-Temperature SR Diffraction Measurements for As-Prepared Nano-Cr. The SR diffraction pattern of the asprepared nano-Cr at room temperature measured with a scan of 20 min is shown in Figure 1 together with the result of the Rietveld analysis obtained with the program Fullprof.21 The observed SR diffraction pattern agrees well with the standard Cr structure (i.e., bcc (R-Cr)).30 The refined lattice constant a ) 2.89601 ( 0.00006Å differs slightly from the reference value for polycrystalline Cr (i.e., ap ) 2.88313Å).31 The microstructural parameters were the following: the volume-averaged grain diameter 〈D〉vol ) 28.9 ( (0.1) nm, the area-averaged grain diameter 〈D〉area ) 26.4 ( (0.2) nm, the distribution median D0 ) 20.9 ( (0.4) nm, and the variance σ0 ) 1.36. The volume-averaged interplanar d-spacing fluctuation due to strains ∆d/d ) 0.29%. These values are close to those reported earlier from laboratory X-ray nano-Cr studies.7 B. Room-Temperature SAXS Measurements for AsPrepared Nano-Cr. The SAXS measurements of the asprepared nano-Cr thin foil samples were done at rt by pointing the SR beam with 0.3 × 0.3 mm size to five different points of the sample separated by at least 0.6 mm from each other. The
radial intensity distribution I(q) measured in all beam positions showed the power-law behavior
I(q) ) A × q-R + B
(2)
The obtained radial intensity distribution for one representative measurement is shown in Figure 2. The values of the exponents R obtained in the measured SAXS signals I(q) are equal to 3.70, 3.47, 3.72, 3.77, and 3.25. These results may be due to a surface fractal-type morphology with fractal dimensions dS ) 6 - R32 as shown in ref 7. The mean 〈R〉 ) 3.58 ( (0.20) value agrees with our earlier SANS result of R ) 3.76 ( (0.03).7 C. Thermal Expansion and Grain Growth on Annealing of As-Prepared Nano-Cr. Changes of the microstructure of as-prepared nano-Cr samples during annealing at several temperatures between 350 and 700 °C were studied by in-situ SR diffraction and SAXS as described above. The temperature dependence of the lattice parameter a has been determined by using the Rietveld method21 at each temperature after at least 60 min annealing. The lattice parameter a of nano-Cr is shown as a function of annealing temperature in Figure 3. The measured values of the lattice constant agree well with data for reference polycrystalline Cr.31 Note that the thermal expansion of nanoCr and poly-Cr are similar though many other studies show important differences of the thermal expansion of polycrystalline and nanocrystalline materials as shown, for example, in the review by Lu.33 The X-ray diffraction peaks observed during annealing become narrower with increasing annealing time showing the largest peak-width changes at the beginning of the annealing process. The volume-averaged grain diameter 〈D〉vol increases
5602 J. Phys. Chem. C, Vol. 111, No. 15, 2007
Chojnowski et al. observed range of scattering vector length q. At temperatures above 300 °C, the value of R1 increased with time and saturated after about 60 min. SAXS data from 400 and 500 °C I(q) cannot be described by a single power-law. Shortly after the beginning of annealing, the SAXS signal shows two different regimes in the low-q and high-q parts as shown in Figure 6. We tried to fit a linear combination of two power-law functions
I(q) ) A × q-R1 + B × q-R2 + C
Figure 5. The volume-averaged lattice constant fluctuations, , due to internal strains in nano-Cr as a function of annealing temperature. The values were determined from SR diffraction data measured after 60 min annealing at each temperature.
with temperature, while the volume-averaged lattice constant fluctuations ∆d/d due to strains decrease with temperature in qualitative agreement with the results of grain growth studies for other nanocrystalline metals (e.g., nano-Fe11,12 or nano-Ni).13 The temperature dependence of the average nano-Cr grain sizes 〈D〉vol and strains ) ∆d/d obtained after 60 min of annealing are shown in Figures 4 and 5, respectively. There are no changes in both quantities between rt and 300 °C. Above 300 °C, the grain diameter increases linearly with temperature. The grain growth rate is much slower than that measured for other nanocrystalline materials (e.g., ball-milled nano-Fe)11 in which the volume-averaged nano-Fe grain diameter 〈D〉vol changed from 19 to 395 nm during annealing at 500 °C. The strain decreases above 300 °C and saturates at a value of ) 0.07% above 600 °C in a similar way as in nano-Fe.11 In annealing at temperatures below 300 °C, the SAXS signal I(q) did not change with time and showed a power-law dependence with one exponent value, denoted as R1 in the whole
(3)
The fit quality was poor with large χ2 values and in many cases the fitted B value was negative. Because the crossover between the two regimes is clearly visible on the SAXS curves, we have made a fit assuming
I(q) ) A × q-R1 + C
for q < q0
(4)
I(q) ) B × q-R2 + C
for q g q0
(5)
where the crossover value q0 is also fitted and the A and B parameter values are such that I(q) is continuous. The resulting fits describe well all the SAXS signals obtained in our annealing measurements as shown in Figure 6. The time dependence of the microstructural changes in nanoCr annealed in situ is shown in Figure 7. The upper panels (a,b) show the time dependence of the exponents R1 and R2 obtained from in situ SAXS studies, while the lower panel (c) shows the time dependence of the volume-averaged grain diameter 〈D〉vol obtained from in situ SR diffraction and the values of the crossover q0 obtained from SAXS (the annealing temperatures are shown on top of Figure 7). The value of R1 (low-q) begins to increase for temperatures above 300 °C and it tends to 4.0 (Porod law) at temperatures above 400 °C. The value of R2 (high-q) tends to about 3.15 for longer annealing times.
Figure 6. Small-angle X-ray scattering signals measured for nano-Cr annealed at 400 and 500 °C are shown on panels a and b, respectively. The low-q and high-q ranges corresponding to different exponents R1 and R2 (see text) are separated by crossover q0 values (vertical arrows). The values of the exponents R1 and R2 are also indicated.
Evolution and Kinetics in Annealed Nanocrystalline Cr
Figure 7. Time dependence of the exponents R1 and R2 determined for nano-Cr from SAXS measurements at various annealing temperatures is shown in panels a and b, respectively. The time dependence of the volume-averaged grain size and the reciprocal of the crossover value ξ ) 2π/q0 (see text) of nano-Cr determined from SR diffraction and SAXS data at various annealing temperatures is shown in panel c.
TABLE 1: Comparison of Exponents r1 and r2 Obtained from SAXS Data at Beamline ID-02 for Nano-Cr Annealed in situ temp °C
R1
R2
25 150 200 250 300 350 400 500
3.58 ( 0.02 3.63 ( 0.03 3.65 ( 0.02 3.64 ( 0.02 3.82 ( 0.04 3.72 ( 0.04 3.93 ( 0.01 4.00 ( 10-3
3.1 ( 0.1 3.19 ( 0.02
The saturation values of the two exponents R1 and R2 obtained by in situ SAXS measurements after at least 60 min annealing are shown in Table 1. IV. Discussion and Conclusions The large differences between the power-law exponent observed in SAXS measurements at different positions of the as-prepared nano-Cr foil (see Section III B) confirm the single power-law in the as-prepared nano-Cr material already reported in SANS studies7 and also indicate important inhomogeneities of the microstructure. Besides the inhomogeneities of the microstructure, SR diffraction shows a crystal structure that agrees well with the
J. Phys. Chem. C, Vol. 111, No. 15, 2007 5603 polycrystalline bcc R-Cr structure. The background is flat and the peak-to-background ratio is 230:1 for the strongest Bragg peak. One may conclude that a majority of the sample volume is composed of grains with well-defined bcc R-Cr crystal structure. The changes of the annealed nano-Cr microstructure may be summarized as follows: At temperatures above 350 °C, the average crystallite diameter grows and saturates after about 60 min of annealing. The final grain diameter increases linearly with annealing temperature (see Figure 4) and the maximum value of 〈D〉vol ≈ 100 nm is observed after annealing at 700 °C. The time dependence of the fast grain growth is similar to that observed in nano-Ni13 and could be described with the grain growth model with sizedependent or size-independent impediment as discussed in refs 13 and 34. The saturation of 〈D〉vol for nano-Cr at relatively low values ≈ 100 nm maybe due to, for example, the presence of oxide impurities that act as grain growth inhibitors as it was shown in nano-Ni.13 The SAXS signal measured for in situ annealed nano-Cr (see Figure 6) shows that, depending on the annealing temperature, one can obtain two possible final states of the nano-Cr microstructure. (i) Below 400 °C, one observes modest changes of both crystallite sizes and of the character of the SAXS signal (i.e., the power-law exponent slightly increases and saturates at values R1 < 4) after about 1 h annealing time. The microstructural correlations leading to this power-law behavior extend beyond the largest distances probed by SAXS (above 200 nm). At these temperatures, it is possible that the microstructure is described by the same model as in the as-prepared nano-Cr sample. That means it may be due to roughness or surface fractal-type correlations29 or alternatively to a specific arrangement of pores.37 (ii) Above 400 °C, the grain growth is more pronounced and the SAXS curves show a power-law behavior with different exponents. At low q, the exponent R1 tends to 4.0 (Porod law), while at large q the exponent R2 tends to about 3.15. The crossover value (denoted as q0) between the low-q and high-q parts changes with time. The reciprocal ξ ) 2π/q0 (see Figure 7c) increases with time in a similar way as the grain size. This may suggest that during annealing at temperatures above 400 °C there are changes of the grain surface roughness. The microstructure of nano-Cr after long annealing at temperatures above 400 °C is similar to that observed in SANS studies of carbonate rocks39 with R ) 4 at low-q and R ≈ 3 at high-q. The SAXS signal observed in the present studies is probably due to the contrast between empty pores and the compact metallic Cr host structure (note the relative density is about 96%). It is, however, possible that there are also small particles (ca. 0.5-1 nm) of Cr oxides located around the crystallites in the as-prepared nano-Cr samples. These Cr oxide particles inhibit grain growth and at annealing temperatures above 400 °C they may migrate and contribute to the changes of the SAXS signal. It is also possible that the non-Porod power-law in SAXS is observed from smooth particles with a “fitting” distribution of particle size. The interpretation of SAXS data is not unique in this respect; however, we think that our interpretation of surface roughness is the least arbitrary and is corroborated by the determination of particle size from the diffraction data. Acknowledgment. Thanks are due to W. Sławin´ski (Warsaw University) for his help in carrying out the measurements, and
5604 J. Phys. Chem. C, Vol. 111, No. 15, 2007 to R. Ghosh (ILL Grenoble) for his help in the SAXS data reduction process. Thanks are due to ESRF for providing beamtime. This work has been supported in part by the Ministry of Education and Science (Poland). V.U. acknowledges partial support by the Division of Materials Sciences and Engineering, Office of Basic Energy Sciences, U.S. Department of Energy, under contract DE-AC05-00OR22725 with Oak Ridge National Laboratory, managed and operated by UT-Batelle, LLC. References and Notes (1) Beer, P. Tribol. Lett. 2005, 18, 373. (2) Ortiz-Moncilla, M.; Marino-Benoteron, C.; Benitez-Ortiz, J.; Mesmacque, G.; Puchi-Cabrera, E. Surf. Eng. 2004, 20, 345. (3) Tsunoda, Y.; Nakano, H.; Matsuo, S. J. Phys.: Condens. Matter 1993, 5, L29. (4) Ishibashi, K.; Nakahigashi, H.; Tsunoda, Y. J. Phys.: Condens. Matter 1993, 5, L415. (5) Przeniosło, R.; Sosnowska, I.; Rousse, G.; Hempelmann, R. Phys. ReV. B 2002, 66, 014404. (6) Fawcett, E. ReV. Mod. Phys. 1988, 60, 209. (7) Przeniosło, R.; Wagner, J.; Natter, H.; Hempelmann, R.; Wagner, W. J. Alloys Compd. 2001, 328, 259. (8) Przeniosło, R.; Winter, R.; Natter, H.; Schmelzer, M.; Hempelmann, R.; Wagner, W. Phys. ReV. B 2001, 63, 054408. (9) Loeffler, J.; Wagner, W.; van Swygenhoven, H.; Wiedenmann, A. Nanostruct. Mater. 1997, 9, 331. (10) Loeffler, J.; Wagner, W.; Kostorz, G. J. Appl. Crystallogr. 2000, 33, 451. (11) Natter, H.; Schmelzer, M.; Lo¨ffler, M.; Krill, C.; Fitch, A.; Hempelmann, R. J. Phys. Chem. B 2000, 104, 2467. (12) Krill, C.; Helfen, L.; Michels, D.; Natter, H.; Fitch, A.; Birringer, R. Phys. ReV. Lett. 2001, 85 (5), 842. (13) Natter, H.; Lo¨ffler, M.; Krill, C.; Hempelmann, R. Scr. Mater. 2001, 44, 2321. (14) Liu, W.; Johnson, W.; Schneider, S.; Geyer, U.; Thiyagarajan, P. Phys. ReV. B 1999, 59, 11 755.
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