Abstract

The capability of operating a separate crystal x-ray interferometer over centimeter displacements has made it possible to observe minute strain fields of a bent crystal at the atomic scale resolution by means of phase-contrast x-ray topography. Measurement and predictive capabilities of lattice strain are key ingredients of a highly accurate measurement of the Si lattice parameter and of a determination of the number of atoms in a realization of the mass unit based on an atom mass. Here we show that the observed strain can be accurately predicted by a finite-element analysis of the crystal deformation.

© 2009 Optical Society of America

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References

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  1. J. Flowers, "The route to atomic and quantum standards," Science 306, 1324-1330 (2004).
    [CrossRef] [PubMed]
  2. J. Robinson, "Weighty matter," Scientific American, 102-109 (December 2006).
    [CrossRef] [PubMed]
  3. E. Massa, G. Mana, G. U. Kuetgens, and L. Ferroglio, "Measurement of the lattice parameter of a silicon crystal," New J. Phys. 11 (2009) 053013.
    [CrossRef]
  4. U. Bonse and W. Graeff, "X-ray and neutron interferometry," in: Topics in applied physics, 22 X-ray optics, applications to solids, H.-J. Queisser ed. (Berlin, Springer, 1997).
  5. G. Mana and E. Vittone, "Scanning LLL x-ray interferometry I. Theory," Z. Phys. B 102, 197-206 (1997)
    [CrossRef]
  6. U. Bonse, W. Graeff, and G. Materlik, "X-ray interferometry and lattice parameter investigations," Rev. Phys. Appl. 11, 83-87 (1976).
    [CrossRef]
  7. L. Ferroglio, G. Mana, and E. Massa, "Si lattice parameter measurement by centimeter x-ray interferometry," Opt. Express 16, 16877-16888 (2008).
    [CrossRef] [PubMed]
  8. A. G. Every and A. K. Mc Curdy, Low Frequency Properties of Dielectric Crystals: Second and Higher Order Elastic Constants in: Landolt-B¨ornstein III/29a, Nelson, D. F. ed. (Springer, Berlin, 1992).

2009 (1)

E. Massa, G. Mana, G. U. Kuetgens, and L. Ferroglio, "Measurement of the lattice parameter of a silicon crystal," New J. Phys. 11 (2009) 053013.
[CrossRef]

2008 (1)

2006 (1)

J. Robinson, "Weighty matter," Scientific American, 102-109 (December 2006).
[CrossRef] [PubMed]

2004 (1)

J. Flowers, "The route to atomic and quantum standards," Science 306, 1324-1330 (2004).
[CrossRef] [PubMed]

1997 (1)

G. Mana and E. Vittone, "Scanning LLL x-ray interferometry I. Theory," Z. Phys. B 102, 197-206 (1997)
[CrossRef]

1976 (1)

U. Bonse, W. Graeff, and G. Materlik, "X-ray interferometry and lattice parameter investigations," Rev. Phys. Appl. 11, 83-87 (1976).
[CrossRef]

Bonse, U.

U. Bonse, W. Graeff, and G. Materlik, "X-ray interferometry and lattice parameter investigations," Rev. Phys. Appl. 11, 83-87 (1976).
[CrossRef]

Ferroglio, L.

Flowers, J.

J. Flowers, "The route to atomic and quantum standards," Science 306, 1324-1330 (2004).
[CrossRef] [PubMed]

Graeff, W.

U. Bonse, W. Graeff, and G. Materlik, "X-ray interferometry and lattice parameter investigations," Rev. Phys. Appl. 11, 83-87 (1976).
[CrossRef]

Mana, G.

E. Massa, G. Mana, G. U. Kuetgens, and L. Ferroglio, "Measurement of the lattice parameter of a silicon crystal," New J. Phys. 11 (2009) 053013.
[CrossRef]

L. Ferroglio, G. Mana, and E. Massa, "Si lattice parameter measurement by centimeter x-ray interferometry," Opt. Express 16, 16877-16888 (2008).
[CrossRef] [PubMed]

G. Mana and E. Vittone, "Scanning LLL x-ray interferometry I. Theory," Z. Phys. B 102, 197-206 (1997)
[CrossRef]

Massa, E.

E. Massa, G. Mana, G. U. Kuetgens, and L. Ferroglio, "Measurement of the lattice parameter of a silicon crystal," New J. Phys. 11 (2009) 053013.
[CrossRef]

L. Ferroglio, G. Mana, and E. Massa, "Si lattice parameter measurement by centimeter x-ray interferometry," Opt. Express 16, 16877-16888 (2008).
[CrossRef] [PubMed]

Materlik, G.

U. Bonse, W. Graeff, and G. Materlik, "X-ray interferometry and lattice parameter investigations," Rev. Phys. Appl. 11, 83-87 (1976).
[CrossRef]

Robinson, J.

J. Robinson, "Weighty matter," Scientific American, 102-109 (December 2006).
[CrossRef] [PubMed]

Vittone, E.

G. Mana and E. Vittone, "Scanning LLL x-ray interferometry I. Theory," Z. Phys. B 102, 197-206 (1997)
[CrossRef]

New J. Phys. (1)

E. Massa, G. Mana, G. U. Kuetgens, and L. Ferroglio, "Measurement of the lattice parameter of a silicon crystal," New J. Phys. 11 (2009) 053013.
[CrossRef]

Opt. Express (1)

Rev. Phys. Appl. (1)

U. Bonse, W. Graeff, and G. Materlik, "X-ray interferometry and lattice parameter investigations," Rev. Phys. Appl. 11, 83-87 (1976).
[CrossRef]

Science (1)

J. Flowers, "The route to atomic and quantum standards," Science 306, 1324-1330 (2004).
[CrossRef] [PubMed]

Scientific American (1)

J. Robinson, "Weighty matter," Scientific American, 102-109 (December 2006).
[CrossRef] [PubMed]

Z. Phys. B (1)

G. Mana and E. Vittone, "Scanning LLL x-ray interferometry I. Theory," Z. Phys. B 102, 197-206 (1997)
[CrossRef]

Other (2)

A. G. Every and A. K. Mc Curdy, Low Frequency Properties of Dielectric Crystals: Second and Higher Order Elastic Constants in: Landolt-B¨ornstein III/29a, Nelson, D. F. ed. (Springer, Berlin, 1992).

U. Bonse and W. Graeff, "X-ray and neutron interferometry," in: Topics in applied physics, 22 X-ray optics, applications to solids, H.-J. Queisser ed. (Berlin, Springer, 1997).

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Figures (5)

Fig. 1.
Fig. 1.

Combined x-ray and optical interferometer. The analyzer crystal is moved retaining the angular alignment of the diffracting planes. The intensity of the transmitted x rays varies according to the relative positions of the moving and fixed crystals and allows the number of atoms the crystal has shifted to be counted. The translation distance, angular attitude, and transverse motion are measured using a laser interferometer and capacitive transducer (not shown in the figure).

Fig. 2.
Fig. 2.

Analyzer crystal. The two weights illustrate where the crystal has been loaded; in practice a Si bridge was put between the weights and the crystal end-pillars. Crystal dimensions are (70.5×32.2×15.0) mm3. The dashed rectangle is the surveyed part of the (55×20) mm2 lamella. The two arrows – and those symmetrical, not visible, on the opposite pillar – indicate where the load was applied; the stars indicate the support lines. The reference frame and crystal orientation are also indicated.

Fig. 3.
Fig. 3.

Finite-element analysis of the analyzer self-weight strain. In the crystal top it is below 10-9 d 220. The contour levels are spaced by 1 nm/m

Fig. 4.
Fig. 4.

Comparison of phase contrast topography (left) and finite-element analysis (right) of the loaded lattice planes; displacement (top), strain (middle), and tilt (bottom). The contour levels are spaced by 0.25 nm (top), 10 nm/m (middle), and 5 nrad (bottom). The dashed rectangle is the surveyed part of the (55×20) mm2 lamella.

Fig. 5.
Fig. 5.

Comparison between the simulated (solid line) and observed strain in the loaded analyzer. The red line (dashed) is the relevant section of the polynomial approximation (3).

Equations (5)

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I ( x ) cos [ 2 π x d 220 2 π u x x z d 220 ] ,
d eff x z = ( m n ) λ 2 ,
ε x x i , j = 0 i = 4 , j = 3 a i j x i z j
u x x z = x 0 x ε x x τ z ,
ε x z = z u x = x 0 x z ε xx τ z ,

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