Abstract

To determine Avogadro constant with a relative uncertainty of better than 2 × 10−8, the mean thickness of the oxide layer grown non-uniformly on the silicon sphere should be determined with about 0.1 nm uncertainty. An effective and flexible mapping strategy is proposed, which is insensitive to the angle resolution of the sphere-rotating mechanism. In this method, a sphere-rotating mechanism is associated with spectroscopic ellipsometer to determine the distribution of the layer, and a weighted mean method based on equal-area projection theory is applied to estimate the mean thickness. The spectroscopic ellipsometer is calibrated by X-ray reflectivity method. Within 12 hours, eight hundred positions on the silicon sphere are measured twice. The mean thickness is determined to be 4.23 nm with an uncertainty of 0.13 nm, which is in the acceptable level for the Avogadro project.

© 2010 OSA

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  1. P. Becker, “History and progress in the accurate determination of the Avogadro constant,” Rep. Prog. Phys. 64(12), 1945–2008 (2001).
    [CrossRef]
  2. P. Becker, P. D. Bièvre, K. Fujii, M. Gläser, B. Inglis, H. Luebbig, and G. Mana, “Considerations on the future redefinitions of the kilogram, the mole and of other units,” Metrologia 44(1), 1–14 (2007).
    [CrossRef]
  3. P. Becker, H. Bettin, H.-U. Danzebrink, M. Gläser, U. Kuetgens, A. Nicolaus, D. Schiel, P. D. Bièvre, S. Valkiers, and P. Taylor, “Determination of the Avogadro constant via the silicon route,” Metrologia 40(5), 271–287 (2003).
    [CrossRef]
  4. R. A. Nicolaus and K. Fujii, “Primary calibration of the volume of silicon sphere,” Meas. Sci. Technol. 17(10), 2527–2539 (2006).
    [CrossRef]
  5. K. Fujii, M. Tanaka, Y. Nezu, K. Nakayama, H. Fujimoto, P. D. Bièvre, and S. Valkiers, “Determination of the Avogadro constant by accurate measurement of the molar volume of a silicon crystal,” Metrologia 36(5), 455–464 (1999).
    [CrossRef]
  6. N. Kuramoto, K. Fujii, Y. Azuma, S. Mizushima, and Y. Toyoshima, “Density determination of silicon spheres using an interferometer with optical frequency tuning,” IEEE Trans. Instrum. Meas. 56(2), 476–480 (2007).
    [CrossRef]
  7. P. Becker, H. Friedrich, K. Fujii, W. Giardini, G. Mana, A. Picard, H.-J. Pohl, H. Riemann, and S. Valkiers, “The Avogadro constant determination via enriched silicon-28,” Meas. Sci. Technol. 20(9), 092002 (2009).
    [CrossRef]
  8. M. J. Kenny, R. P. Netterfield, L. S. Wielunski, and D. Beaglehole, “Surface layer impurities on silicon spheres used in determination of the Avogadro constant,” IEEE Trans. Instrum. Meas. 48(2), 233–237 (1999).
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    [CrossRef]
  13. D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27(2), 985–1009 (1983).
    [CrossRef]
  14. D. P. Johnson, “Geometrical considerations in the measurement of the volume of an approximate sphere,” J. Res. Natl. Bur. Stand. A 78, 41–48 (1974).
  15. G. Mana, “Volume of quasi-spherical solid density standards,” Metrologia 31(4), 289–300 (1994).
    [CrossRef]
  16. W. Giardini and J. Ha, “Measurement, characterization and volume determination of approximately spherical objects,” Meas. Sci. Technol. 5(9), 1048–1052 (1994).
    [CrossRef]
  17. J. P. Snyder, Map Projections- a working manual (US Government Printing Office, 1987), pp. 76–81.
  18. J. Zhang, Y. Li, and Z. Luo, “A traceable calibration method for spectroscopic ellipsometry,” Acta. Physica. Sinca. 59, 186–191 (2010).

2010 (1)

J. Zhang, Y. Li, and Z. Luo, “A traceable calibration method for spectroscopic ellipsometry,” Acta. Physica. Sinca. 59, 186–191 (2010).

2009 (1)

P. Becker, H. Friedrich, K. Fujii, W. Giardini, G. Mana, A. Picard, H.-J. Pohl, H. Riemann, and S. Valkiers, “The Avogadro constant determination via enriched silicon-28,” Meas. Sci. Technol. 20(9), 092002 (2009).
[CrossRef]

2007 (2)

N. Kuramoto, K. Fujii, Y. Azuma, S. Mizushima, and Y. Toyoshima, “Density determination of silicon spheres using an interferometer with optical frequency tuning,” IEEE Trans. Instrum. Meas. 56(2), 476–480 (2007).
[CrossRef]

P. Becker, P. D. Bièvre, K. Fujii, M. Gläser, B. Inglis, H. Luebbig, and G. Mana, “Considerations on the future redefinitions of the kilogram, the mole and of other units,” Metrologia 44(1), 1–14 (2007).
[CrossRef]

2006 (1)

R. A. Nicolaus and K. Fujii, “Primary calibration of the volume of silicon sphere,” Meas. Sci. Technol. 17(10), 2527–2539 (2006).
[CrossRef]

2005 (1)

Y. Azuma, J. Fan, I. Kojima, and S. Wei, “Physical structures of SiO2 ultrathin films probed by grazing incidence x-ray reflectivity,” J. Appl. Phys. 97(12), 123522 (2005).
[CrossRef]

2003 (1)

P. Becker, H. Bettin, H.-U. Danzebrink, M. Gläser, U. Kuetgens, A. Nicolaus, D. Schiel, P. D. Bièvre, S. Valkiers, and P. Taylor, “Determination of the Avogadro constant via the silicon route,” Metrologia 40(5), 271–287 (2003).
[CrossRef]

2001 (1)

P. Becker, “History and progress in the accurate determination of the Avogadro constant,” Rep. Prog. Phys. 64(12), 1945–2008 (2001).
[CrossRef]

1999 (2)

M. J. Kenny, R. P. Netterfield, L. S. Wielunski, and D. Beaglehole, “Surface layer impurities on silicon spheres used in determination of the Avogadro constant,” IEEE Trans. Instrum. Meas. 48(2), 233–237 (1999).
[CrossRef]

K. Fujii, M. Tanaka, Y. Nezu, K. Nakayama, H. Fujimoto, P. D. Bièvre, and S. Valkiers, “Determination of the Avogadro constant by accurate measurement of the molar volume of a silicon crystal,” Metrologia 36(5), 455–464 (1999).
[CrossRef]

1994 (2)

G. Mana, “Volume of quasi-spherical solid density standards,” Metrologia 31(4), 289–300 (1994).
[CrossRef]

W. Giardini and J. Ha, “Measurement, characterization and volume determination of approximately spherical objects,” Meas. Sci. Technol. 5(9), 1048–1052 (1994).
[CrossRef]

1983 (1)

D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27(2), 985–1009 (1983).
[CrossRef]

1982 (1)

D. E. Aspnes, “Optical properties of thin films,” Thin Solid Films 89(3), 249–262 (1982).
[CrossRef]

1974 (1)

D. P. Johnson, “Geometrical considerations in the measurement of the volume of an approximate sphere,” J. Res. Natl. Bur. Stand. A 78, 41–48 (1974).

Aspnes, D. E.

D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27(2), 985–1009 (1983).
[CrossRef]

D. E. Aspnes, “Optical properties of thin films,” Thin Solid Films 89(3), 249–262 (1982).
[CrossRef]

Azuma, Y.

N. Kuramoto, K. Fujii, Y. Azuma, S. Mizushima, and Y. Toyoshima, “Density determination of silicon spheres using an interferometer with optical frequency tuning,” IEEE Trans. Instrum. Meas. 56(2), 476–480 (2007).
[CrossRef]

Y. Azuma, J. Fan, I. Kojima, and S. Wei, “Physical structures of SiO2 ultrathin films probed by grazing incidence x-ray reflectivity,” J. Appl. Phys. 97(12), 123522 (2005).
[CrossRef]

Beaglehole, D.

M. J. Kenny, R. P. Netterfield, L. S. Wielunski, and D. Beaglehole, “Surface layer impurities on silicon spheres used in determination of the Avogadro constant,” IEEE Trans. Instrum. Meas. 48(2), 233–237 (1999).
[CrossRef]

Becker, P.

P. Becker, H. Friedrich, K. Fujii, W. Giardini, G. Mana, A. Picard, H.-J. Pohl, H. Riemann, and S. Valkiers, “The Avogadro constant determination via enriched silicon-28,” Meas. Sci. Technol. 20(9), 092002 (2009).
[CrossRef]

P. Becker, P. D. Bièvre, K. Fujii, M. Gläser, B. Inglis, H. Luebbig, and G. Mana, “Considerations on the future redefinitions of the kilogram, the mole and of other units,” Metrologia 44(1), 1–14 (2007).
[CrossRef]

P. Becker, H. Bettin, H.-U. Danzebrink, M. Gläser, U. Kuetgens, A. Nicolaus, D. Schiel, P. D. Bièvre, S. Valkiers, and P. Taylor, “Determination of the Avogadro constant via the silicon route,” Metrologia 40(5), 271–287 (2003).
[CrossRef]

P. Becker, “History and progress in the accurate determination of the Avogadro constant,” Rep. Prog. Phys. 64(12), 1945–2008 (2001).
[CrossRef]

Bettin, H.

P. Becker, H. Bettin, H.-U. Danzebrink, M. Gläser, U. Kuetgens, A. Nicolaus, D. Schiel, P. D. Bièvre, S. Valkiers, and P. Taylor, “Determination of the Avogadro constant via the silicon route,” Metrologia 40(5), 271–287 (2003).
[CrossRef]

Bièvre, P. D.

P. Becker, P. D. Bièvre, K. Fujii, M. Gläser, B. Inglis, H. Luebbig, and G. Mana, “Considerations on the future redefinitions of the kilogram, the mole and of other units,” Metrologia 44(1), 1–14 (2007).
[CrossRef]

P. Becker, H. Bettin, H.-U. Danzebrink, M. Gläser, U. Kuetgens, A. Nicolaus, D. Schiel, P. D. Bièvre, S. Valkiers, and P. Taylor, “Determination of the Avogadro constant via the silicon route,” Metrologia 40(5), 271–287 (2003).
[CrossRef]

K. Fujii, M. Tanaka, Y. Nezu, K. Nakayama, H. Fujimoto, P. D. Bièvre, and S. Valkiers, “Determination of the Avogadro constant by accurate measurement of the molar volume of a silicon crystal,” Metrologia 36(5), 455–464 (1999).
[CrossRef]

Danzebrink, H.-U.

P. Becker, H. Bettin, H.-U. Danzebrink, M. Gläser, U. Kuetgens, A. Nicolaus, D. Schiel, P. D. Bièvre, S. Valkiers, and P. Taylor, “Determination of the Avogadro constant via the silicon route,” Metrologia 40(5), 271–287 (2003).
[CrossRef]

Fan, J.

Y. Azuma, J. Fan, I. Kojima, and S. Wei, “Physical structures of SiO2 ultrathin films probed by grazing incidence x-ray reflectivity,” J. Appl. Phys. 97(12), 123522 (2005).
[CrossRef]

Friedrich, H.

P. Becker, H. Friedrich, K. Fujii, W. Giardini, G. Mana, A. Picard, H.-J. Pohl, H. Riemann, and S. Valkiers, “The Avogadro constant determination via enriched silicon-28,” Meas. Sci. Technol. 20(9), 092002 (2009).
[CrossRef]

Fujii, K.

P. Becker, H. Friedrich, K. Fujii, W. Giardini, G. Mana, A. Picard, H.-J. Pohl, H. Riemann, and S. Valkiers, “The Avogadro constant determination via enriched silicon-28,” Meas. Sci. Technol. 20(9), 092002 (2009).
[CrossRef]

N. Kuramoto, K. Fujii, Y. Azuma, S. Mizushima, and Y. Toyoshima, “Density determination of silicon spheres using an interferometer with optical frequency tuning,” IEEE Trans. Instrum. Meas. 56(2), 476–480 (2007).
[CrossRef]

P. Becker, P. D. Bièvre, K. Fujii, M. Gläser, B. Inglis, H. Luebbig, and G. Mana, “Considerations on the future redefinitions of the kilogram, the mole and of other units,” Metrologia 44(1), 1–14 (2007).
[CrossRef]

R. A. Nicolaus and K. Fujii, “Primary calibration of the volume of silicon sphere,” Meas. Sci. Technol. 17(10), 2527–2539 (2006).
[CrossRef]

K. Fujii, M. Tanaka, Y. Nezu, K. Nakayama, H. Fujimoto, P. D. Bièvre, and S. Valkiers, “Determination of the Avogadro constant by accurate measurement of the molar volume of a silicon crystal,” Metrologia 36(5), 455–464 (1999).
[CrossRef]

Fujimoto, H.

K. Fujii, M. Tanaka, Y. Nezu, K. Nakayama, H. Fujimoto, P. D. Bièvre, and S. Valkiers, “Determination of the Avogadro constant by accurate measurement of the molar volume of a silicon crystal,” Metrologia 36(5), 455–464 (1999).
[CrossRef]

Giardini, W.

P. Becker, H. Friedrich, K. Fujii, W. Giardini, G. Mana, A. Picard, H.-J. Pohl, H. Riemann, and S. Valkiers, “The Avogadro constant determination via enriched silicon-28,” Meas. Sci. Technol. 20(9), 092002 (2009).
[CrossRef]

W. Giardini and J. Ha, “Measurement, characterization and volume determination of approximately spherical objects,” Meas. Sci. Technol. 5(9), 1048–1052 (1994).
[CrossRef]

Gläser, M.

P. Becker, P. D. Bièvre, K. Fujii, M. Gläser, B. Inglis, H. Luebbig, and G. Mana, “Considerations on the future redefinitions of the kilogram, the mole and of other units,” Metrologia 44(1), 1–14 (2007).
[CrossRef]

P. Becker, H. Bettin, H.-U. Danzebrink, M. Gläser, U. Kuetgens, A. Nicolaus, D. Schiel, P. D. Bièvre, S. Valkiers, and P. Taylor, “Determination of the Avogadro constant via the silicon route,” Metrologia 40(5), 271–287 (2003).
[CrossRef]

Ha, J.

W. Giardini and J. Ha, “Measurement, characterization and volume determination of approximately spherical objects,” Meas. Sci. Technol. 5(9), 1048–1052 (1994).
[CrossRef]

Inglis, B.

P. Becker, P. D. Bièvre, K. Fujii, M. Gläser, B. Inglis, H. Luebbig, and G. Mana, “Considerations on the future redefinitions of the kilogram, the mole and of other units,” Metrologia 44(1), 1–14 (2007).
[CrossRef]

Johnson, D. P.

D. P. Johnson, “Geometrical considerations in the measurement of the volume of an approximate sphere,” J. Res. Natl. Bur. Stand. A 78, 41–48 (1974).

Kenny, M. J.

M. J. Kenny, R. P. Netterfield, L. S. Wielunski, and D. Beaglehole, “Surface layer impurities on silicon spheres used in determination of the Avogadro constant,” IEEE Trans. Instrum. Meas. 48(2), 233–237 (1999).
[CrossRef]

Kojima, I.

Y. Azuma, J. Fan, I. Kojima, and S. Wei, “Physical structures of SiO2 ultrathin films probed by grazing incidence x-ray reflectivity,” J. Appl. Phys. 97(12), 123522 (2005).
[CrossRef]

Kuetgens, U.

P. Becker, H. Bettin, H.-U. Danzebrink, M. Gläser, U. Kuetgens, A. Nicolaus, D. Schiel, P. D. Bièvre, S. Valkiers, and P. Taylor, “Determination of the Avogadro constant via the silicon route,” Metrologia 40(5), 271–287 (2003).
[CrossRef]

Kuramoto, N.

N. Kuramoto, K. Fujii, Y. Azuma, S. Mizushima, and Y. Toyoshima, “Density determination of silicon spheres using an interferometer with optical frequency tuning,” IEEE Trans. Instrum. Meas. 56(2), 476–480 (2007).
[CrossRef]

Li, Y.

J. Zhang, Y. Li, and Z. Luo, “A traceable calibration method for spectroscopic ellipsometry,” Acta. Physica. Sinca. 59, 186–191 (2010).

Luebbig, H.

P. Becker, P. D. Bièvre, K. Fujii, M. Gläser, B. Inglis, H. Luebbig, and G. Mana, “Considerations on the future redefinitions of the kilogram, the mole and of other units,” Metrologia 44(1), 1–14 (2007).
[CrossRef]

Luo, Z.

J. Zhang, Y. Li, and Z. Luo, “A traceable calibration method for spectroscopic ellipsometry,” Acta. Physica. Sinca. 59, 186–191 (2010).

Mana, G.

P. Becker, H. Friedrich, K. Fujii, W. Giardini, G. Mana, A. Picard, H.-J. Pohl, H. Riemann, and S. Valkiers, “The Avogadro constant determination via enriched silicon-28,” Meas. Sci. Technol. 20(9), 092002 (2009).
[CrossRef]

P. Becker, P. D. Bièvre, K. Fujii, M. Gläser, B. Inglis, H. Luebbig, and G. Mana, “Considerations on the future redefinitions of the kilogram, the mole and of other units,” Metrologia 44(1), 1–14 (2007).
[CrossRef]

G. Mana, “Volume of quasi-spherical solid density standards,” Metrologia 31(4), 289–300 (1994).
[CrossRef]

Mizushima, S.

N. Kuramoto, K. Fujii, Y. Azuma, S. Mizushima, and Y. Toyoshima, “Density determination of silicon spheres using an interferometer with optical frequency tuning,” IEEE Trans. Instrum. Meas. 56(2), 476–480 (2007).
[CrossRef]

Nakayama, K.

K. Fujii, M. Tanaka, Y. Nezu, K. Nakayama, H. Fujimoto, P. D. Bièvre, and S. Valkiers, “Determination of the Avogadro constant by accurate measurement of the molar volume of a silicon crystal,” Metrologia 36(5), 455–464 (1999).
[CrossRef]

Netterfield, R. P.

M. J. Kenny, R. P. Netterfield, L. S. Wielunski, and D. Beaglehole, “Surface layer impurities on silicon spheres used in determination of the Avogadro constant,” IEEE Trans. Instrum. Meas. 48(2), 233–237 (1999).
[CrossRef]

Nezu, Y.

K. Fujii, M. Tanaka, Y. Nezu, K. Nakayama, H. Fujimoto, P. D. Bièvre, and S. Valkiers, “Determination of the Avogadro constant by accurate measurement of the molar volume of a silicon crystal,” Metrologia 36(5), 455–464 (1999).
[CrossRef]

Nicolaus, A.

P. Becker, H. Bettin, H.-U. Danzebrink, M. Gläser, U. Kuetgens, A. Nicolaus, D. Schiel, P. D. Bièvre, S. Valkiers, and P. Taylor, “Determination of the Avogadro constant via the silicon route,” Metrologia 40(5), 271–287 (2003).
[CrossRef]

Nicolaus, R. A.

R. A. Nicolaus and K. Fujii, “Primary calibration of the volume of silicon sphere,” Meas. Sci. Technol. 17(10), 2527–2539 (2006).
[CrossRef]

Picard, A.

P. Becker, H. Friedrich, K. Fujii, W. Giardini, G. Mana, A. Picard, H.-J. Pohl, H. Riemann, and S. Valkiers, “The Avogadro constant determination via enriched silicon-28,” Meas. Sci. Technol. 20(9), 092002 (2009).
[CrossRef]

Pohl, H.-J.

P. Becker, H. Friedrich, K. Fujii, W. Giardini, G. Mana, A. Picard, H.-J. Pohl, H. Riemann, and S. Valkiers, “The Avogadro constant determination via enriched silicon-28,” Meas. Sci. Technol. 20(9), 092002 (2009).
[CrossRef]

Riemann, H.

P. Becker, H. Friedrich, K. Fujii, W. Giardini, G. Mana, A. Picard, H.-J. Pohl, H. Riemann, and S. Valkiers, “The Avogadro constant determination via enriched silicon-28,” Meas. Sci. Technol. 20(9), 092002 (2009).
[CrossRef]

Schiel, D.

P. Becker, H. Bettin, H.-U. Danzebrink, M. Gläser, U. Kuetgens, A. Nicolaus, D. Schiel, P. D. Bièvre, S. Valkiers, and P. Taylor, “Determination of the Avogadro constant via the silicon route,” Metrologia 40(5), 271–287 (2003).
[CrossRef]

Studna, A. A.

D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27(2), 985–1009 (1983).
[CrossRef]

Tanaka, M.

K. Fujii, M. Tanaka, Y. Nezu, K. Nakayama, H. Fujimoto, P. D. Bièvre, and S. Valkiers, “Determination of the Avogadro constant by accurate measurement of the molar volume of a silicon crystal,” Metrologia 36(5), 455–464 (1999).
[CrossRef]

Taylor, P.

P. Becker, H. Bettin, H.-U. Danzebrink, M. Gläser, U. Kuetgens, A. Nicolaus, D. Schiel, P. D. Bièvre, S. Valkiers, and P. Taylor, “Determination of the Avogadro constant via the silicon route,” Metrologia 40(5), 271–287 (2003).
[CrossRef]

Toyoshima, Y.

N. Kuramoto, K. Fujii, Y. Azuma, S. Mizushima, and Y. Toyoshima, “Density determination of silicon spheres using an interferometer with optical frequency tuning,” IEEE Trans. Instrum. Meas. 56(2), 476–480 (2007).
[CrossRef]

Valkiers, S.

P. Becker, H. Friedrich, K. Fujii, W. Giardini, G. Mana, A. Picard, H.-J. Pohl, H. Riemann, and S. Valkiers, “The Avogadro constant determination via enriched silicon-28,” Meas. Sci. Technol. 20(9), 092002 (2009).
[CrossRef]

P. Becker, H. Bettin, H.-U. Danzebrink, M. Gläser, U. Kuetgens, A. Nicolaus, D. Schiel, P. D. Bièvre, S. Valkiers, and P. Taylor, “Determination of the Avogadro constant via the silicon route,” Metrologia 40(5), 271–287 (2003).
[CrossRef]

K. Fujii, M. Tanaka, Y. Nezu, K. Nakayama, H. Fujimoto, P. D. Bièvre, and S. Valkiers, “Determination of the Avogadro constant by accurate measurement of the molar volume of a silicon crystal,” Metrologia 36(5), 455–464 (1999).
[CrossRef]

Wei, S.

Y. Azuma, J. Fan, I. Kojima, and S. Wei, “Physical structures of SiO2 ultrathin films probed by grazing incidence x-ray reflectivity,” J. Appl. Phys. 97(12), 123522 (2005).
[CrossRef]

Wielunski, L. S.

M. J. Kenny, R. P. Netterfield, L. S. Wielunski, and D. Beaglehole, “Surface layer impurities on silicon spheres used in determination of the Avogadro constant,” IEEE Trans. Instrum. Meas. 48(2), 233–237 (1999).
[CrossRef]

Zhang, J.

J. Zhang, Y. Li, and Z. Luo, “A traceable calibration method for spectroscopic ellipsometry,” Acta. Physica. Sinca. 59, 186–191 (2010).

Acta. Physica. Sinca. (1)

J. Zhang, Y. Li, and Z. Luo, “A traceable calibration method for spectroscopic ellipsometry,” Acta. Physica. Sinca. 59, 186–191 (2010).

IEEE Trans. Instrum. Meas. (2)

N. Kuramoto, K. Fujii, Y. Azuma, S. Mizushima, and Y. Toyoshima, “Density determination of silicon spheres using an interferometer with optical frequency tuning,” IEEE Trans. Instrum. Meas. 56(2), 476–480 (2007).
[CrossRef]

M. J. Kenny, R. P. Netterfield, L. S. Wielunski, and D. Beaglehole, “Surface layer impurities on silicon spheres used in determination of the Avogadro constant,” IEEE Trans. Instrum. Meas. 48(2), 233–237 (1999).
[CrossRef]

J. Appl. Phys. (1)

Y. Azuma, J. Fan, I. Kojima, and S. Wei, “Physical structures of SiO2 ultrathin films probed by grazing incidence x-ray reflectivity,” J. Appl. Phys. 97(12), 123522 (2005).
[CrossRef]

J. Res. Natl. Bur. Stand. A (1)

D. P. Johnson, “Geometrical considerations in the measurement of the volume of an approximate sphere,” J. Res. Natl. Bur. Stand. A 78, 41–48 (1974).

Meas. Sci. Technol. (3)

W. Giardini and J. Ha, “Measurement, characterization and volume determination of approximately spherical objects,” Meas. Sci. Technol. 5(9), 1048–1052 (1994).
[CrossRef]

P. Becker, H. Friedrich, K. Fujii, W. Giardini, G. Mana, A. Picard, H.-J. Pohl, H. Riemann, and S. Valkiers, “The Avogadro constant determination via enriched silicon-28,” Meas. Sci. Technol. 20(9), 092002 (2009).
[CrossRef]

R. A. Nicolaus and K. Fujii, “Primary calibration of the volume of silicon sphere,” Meas. Sci. Technol. 17(10), 2527–2539 (2006).
[CrossRef]

Metrologia (4)

K. Fujii, M. Tanaka, Y. Nezu, K. Nakayama, H. Fujimoto, P. D. Bièvre, and S. Valkiers, “Determination of the Avogadro constant by accurate measurement of the molar volume of a silicon crystal,” Metrologia 36(5), 455–464 (1999).
[CrossRef]

P. Becker, P. D. Bièvre, K. Fujii, M. Gläser, B. Inglis, H. Luebbig, and G. Mana, “Considerations on the future redefinitions of the kilogram, the mole and of other units,” Metrologia 44(1), 1–14 (2007).
[CrossRef]

P. Becker, H. Bettin, H.-U. Danzebrink, M. Gläser, U. Kuetgens, A. Nicolaus, D. Schiel, P. D. Bièvre, S. Valkiers, and P. Taylor, “Determination of the Avogadro constant via the silicon route,” Metrologia 40(5), 271–287 (2003).
[CrossRef]

G. Mana, “Volume of quasi-spherical solid density standards,” Metrologia 31(4), 289–300 (1994).
[CrossRef]

Phys. Rev. B (1)

D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27(2), 985–1009 (1983).
[CrossRef]

Rep. Prog. Phys. (1)

P. Becker, “History and progress in the accurate determination of the Avogadro constant,” Rep. Prog. Phys. 64(12), 1945–2008 (2001).
[CrossRef]

Thin Solid Films (1)

D. E. Aspnes, “Optical properties of thin films,” Thin Solid Films 89(3), 249–262 (1982).
[CrossRef]

Other (3)

D. F. Edwards, “Silicon(Si),” in Handbook of Optical Constants of Solid, E. D. Palik ed., (Academic, 1985).

J. P. Snyder, Map Projections- a working manual (US Government Printing Office, 1987), pp. 76–81.

H. Fujiwara, Spectroscopic Ellipsometry: principles and applications (Wiley, 2007).

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Figures (7)

Fig. 1
Fig. 1

Optical model of the native oxide layer on the silicon sphere, c-Si is single crystal silicon.

Fig. 2
Fig. 2

Two sampling methods. (a) equal-area sampling, (b) equal-spherical angle sampling.

Fig. 3
Fig. 3

Experimental set-up. (L) light source, (P) polarizer, (C) compensator, (A) analyzer, (S1), (S2), and (S3) slits, (D) detector, (1) supporting pins, (2) supporting stage, (3) rotating wheels, (4) rotating stage, (5) lifting stage, (6) supporting pillars, (7) adjusting screws, (8) baseboard, (9) adjustable board.

Fig. 4
Fig. 4

Calibration curve of the SE by XRR. dXRR and dSE are measurement results of the XRR and SE, and b1 is the slope of the fitted line.

Fig. 5
Fig. 5

Measurement result of the layer thickness on one position of the silicon sphere by SE. The MSE of the fitting between measured and theory data is 0.85. PSI and DELTA are the parameters that SE measures directly.

Fig. 6
Fig. 6

The mapped results of the oxide layer on 1600 positions of the silicon sphere by SE. Theta and Phi are spherical angles.(a) is the result of 800 positions, and (b) is that of the other 800 positions by changing the initial position of the silicon sphere randomly before the mapping.

Fig. 7
Fig. 7

Repeated measurement results by SE versus the temperature in 10 hours. The sampling time interval was one minute.

Tables (1)

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Table 1 Uncertainty budget of the mean thickness of the oxide layer

Equations (5)

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d = λ / 2 [ ( Θ m + 1 2 2 δ ) 1 / 2 ( Θ m 2 2 δ ) 1 / 2 ] ,
ϕ = x / sin ( y ) θ = y } ,
t ¯ = i = 1 N t i / N ,
t ¯ = i = 1 N w i t i / i = 1 N w i ,
t ¯ = ( j = 1 N i = 1 M t ( θ j , ϕ i ) sin θ j ) / ( M j = 1 N sin θ j ) ,

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