Local Symmetry Effects in Actinide 4f X-ray Absorption in Oxides

Mar 23, 2016 - ... gap with Np 6d states was observed thus supporting a phase coexistence of Np metal and stoichiometric NpO2 which is in agreement wi...
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Local Symmetry Effects in Actinide 4f X‑ray Absorption in Oxides Sergei M. Butorin,*,† Anders Modin,† Johan R. Vegelius,† Michi-To Suzuki,†,∥ Peter M. Oppeneer,† David A. Andersson,‡ and David K. Shuh§ †

Department of Physics and Astronomy, Uppsala University, P.O. Box 516, SE-751 20 Uppsala, Sweden Materials Science in Radiation and Dynamical Extremes, Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, United States § Chemical Sciences Division, Lawrence Berkeley National Laboratory, MS 70A1150, One Cyclotron Road, Berkeley, California 94720, United States ∥ RIKEN Center for Emergent Matter Science, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan ‡

S Supporting Information *

ABSTRACT: A systematic X-ray absorption study at actinide N6,7 (4f → 6d transitions) edges was performed for light-actinide oxides including data obtained for the first time for NpO2, PuO2, and UO3. The measurements were supported by ab initio calculations based on local-densityapproximation with added 5f−5f Coulomb interaction (LDA+U). Improved energy resolution compared to common experiments at actinide L2,3 (2p → 6d transitions) edges allowed us to resolve the major structures of the unoccupied 6d density of states (DOS) and estimate the crystal-field splittings in the 6d shell directly from the spectra of light-actinide dioxides. The measurements demonstrated an enhanced sensitivity of the N6,7 spectral shape to changes in the compound crystal structure. For nonstoichiometric NpO2−x, the filling of the entire band gap with Np 6d states was observed thus supporting a phase coexistence of Np metal and stoichiometric NpO2 which is in agreement with the tentative Np−O phase diagram.

I

Doing experiments at shallow edges can provide a reduced core-hole lifetime broadening and therefore a higher energy resolution. In particular, for the uranium 4f core levels at which the transitions to the 6d states can be probed by XAS, the lifetime broadening is estimated to be between 0.4 eV4 and 0.8 eV.5 This is significantly small in comparison to the uranium 2p levels. However, so far, only a couple of X-ray absorption measurements at the N6,7 edges were performed: for Th and U metals using electron-energy-loss spectroscopy6 and for UO2 using conventional XAS.7 Here, we present a systematic study of light-actinide dioxides UO2, NpO2, and PuO2, using XAS at the N6,7 edges. In addition, UO3 was investigated as a system with the different crystal structure and different oxidation state for uranium. It is important to note that the N6,7 XAS data were obtained for the first time for NpO2, PuO2, and UO3. The study also addresses the issue of the Np−O phase diagram concerning the substoichiometric range (NpO2−x). For this range, Richter and Sari8 proposed a coexistence of Np metal and stoichiometric NpO2 phases and did not confirm the existence of the Np2O3 phase. However, based on X-ray photoelectron spectroscopy (XPS) measurements during the

n the X-ray spectroscopic research of actinides, the experiments are usually conducted in the hard X-ray range. The large penetration depth of high-energy X-rays makes it easier to fulfill the safety requirements and allows for easier sample handling. The most common experiments are X-ray absorption measurements at the actinide L2,3 edges which are mainly a representation of the 2p → 6d transitions and probe the unoccupied 6d states of actinides. The drawback is however a significant core-hole lifetime broadening of the spectra, for example, as high as 8.4 eV for the uranium 2p3/2 level.1 The use of the advanced technique such as high-energy-resolution fluorescence-detection X-ray absorption spectroscopy (HERFD-XAS)2,3 to reduce the core-hole lifetime broadening has improved the situation but the improvement is still not sufficient to resolve the full details in the distribution of the unoccupied 6d states using measurements at the actinide L2,3 edges. While the observed changes in the actinide L2,3 XAS on going from one system to another are usually interpreted in terms of chemical shift and consequently in terms of evolution of the chemical state of actinide in question, the shifts of the absorption edges and the change of the white line are not discussed in terms of modifications in the unoccupied density of states (DOS) and the DOS redistribution is not viewed as a reason for the observed changes in the experiment, partly due to lack of resolution. © 2016 American Chemical Society

Received: November 18, 2015 Accepted: March 23, 2016 Published: March 23, 2016 4169

DOI: 10.1021/acs.analchem.5b04380 Anal. Chem. 2016, 88, 4169−4173

Letter

Analytical Chemistry

was inductively heated to approximately 650 °C under atmosphere to oxidize material to stoichiometric NpO2 and then studied with XAS again. The polycrystalline UO2 and UO3 (γ-phase) reference samples were powders imbedded into indium, with the powders from Alfa Aesar. The studied samples were characterized by the resonant inelastic X-ray scattering (RIXS) technique at the actinide 5d edges as described in refs 15 and 16 (see the Supporting Information). The measured XAS data were compared with unoccupied d DOS of actinides in oxides which were obtained from the density-functional theory calculations, employing the localdensity-approximation with added 5f-5f Coulomb interaction (LDA+U). The details of our calculations for UO2, NpO2 and PuO2 are described in refs.17,18 and for UO3 in ref.19 Figure 1 displays the N6,7 XAS spectra of light-actinide dioxides measured in the TEY mode. The spectra probe the

slow oxidation of clean Np-metal surfaces in situ, the intermediate formation of Np2O3 was argued to have been detected.9,10 Furthermore, Beauvy et al.11 observed no neptunium metal by X-ray diffraction method in the Np−O composites for the used oxygen potential at which the phase diagram predicts an existence of the Np metal precipitates. On the contrary, our results support the predictions by the phase diagram. This paper focuses on the advances in studying of the delocalized component in the electronic structure of the actinide systems, such as actinide 6d states while the recent progress in probing the localized component, such as actinide 5f states, using the HERFD-XAS technique has been discussed in ref 12. Experiments in the energy range of the U, Np and Pu N6,7 (4f → 5g,6d transitions) X-ray absorption edges of UO2, NpO2, and PuO2, respectively, were performed at beamline 7.0.113 of the Advanced Light Source, Lawrence Berkeley National Laboratory (LBNL). U, Np, and Pu 4f XAS data were measured in the total electron yield (TEY) mode using drain current on the samples. The incidence angle of the incoming photons was close to 90° to the surface of the samples. The monochromator resolution was set to to ∼500 meV at 385 eV, to ∼400 meV at 405 eV, and to ∼500 meV at 430 eV during measurements at the U, Np, and Pu N6,7 edges, respectively. Because of strong charging effects, the U N6,7 X-ray absorption spectra of the UO3 sample could be only recorded in the total fluorescence yield (TFY) mode using a channeltron. Therefore, for the purpose of comparison, the U N6,7 spectra of UO2 were also measured in this mode. The plutonium-242 dioxide sample used for measurements was fabricated by standard techniques used to prepare radionuclide counting plates at LBNL (see the “Preparation of counting sources” subsection in ref 14). The sample was prepared from an aqueous solution of about 0.8 mM plutonium-242 (in the Pu(IV) state) in approximately 0.1 M HCl that was localized onto the surface of high purity platinum substrate (25.4 mm diameter, 0.05 mm thickness) by successive, partial micropipet aliquots into an area of about 2 mm2. The isotopic composition of the plutonium solution was about 99.9% plutonium-242 and 0.1% in the suite of Pu isotopes (238−241) by mass. The aqueous solution was allowed to dry, and the resulting solid residue was distributed in a ring-shaped manner. This structure was inductively heated to 700 °C under atmosphere to oxidize the material and to fix the material onto the substrate to preclude loss in the ultrahigh vacuum spectrometer chamber during the measurements. In 2 months, the sample was reheated again. The final amount of Pu-242 in the sample was found to be about 1.2 μg. The portion of the platinum substrate containing the plutonium was cut into a to 4 mm × 4 mm square and mounted with conductive tape on a rectangular sample holder. The neptunium-237 dioxide sample was fabricated by the same techniques. The sample was prepared from an aqueous solution of about 5.0 mM Np-237 (in the Np(IV) state) in approximately 0.1 M HCl that was localized onto the surface of a Pt substrate by successive, partial micropipet aliquots into an area of 4 mm2. The concentration and isotopic composition of the Np solution was determined by nuclear counting methods. The aqueous solution was allowed to dry on the substrate and formed a ring-shaped deposit. This residue which represented partially oxidized NpO2−x was studied with XAS, after that it

Figure 1. X-ray absorption spectra at N6,7 (4f-to-5g, 6d transitions) edges of UO2, NpO2, and PuO2, measured in the total electron yield (TEY) mode.

distribution of the unoccupied 6d states of actinides in these oxides. Although, transitions to the actinide 5g states are allowed by the dipole selection rules, these states are located high in energy and therefore are not expected to give a significant contribution here. Indeed, according to determinations with various methods (see e.g., ref 20 and references therein), the photoabsorption cross-section of uranium in the energy range of the 4f edges shows a steep increase at around 420 eV toward higher energies and represents a hump with a broad maximum at around 500 eV. A survey scan taken with a large energy step over the extended energy range confirms the existence of such a hump (see Figure 2) thus pointing toward the energies of the 5g main contribution and thus strongly suggesting that the spectra in Figure 1 mainly result from the contribution of the transition to the 6d states. It is interesting to note that the N6,7 spectra of UO2, NpO2 and PuO2 have very similar shapes. The N7 and N6 edges each exhibit three maxima. The origin of these maxima becomes clear below, upon a comparison with the results of the LDA+U calculations. In Figures 3−5, the N7 X-ray absorption edges of UO2, NpO2, and PuO2 are displayed on the binding energy scale along with the LDA+U calculated unoccupied U, Np, and Pu d DOSs, respectively, in these oxides. To bring the spectra to the binding energy scale we used the energies of the actinide 4f levels determined from X-ray photoelectron spectroscopy 4170

DOI: 10.1021/acs.analchem.5b04380 Anal. Chem. 2016, 88, 4169−4173

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Figure 2. Part of the survey X-ray absorption scan for UO2 recorded in the TEY mode with a large energy step around N6,7 edges.

Figure 4. N7 X-ray absorption spectrum of NpO2−x and NpO2 (TEY mode) compared with unoccupied Np d density of states of NpO2 calculated with the LDA+U approach.

Figure 3. N7 X-ray absorption spectrum of UO2 (TEY mode) compared with unoccupied U d density of states of UO2 calculated with the LDA+U approach.

(XPS). The binding energies of the main 4f7/2 XPS lines which were taken as 380.0 eV21 for UO2, 402.9 eV10 for NpO2, and 426.6 eV22 for PuO2 were subtracted from the photon energy scale of the corresponding N6,7 spectra. Note that in the calculated DOS, the binding energy zero is set to the top of the valence band by convention. A comparison of the N7 edge with the unoccupied d DOS of actinide in UO2, NpO2, and PuO2 indicates that the three structures/maxima appearing in the N7 spectrum of these dioxides correspond to the three groups of bands in the d DOS. For example in UO2, these three groups of bands can be identified in Figure 3 in the energy ranges of 2−6 eV, 6−9 eV, and 9−11 eV which serve as origins for the structures with maxima at ∼4.5, ∼ 7.5, and ∼9.5 eV in the measured spectrum. Similar three groups of bands in the d DOS can be seen for NpO2 and PuO2 (Figures 4 and 5) although their energy positions, spread, and separation in energy vary from one dioxide to another. Such a band grouping is a result of the cubic fluorite crystal structure of UO2, NpO2, and PuO2 and the 8fold coordination of the actinide atoms in these dioxides while the distribution in energy depends somewhat on the U value applied to the 5f states in the calculations.17,18 For example in UO2, our orbital-resolved analysis of the U d DOS reveals that the states in the energy range between ∼2 eV and ∼6 eV belong to the eg orbitals and the states in the 6−11

Figure 5. N7 X-ray absorption edge of PuO2 (measured in TEY mode) compared with unoccupied Pu d density of states of PuO2 calculated with the LDA+U approach.

eV range belong to the t2g orbitals. As a result, the ∼4.5 eV structure in the U N7 spectrum can be assigned to the contribution of the eg-derived states and the ∼7.5 eV and ∼9.5 eV structures to the t2g-derived states. The splitting between eg and t2g orbitals of the U 6d shell caused by the crystal-field interaction was estimated to be about 3.5 eV in UO2 from resonant inelastic X-ray scattering (RIXS) measurements23 at the U L3 edge (10Dq = −3.5 eV for the cubic (Oh) crystal-field symmetry). We derived a similar value of 3.7 eV from the N7 spectrum of UO2 using the maximum of the ∼4.5 eV structure and the center of gravity for the ∼7.5 eV and ∼9.5 eV structures. The same value for the crystal-field splitting between the eg and t2g orbitals was estimated for NpO2 (Figure 4) while the splitting was found to slightly increase in PuO2 (Figure 5). The summarized 10Dq values for the actinide 6d shell in the dioxides are collected in Table 1. The metal− ligand distance on going from UO2 to PuO2 changes only 4171

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our measurements support the corresponding part of the proposed Np−O phase diagram. Additional support comes from the interpretation of the RIXS data of the NpO2−x sample measured at the Np 5d edge (see the Supporting Information). Figure 6 compares the TFY spectra at the U N6,7 X-ray absorption edges of UO2 and UO3. The TFY spectrum of UO2

insignificantly and therefore the crystal-field strength is nearly the same in the light-actinide oxides. Table 1. Values of 10Dq Parameter (in eV) for the eg/t2g Splitting of 6d Shell in Cubic (Oh) Crystal Field Derived from the N7 X-ray Absorption Spectra of Light-Actinide Dioxides cmpd

10Dq value

UO2 NpO2 PuO2

−3.7 −3.7 −3.8

As mentioned before, the energy positions, spread and separation of the groups of bands in the actinide 6d DOS are sensitive to the U value used in the calculations because, although the Coulomb U is applied to the 5f states, the 6d states are hybridized with the 5f states. Therefore, an analysis of the actinide N7 spectra can help in an estimation of the Coulomb interaction strength between 5f electrons. Unfortunately, the measured spectra can be compared with the results of the LDA+U calculations only rather qualitatively instead of quantitatively, because the shape of the spectra is not as well described by a simple broadening of the calculated 6d DOS as it was in case of O K (1s → 2p transitions)18,24 and actinide L323 edges. This suggests some influence of the core hole in the final state of the spectroscopic process and a contribution of the matrix elements for the 4f → 6d transitions in defining the spectral shape. The interaction with the core hole of the valence electrons usually changes the energy positions of the major structures in the DOS so that they shift to lower energies while the matrix elements affect the relative intensities of the major DOS structures contributing to the spectra. However, when comparing the N7 XAS spectra of UO2, NpO2 and PuO2 with the LDA+U-calculated actinide d DOSs for the ground state in Figures 3−5, we find a relatively good correspondence between the main spectral structures and major DOS structures energywise and some discrepancy intensity-wise. Therefore, the main reason for observed discrepancies between the spectral shape and broadened d DOS is probably the behavior of the matrix elements rather than the core-hole effects. The fact that the crystal-field splitting value derived from U 4f XAS is very close to that estimated from RIXS at the U L3 edge where the corehole is absent in the final state of the spectroscopic process indicates the insignificance of the 4f core-hole effects in our case. In Figure 4, besides the Np N7 spectrum of NpO2, the spectrum of nonstoichiometric, oxygen-deficient NpO2−x is also displayed. This spectrum of NpO2−x reveals additional Np 6d states appearing throughout the band gap with a bump at around 0 eV. This bump, band gap filled with states, and smooth statistics of the spectrum indicate the metallic character of the NpO2−x sample with definite participation of the Np 6d electrons in metallicity of the system. These effects cannot be explained by the presence of the Np3+ species, such as, e.g., Np2O3, because they are expected to be insulating and the 6d states cannot fill the band gap in such a case and the chemical shift cannot be that large.9 According to the proposed Np−O phase diagram,8 NpO2−x is not a single phase but a combination of Np metal and stoichiometric NpO2. Such a system would indeed provide the 6d states in the band gap and at the Fermi level due to the Np-metal counterpart. Therefore,

Figure 6. N6,7 X-ray absorption spectra of UO2 and UO3 measured in the total-fluorescence-yield (TFY) mode.

shows the same structures as the TEY spectrum, although the background behaves differently. As pointed out, such spectral structures can be viewed as a signature of the cubic crystal structure and 8-fold coordination of the actinide atom with the U−O distances of 2.37 Å to the closest O neighbors. On the other hand, the spectrum of UO3 shows a different shape and is shifted to higher energies. The observed changes are not only due to the chemical shift on going from the U(IV) compound to the U(VI) compound as it becomes clear from a comparison of the experimental data with results of the LDA+U calculations for UO3 (γ-phase).19 In Figure 7, the U N7 spectrum of UO3 is displayed on the binding energy scale together with the LDA+U-calculated U d DOS of this oxide (for U = 4.5 eV and J = 0.5 eV). The N7 spectrum was brought to the binding energy scale by subtracting the energy of the U 4f7/2 XPS line of UO3 which was 381.8 eV.25 The distribution of the unoccupied U d states

Figure 7. N7 X-ray absorption spectrum of UO3 (TFY mode) compared with unoccupied U d density of states of UO3 calculated with the LDA+U approach. 4172

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(7) Tobin, J. G.; Yu, S.-W. Phys. Rev. Lett. 2011, 107, 167406. Yu, S.W; Tobin, J. G.; Crowhurst, J. C.; Sharma, S.; Dewhurst, J. K.; Olalde Velasco, P.; Yang, W. L.; Siekhaus, W. J. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 83, 165102. (8) Richter, K.; Sari, C. J. Nucl. Mater. 1987, 148, 266−271. (9) Naegele, R.; Cox, L. E.; Ward, J. W. Inorg. Chim. Acta 1987, 139, 327−329. (10) Seibert, A.; Gouder, T.; Huber, F. J. Nucl. Mater. 2009, 389, 470−478. (11) Beauvy, M.; Duverneix, T.; Berlanga, C.; Mazoyer, R.; Duriez, C. J. Alloys Compd. 1998, 271-273, 557−562. (12) Kvashnina, K. O.; Butorin, S. M.; Martin, P.; Glatzel, P. Phys. Rev. Lett. 2013, 111, 253002. (13) Warwick, T.; Heimann, P.; Mossessian, D.; McKinney, W.; Padmore, H. Rev. Sci. Instrum. 1995, 66, 2037−2040. (14) Moody, K. J.; Shaughnessy, D. A.; Casteleyn, K.; Ottmar, H.; Lützenkirchen, K.; Wallenius, M.; Wiss, T. Analytical Chemistry of Plutonium. In The Chemistry of the Actinides and Transactinide Elements; Morss, L. R., Edelstein, N. M., Fuger, J., Eds.; Springer: Dordrecht, The Netherlands, 2010; pp 3889−4003. (15) Butorin, S. M. Resonant Inelastic Soft X-Ray Scattering Spectroscopy of Light-Actinide Materials. In Actinide Nanoparticles Research; Kalmykov, S. N., Denecke, M. A., Eds.; Springer Science: Heidelburg, Germany, 2011; pp 63−104. (16) Butorin, S. M.; Shuh, D. K.; Kvashnina, K. O.; Guo, J.-H.; Werme, L.; Nordgren, J. Anal. Chem. 2013, 85, 11196−11200. (17) Suzuki, M.-T.; Magnani, N.; Oppeneer, P. M. Phys. Rev. B: Condens. Matter Mater. Phys. 2010, 82, 241103; Phys. Rev. B: Condens. Matter Mater. Phys. 2013, 88, 195146. (18) Modin, A.; Suzuki, M.-T.; Vegelius, J.; Yun, Y.; Shuh, D. K.; Werme, L.; Nordgren, J.; Oppeneer, P. M.; Butorin, S. M. J. Phys.: Condens. Matter 2015, 27, 315503. (19) He, H. M.; Andersson, D. A.; Allred, D. D.; Rector, K. D. J. Phys. Chem. C 2013, 117, 16540−16551. (20) Wendin, G.; Del Grande, N. K. Phys. Scr. 1985, 32, 286−290. (21) Idriss, H. Surf. Sci. Rep. 2010, 65, 67−109. (22) Veal, B. W.; Lam, D. J.; Diamond, H.; Hoekstra, H. R. Phys. Rev. B: Condens. Matter Mater. Phys. 1977, 15, 2929−2942. (23) Kvashnina, K. O.; Kvashnin, Y. O.; Vegelius, J. R.; Bosak, A.; Martin, P. M.; Butorin, S. M. Anal. Chem. 2015, 87, 8772−8780. (24) Modin, A.; Yun, Y.; Suzuki, M.-T.; Vegelius, J.; Werme, L.; Nordgren, J.; Oppeneer, P. M.; Butorin, S. M. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 83, 075113. (25) Belai, N.; Frisch, M.; Ilton, E. S.; Ravel, B.; Cahill, C. L. Inorg. Chem. 2008, 47, 10135−10140. (26) Loopstra, B. O.; Taylor, J. C.; Waugh, A. B. J. Solid State Chem. 1977, 20, 9−19. (27) Harrison, W. A. Electronic Structure and the Properties of Solids; Freeman: San Francisco, CA, 1980.

in UO3 appears to be significantly different from that in UO2 due to changes in the crystal structure and hybridization between the U 6d, 5f, and O 2p states, thus driving the changes in the shape of the U N7 spectrum. In orthorhombic γ-UO3, U atoms form both short (∼1.80 and 1.87 Å) and long (∼2.21, 2.26, 2.27, 2.36, and 3.04 Å) bonds26 with O so that the structure consists of parallel chains of edge-fused U(2) octahedra, cross-linked by U(1) dodecahedra. In general, the degree of the covalency of the chemical bonding and the 5f hybridization are expected to increase with an increase in the oxidation state of U and decrease in the cation−anion distance.27 The analysis of the DOS behavior in UO3 supports that (see the Supporting Information). Our results have an important consequence for the actinide L3 XAS commonly used in the actinide research where the energy shifts of the white line are usually interpreted in terms of a change in the actinide oxidation state. However, our study shows that redistribution of the actinide 6d states (which are probed by L3 XAS) due to changes in the crystal structure can also introduce significant shifts of the spectral structures.



ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.analchem.5b04380. Samples characterization and results from the LDA+U calculations for UO3 (PDF)



AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.



ACKNOWLEDGMENTS The work at the Advanced Light Source was supported by the Director, Office of Science, Office of Basic Energy Sciences, and this research (D.K.S.) was supported by the Director, Office of Science, Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences Heavy Element Chemistry Program, both of the U.S. Department of Energy at Lawrence Berkeley National Laboratory under Contract No. DE-AC02-05CH11231. P.M.O. acknowledges support from the Swedish Research Council (VR), the European Atomic Energy Community’s FP7-work programme (Grant No. 269903, “REDUPP”), and the Swedish National Infrastructure for Computing (SNIC).



REFERENCES

(1) Campbell, J. L.; Papp, T. At. Data Nucl. Data Tables 2001, 77, 1. (2) Vitova, T.; Kvashnina, K.; Nocton, G.; Sukharina, G.; Denecke, M.; Butorin, S.; Mazzanti, M.; Caciuffo, R.; Soldatov, A.; Behrends, T.; Geckeis, H. Phys. Rev. B: Condens. Matter Mater. Phys. 2010, 82, 235118. (3) Kvashnina, K. O.; Kvashnin, Y. O.; Butorin, S. M. J. Electron Spectrosc. Relat. Phenom. 2014, 194, 27−36. (4) Mårtensson, N.; Malmquist, P.-A.; Svensson, S.; Johansson, B. J. Chem. Phys. 1984, 80, 5458−5464. (5) Fuggle, J. C.; Alvarado, S. F. Phys. Rev. A: At., Mol., Opt. Phys. 1980, 22, 1615−1624. (6) Moser, H. R.; Delley, B.; Schneider, W. D.; Baer, Y. Phys. Rev. B: Condens. Matter Mater. Phys. 1984, 29, 2947−2955. 4173

DOI: 10.1021/acs.analchem.5b04380 Anal. Chem. 2016, 88, 4169−4173