Abstract

A nearly exact general equation for geometrical angular deviations from the Bragg angle over entire curved crystal surfaces is derived using a toroidally curvilinear coordinate system and applied on the nine conventional crystal geometries. Although the derived formula confirms Wittry’s results for the first five cases, it shows considerable differences for the more important cases, such as 45° point focusing, general point focusing, and Berreman geometries. The effective scattering areas for the mentioned cases have been derived, plotted, and interpreted. A point-to-point focusing crystal geometry is introduced, and it is shown that it approaches Wittry’s and spherical plane–spherical Johansson geometries as θB90° and θB0°, respectively.

© 2012 Optical Society of America

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References

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  1. D. B. Berreman, “Single quartz crystal point focusing x-ray monochromator,” Rev. Sci. Instrum. 26, 1048–1052 (1955).
    [CrossRef]
  2. E. P. Bertin, “Principles and Practice of X-Ray Spectrometric Analysis,” 2nd ed. (Plenum, 1975), p. 200.
  3. D. B. Wittry and S. Sun, “X-ray optics of doubly curved diffractors,” J. Appl. Phys. 67, 1633–1638 (1990).
    [CrossRef]
  4. D. B. Wittry and S. Sun, “X-ray optics of doubly curved diffractors II,” J. Appl. Phys. 71, 564–568 (1992).
    [CrossRef]
  5. D. B. Wittry and D. M. Golijanin, “Large aperture point focusing diffractor for x-rays,” Appl. Phys. Lett. 52, 1381–1382 (1988).
    [CrossRef]
  6. Z. W. Chen, F. Wei, and D. Gibson, “Advance in detection of low sulfur content by wavelenght dispersive XRF,” X-Ray Optical Systems Inc., 30 Corporate Circle, Albany, New York 12203 (2003).
  7. Z. W. Chen, T. N. Mailb, F. Z. Weia, C. A. MacDonald, and W. M. Gibson, “Focused beam total reflection x-ray fluorescence with low power sources coupled to doubly curved crystal optics,” Spectrochim. Acta B 60, 471–478 (2005).
    [CrossRef]
  8. M. Krämer, K. Kuzushita, S. Maeo, T. Utaka, and K. Taniguchi, “Design of a doubly-curved crystal to improve X-ray fluorescence analysis of aerosol particles,” Spectrochim. Acta B 63, 1408–1414 (2008).
    [CrossRef]
  9. Z. W. Chen, W. M. Gibson, and H. Huang, “High definition x-ray fluorescence: principles and techniques,” ID 318171, X-Ray Optics and Instrumentation, Inc., 15 Tech Valley Drive, East Greenbush, New York 12061, USA (2008).
  10. K. Yamamura, K. Ueda, M. Nagano, N. Zettsu, S. Maeo, S. Shimada, T. Utaka, and K. Taniguchi, “Fabrication of damage-free Johansson-type doubly curved crystal spectrometer substrate by numerically controlled local wet etching,” Nucl. Instrum. Methods Phys. Res. A 616, 281–284 (2010).
    [CrossRef]
  11. D. M. Golijanin and D. B. Wittry, “Microprobe x-ray fluorescence: new developments in an old technique,” in Microbeam Analysis 1988: Proceedings of the 23rd Conference of the Microbeam Analysis Society, D.E.Newbury, ed. (San Francisco Press, 1988), p. 397–402.
  12. D. B. Wittry, “Scanning monochrometer crystal and method of formation,” U.S. Patent No. 4,807,268 (21 February 1989).
  13. D. W. Berreman, J. Stamatoff, and S. J. Kennedy, “Doubly curved crystal point-focusing x-ray monochromators: geometrical and practical optics,” Appl. Opt. 16, 2081–2085 (1977).
    [CrossRef] [PubMed]
  14. E. M. Latush and M. I. Mazuritsky, “A focusing x-ray diffractor: effect of the crystal bending parameters on the spectral resolution,” Tech. Phys. Lett. 28, 142–144 (2002).
    [CrossRef]
  15. M. M. Stepanenko, “A spectral resolution of Johann-type x-ray spectrometers,” Plasma Devices Oper. 17, 191–200 (2009).
    [CrossRef]
  16. Z. W. Chen and D. B. Wittry, “Microanalysis by monochromatic microprobe x-ray fluorescence—physical basis, properties and future prospects,” J. Appl. Phys. 84, 1064–1073 (1998).
    [CrossRef]
  17. D. B. Wittry and N. C. Barbi, “X-ray crystal spectrometers and monochromators in microanalysis,” Microsc. Microanal. 7, 124–141 (2001).
  18. N. Gao and Z. Chen, “A microbeam wavelength-dispersive x-ray fluorescence system and its application for thin-film analysis,” Rev. Sci. Instrum. 76, 123104 (2005).
    [CrossRef]
  19. D. B. Wittry, W. Z. Chang, and R.Y. Li, “X-ray optics of diffractors curved to a logarithmic spiral,” J. Appl. Phys 74, 3534–3540(1993).
    [CrossRef]
  20. D. B. Wittry and S. Sun, “Focusing properties of curved x-ray diffractors,” J. Appl. Phys. 68, 387–391 (1990).
    [CrossRef]
  21. W. Z. Chang and D. B. Wittry, “Synthesis of x-ray intensity profiles for x-ray optical systems with curved cliff ractors,” J. Appl. Phys. 74, 2999–3008 (1993).
    [CrossRef]
  22. D. B. Wittry and W. Z. Chang, “Evaluation of crystal diffractor parameters for curved diffractors,” J. Appl. Phys. 72, 3440–3446 (1992).
    [CrossRef]
  23. D. B. Wittry and S. Sun, “Properties of curved x-ray diffractors with stepped surfaces,” J. Appl. Phys 69, 3886–3892(1991).
    [CrossRef]

2010 (1)

K. Yamamura, K. Ueda, M. Nagano, N. Zettsu, S. Maeo, S. Shimada, T. Utaka, and K. Taniguchi, “Fabrication of damage-free Johansson-type doubly curved crystal spectrometer substrate by numerically controlled local wet etching,” Nucl. Instrum. Methods Phys. Res. A 616, 281–284 (2010).
[CrossRef]

2009 (1)

M. M. Stepanenko, “A spectral resolution of Johann-type x-ray spectrometers,” Plasma Devices Oper. 17, 191–200 (2009).
[CrossRef]

2008 (1)

M. Krämer, K. Kuzushita, S. Maeo, T. Utaka, and K. Taniguchi, “Design of a doubly-curved crystal to improve X-ray fluorescence analysis of aerosol particles,” Spectrochim. Acta B 63, 1408–1414 (2008).
[CrossRef]

2005 (2)

Z. W. Chen, T. N. Mailb, F. Z. Weia, C. A. MacDonald, and W. M. Gibson, “Focused beam total reflection x-ray fluorescence with low power sources coupled to doubly curved crystal optics,” Spectrochim. Acta B 60, 471–478 (2005).
[CrossRef]

N. Gao and Z. Chen, “A microbeam wavelength-dispersive x-ray fluorescence system and its application for thin-film analysis,” Rev. Sci. Instrum. 76, 123104 (2005).
[CrossRef]

2002 (1)

E. M. Latush and M. I. Mazuritsky, “A focusing x-ray diffractor: effect of the crystal bending parameters on the spectral resolution,” Tech. Phys. Lett. 28, 142–144 (2002).
[CrossRef]

2001 (1)

D. B. Wittry and N. C. Barbi, “X-ray crystal spectrometers and monochromators in microanalysis,” Microsc. Microanal. 7, 124–141 (2001).

1998 (1)

Z. W. Chen and D. B. Wittry, “Microanalysis by monochromatic microprobe x-ray fluorescence—physical basis, properties and future prospects,” J. Appl. Phys. 84, 1064–1073 (1998).
[CrossRef]

1993 (2)

D. B. Wittry, W. Z. Chang, and R.Y. Li, “X-ray optics of diffractors curved to a logarithmic spiral,” J. Appl. Phys 74, 3534–3540(1993).
[CrossRef]

W. Z. Chang and D. B. Wittry, “Synthesis of x-ray intensity profiles for x-ray optical systems with curved cliff ractors,” J. Appl. Phys. 74, 2999–3008 (1993).
[CrossRef]

1992 (2)

D. B. Wittry and W. Z. Chang, “Evaluation of crystal diffractor parameters for curved diffractors,” J. Appl. Phys. 72, 3440–3446 (1992).
[CrossRef]

D. B. Wittry and S. Sun, “X-ray optics of doubly curved diffractors II,” J. Appl. Phys. 71, 564–568 (1992).
[CrossRef]

1991 (1)

D. B. Wittry and S. Sun, “Properties of curved x-ray diffractors with stepped surfaces,” J. Appl. Phys 69, 3886–3892(1991).
[CrossRef]

1990 (2)

D. B. Wittry and S. Sun, “Focusing properties of curved x-ray diffractors,” J. Appl. Phys. 68, 387–391 (1990).
[CrossRef]

D. B. Wittry and S. Sun, “X-ray optics of doubly curved diffractors,” J. Appl. Phys. 67, 1633–1638 (1990).
[CrossRef]

1988 (1)

D. B. Wittry and D. M. Golijanin, “Large aperture point focusing diffractor for x-rays,” Appl. Phys. Lett. 52, 1381–1382 (1988).
[CrossRef]

1977 (1)

1955 (1)

D. B. Berreman, “Single quartz crystal point focusing x-ray monochromator,” Rev. Sci. Instrum. 26, 1048–1052 (1955).
[CrossRef]

Barbi, N. C.

D. B. Wittry and N. C. Barbi, “X-ray crystal spectrometers and monochromators in microanalysis,” Microsc. Microanal. 7, 124–141 (2001).

Berreman, D. B.

D. B. Berreman, “Single quartz crystal point focusing x-ray monochromator,” Rev. Sci. Instrum. 26, 1048–1052 (1955).
[CrossRef]

Berreman, D. W.

Bertin, E. P.

E. P. Bertin, “Principles and Practice of X-Ray Spectrometric Analysis,” 2nd ed. (Plenum, 1975), p. 200.

Chang, W. Z.

D. B. Wittry, W. Z. Chang, and R.Y. Li, “X-ray optics of diffractors curved to a logarithmic spiral,” J. Appl. Phys 74, 3534–3540(1993).
[CrossRef]

W. Z. Chang and D. B. Wittry, “Synthesis of x-ray intensity profiles for x-ray optical systems with curved cliff ractors,” J. Appl. Phys. 74, 2999–3008 (1993).
[CrossRef]

D. B. Wittry and W. Z. Chang, “Evaluation of crystal diffractor parameters for curved diffractors,” J. Appl. Phys. 72, 3440–3446 (1992).
[CrossRef]

Chen, Z.

N. Gao and Z. Chen, “A microbeam wavelength-dispersive x-ray fluorescence system and its application for thin-film analysis,” Rev. Sci. Instrum. 76, 123104 (2005).
[CrossRef]

Chen, Z. W.

Z. W. Chen, T. N. Mailb, F. Z. Weia, C. A. MacDonald, and W. M. Gibson, “Focused beam total reflection x-ray fluorescence with low power sources coupled to doubly curved crystal optics,” Spectrochim. Acta B 60, 471–478 (2005).
[CrossRef]

Z. W. Chen and D. B. Wittry, “Microanalysis by monochromatic microprobe x-ray fluorescence—physical basis, properties and future prospects,” J. Appl. Phys. 84, 1064–1073 (1998).
[CrossRef]

Z. W. Chen, F. Wei, and D. Gibson, “Advance in detection of low sulfur content by wavelenght dispersive XRF,” X-Ray Optical Systems Inc., 30 Corporate Circle, Albany, New York 12203 (2003).

Z. W. Chen, W. M. Gibson, and H. Huang, “High definition x-ray fluorescence: principles and techniques,” ID 318171, X-Ray Optics and Instrumentation, Inc., 15 Tech Valley Drive, East Greenbush, New York 12061, USA (2008).

Gao, N.

N. Gao and Z. Chen, “A microbeam wavelength-dispersive x-ray fluorescence system and its application for thin-film analysis,” Rev. Sci. Instrum. 76, 123104 (2005).
[CrossRef]

Gibson, D.

Z. W. Chen, F. Wei, and D. Gibson, “Advance in detection of low sulfur content by wavelenght dispersive XRF,” X-Ray Optical Systems Inc., 30 Corporate Circle, Albany, New York 12203 (2003).

Gibson, W. M.

Z. W. Chen, T. N. Mailb, F. Z. Weia, C. A. MacDonald, and W. M. Gibson, “Focused beam total reflection x-ray fluorescence with low power sources coupled to doubly curved crystal optics,” Spectrochim. Acta B 60, 471–478 (2005).
[CrossRef]

Z. W. Chen, W. M. Gibson, and H. Huang, “High definition x-ray fluorescence: principles and techniques,” ID 318171, X-Ray Optics and Instrumentation, Inc., 15 Tech Valley Drive, East Greenbush, New York 12061, USA (2008).

Golijanin, D. M.

D. B. Wittry and D. M. Golijanin, “Large aperture point focusing diffractor for x-rays,” Appl. Phys. Lett. 52, 1381–1382 (1988).
[CrossRef]

D. M. Golijanin and D. B. Wittry, “Microprobe x-ray fluorescence: new developments in an old technique,” in Microbeam Analysis 1988: Proceedings of the 23rd Conference of the Microbeam Analysis Society, D.E.Newbury, ed. (San Francisco Press, 1988), p. 397–402.

Huang, H.

Z. W. Chen, W. M. Gibson, and H. Huang, “High definition x-ray fluorescence: principles and techniques,” ID 318171, X-Ray Optics and Instrumentation, Inc., 15 Tech Valley Drive, East Greenbush, New York 12061, USA (2008).

Kennedy, S. J.

Krämer, M.

M. Krämer, K. Kuzushita, S. Maeo, T. Utaka, and K. Taniguchi, “Design of a doubly-curved crystal to improve X-ray fluorescence analysis of aerosol particles,” Spectrochim. Acta B 63, 1408–1414 (2008).
[CrossRef]

Kuzushita, K.

M. Krämer, K. Kuzushita, S. Maeo, T. Utaka, and K. Taniguchi, “Design of a doubly-curved crystal to improve X-ray fluorescence analysis of aerosol particles,” Spectrochim. Acta B 63, 1408–1414 (2008).
[CrossRef]

Latush, E. M.

E. M. Latush and M. I. Mazuritsky, “A focusing x-ray diffractor: effect of the crystal bending parameters on the spectral resolution,” Tech. Phys. Lett. 28, 142–144 (2002).
[CrossRef]

Li, R. Y.

D. B. Wittry, W. Z. Chang, and R.Y. Li, “X-ray optics of diffractors curved to a logarithmic spiral,” J. Appl. Phys 74, 3534–3540(1993).
[CrossRef]

MacDonald, C. A.

Z. W. Chen, T. N. Mailb, F. Z. Weia, C. A. MacDonald, and W. M. Gibson, “Focused beam total reflection x-ray fluorescence with low power sources coupled to doubly curved crystal optics,” Spectrochim. Acta B 60, 471–478 (2005).
[CrossRef]

Maeo, S.

K. Yamamura, K. Ueda, M. Nagano, N. Zettsu, S. Maeo, S. Shimada, T. Utaka, and K. Taniguchi, “Fabrication of damage-free Johansson-type doubly curved crystal spectrometer substrate by numerically controlled local wet etching,” Nucl. Instrum. Methods Phys. Res. A 616, 281–284 (2010).
[CrossRef]

M. Krämer, K. Kuzushita, S. Maeo, T. Utaka, and K. Taniguchi, “Design of a doubly-curved crystal to improve X-ray fluorescence analysis of aerosol particles,” Spectrochim. Acta B 63, 1408–1414 (2008).
[CrossRef]

Mailb, T. N.

Z. W. Chen, T. N. Mailb, F. Z. Weia, C. A. MacDonald, and W. M. Gibson, “Focused beam total reflection x-ray fluorescence with low power sources coupled to doubly curved crystal optics,” Spectrochim. Acta B 60, 471–478 (2005).
[CrossRef]

Mazuritsky, M. I.

E. M. Latush and M. I. Mazuritsky, “A focusing x-ray diffractor: effect of the crystal bending parameters on the spectral resolution,” Tech. Phys. Lett. 28, 142–144 (2002).
[CrossRef]

Nagano, M.

K. Yamamura, K. Ueda, M. Nagano, N. Zettsu, S. Maeo, S. Shimada, T. Utaka, and K. Taniguchi, “Fabrication of damage-free Johansson-type doubly curved crystal spectrometer substrate by numerically controlled local wet etching,” Nucl. Instrum. Methods Phys. Res. A 616, 281–284 (2010).
[CrossRef]

Shimada, S.

K. Yamamura, K. Ueda, M. Nagano, N. Zettsu, S. Maeo, S. Shimada, T. Utaka, and K. Taniguchi, “Fabrication of damage-free Johansson-type doubly curved crystal spectrometer substrate by numerically controlled local wet etching,” Nucl. Instrum. Methods Phys. Res. A 616, 281–284 (2010).
[CrossRef]

Stamatoff, J.

Stepanenko, M. M.

M. M. Stepanenko, “A spectral resolution of Johann-type x-ray spectrometers,” Plasma Devices Oper. 17, 191–200 (2009).
[CrossRef]

Sun, S.

D. B. Wittry and S. Sun, “X-ray optics of doubly curved diffractors II,” J. Appl. Phys. 71, 564–568 (1992).
[CrossRef]

D. B. Wittry and S. Sun, “Properties of curved x-ray diffractors with stepped surfaces,” J. Appl. Phys 69, 3886–3892(1991).
[CrossRef]

D. B. Wittry and S. Sun, “Focusing properties of curved x-ray diffractors,” J. Appl. Phys. 68, 387–391 (1990).
[CrossRef]

D. B. Wittry and S. Sun, “X-ray optics of doubly curved diffractors,” J. Appl. Phys. 67, 1633–1638 (1990).
[CrossRef]

Taniguchi, K.

K. Yamamura, K. Ueda, M. Nagano, N. Zettsu, S. Maeo, S. Shimada, T. Utaka, and K. Taniguchi, “Fabrication of damage-free Johansson-type doubly curved crystal spectrometer substrate by numerically controlled local wet etching,” Nucl. Instrum. Methods Phys. Res. A 616, 281–284 (2010).
[CrossRef]

M. Krämer, K. Kuzushita, S. Maeo, T. Utaka, and K. Taniguchi, “Design of a doubly-curved crystal to improve X-ray fluorescence analysis of aerosol particles,” Spectrochim. Acta B 63, 1408–1414 (2008).
[CrossRef]

Ueda, K.

K. Yamamura, K. Ueda, M. Nagano, N. Zettsu, S. Maeo, S. Shimada, T. Utaka, and K. Taniguchi, “Fabrication of damage-free Johansson-type doubly curved crystal spectrometer substrate by numerically controlled local wet etching,” Nucl. Instrum. Methods Phys. Res. A 616, 281–284 (2010).
[CrossRef]

Utaka, T.

K. Yamamura, K. Ueda, M. Nagano, N. Zettsu, S. Maeo, S. Shimada, T. Utaka, and K. Taniguchi, “Fabrication of damage-free Johansson-type doubly curved crystal spectrometer substrate by numerically controlled local wet etching,” Nucl. Instrum. Methods Phys. Res. A 616, 281–284 (2010).
[CrossRef]

M. Krämer, K. Kuzushita, S. Maeo, T. Utaka, and K. Taniguchi, “Design of a doubly-curved crystal to improve X-ray fluorescence analysis of aerosol particles,” Spectrochim. Acta B 63, 1408–1414 (2008).
[CrossRef]

Wei, F.

Z. W. Chen, F. Wei, and D. Gibson, “Advance in detection of low sulfur content by wavelenght dispersive XRF,” X-Ray Optical Systems Inc., 30 Corporate Circle, Albany, New York 12203 (2003).

Weia, F. Z.

Z. W. Chen, T. N. Mailb, F. Z. Weia, C. A. MacDonald, and W. M. Gibson, “Focused beam total reflection x-ray fluorescence with low power sources coupled to doubly curved crystal optics,” Spectrochim. Acta B 60, 471–478 (2005).
[CrossRef]

Wittry, D. B.

D. B. Wittry and N. C. Barbi, “X-ray crystal spectrometers and monochromators in microanalysis,” Microsc. Microanal. 7, 124–141 (2001).

Z. W. Chen and D. B. Wittry, “Microanalysis by monochromatic microprobe x-ray fluorescence—physical basis, properties and future prospects,” J. Appl. Phys. 84, 1064–1073 (1998).
[CrossRef]

W. Z. Chang and D. B. Wittry, “Synthesis of x-ray intensity profiles for x-ray optical systems with curved cliff ractors,” J. Appl. Phys. 74, 2999–3008 (1993).
[CrossRef]

D. B. Wittry, W. Z. Chang, and R.Y. Li, “X-ray optics of diffractors curved to a logarithmic spiral,” J. Appl. Phys 74, 3534–3540(1993).
[CrossRef]

D. B. Wittry and W. Z. Chang, “Evaluation of crystal diffractor parameters for curved diffractors,” J. Appl. Phys. 72, 3440–3446 (1992).
[CrossRef]

D. B. Wittry and S. Sun, “X-ray optics of doubly curved diffractors II,” J. Appl. Phys. 71, 564–568 (1992).
[CrossRef]

D. B. Wittry and S. Sun, “Properties of curved x-ray diffractors with stepped surfaces,” J. Appl. Phys 69, 3886–3892(1991).
[CrossRef]

D. B. Wittry and S. Sun, “Focusing properties of curved x-ray diffractors,” J. Appl. Phys. 68, 387–391 (1990).
[CrossRef]

D. B. Wittry and S. Sun, “X-ray optics of doubly curved diffractors,” J. Appl. Phys. 67, 1633–1638 (1990).
[CrossRef]

D. B. Wittry and D. M. Golijanin, “Large aperture point focusing diffractor for x-rays,” Appl. Phys. Lett. 52, 1381–1382 (1988).
[CrossRef]

D. M. Golijanin and D. B. Wittry, “Microprobe x-ray fluorescence: new developments in an old technique,” in Microbeam Analysis 1988: Proceedings of the 23rd Conference of the Microbeam Analysis Society, D.E.Newbury, ed. (San Francisco Press, 1988), p. 397–402.

D. B. Wittry, “Scanning monochrometer crystal and method of formation,” U.S. Patent No. 4,807,268 (21 February 1989).

Yamamura, K.

K. Yamamura, K. Ueda, M. Nagano, N. Zettsu, S. Maeo, S. Shimada, T. Utaka, and K. Taniguchi, “Fabrication of damage-free Johansson-type doubly curved crystal spectrometer substrate by numerically controlled local wet etching,” Nucl. Instrum. Methods Phys. Res. A 616, 281–284 (2010).
[CrossRef]

Zettsu, N.

K. Yamamura, K. Ueda, M. Nagano, N. Zettsu, S. Maeo, S. Shimada, T. Utaka, and K. Taniguchi, “Fabrication of damage-free Johansson-type doubly curved crystal spectrometer substrate by numerically controlled local wet etching,” Nucl. Instrum. Methods Phys. Res. A 616, 281–284 (2010).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

D. B. Wittry and D. M. Golijanin, “Large aperture point focusing diffractor for x-rays,” Appl. Phys. Lett. 52, 1381–1382 (1988).
[CrossRef]

J. Appl. Phys (2)

D. B. Wittry, W. Z. Chang, and R.Y. Li, “X-ray optics of diffractors curved to a logarithmic spiral,” J. Appl. Phys 74, 3534–3540(1993).
[CrossRef]

D. B. Wittry and S. Sun, “Properties of curved x-ray diffractors with stepped surfaces,” J. Appl. Phys 69, 3886–3892(1991).
[CrossRef]

J. Appl. Phys. (6)

Z. W. Chen and D. B. Wittry, “Microanalysis by monochromatic microprobe x-ray fluorescence—physical basis, properties and future prospects,” J. Appl. Phys. 84, 1064–1073 (1998).
[CrossRef]

D. B. Wittry and S. Sun, “Focusing properties of curved x-ray diffractors,” J. Appl. Phys. 68, 387–391 (1990).
[CrossRef]

W. Z. Chang and D. B. Wittry, “Synthesis of x-ray intensity profiles for x-ray optical systems with curved cliff ractors,” J. Appl. Phys. 74, 2999–3008 (1993).
[CrossRef]

D. B. Wittry and W. Z. Chang, “Evaluation of crystal diffractor parameters for curved diffractors,” J. Appl. Phys. 72, 3440–3446 (1992).
[CrossRef]

D. B. Wittry and S. Sun, “X-ray optics of doubly curved diffractors,” J. Appl. Phys. 67, 1633–1638 (1990).
[CrossRef]

D. B. Wittry and S. Sun, “X-ray optics of doubly curved diffractors II,” J. Appl. Phys. 71, 564–568 (1992).
[CrossRef]

Microsc. Microanal. (1)

D. B. Wittry and N. C. Barbi, “X-ray crystal spectrometers and monochromators in microanalysis,” Microsc. Microanal. 7, 124–141 (2001).

Nucl. Instrum. Methods Phys. Res. A (1)

K. Yamamura, K. Ueda, M. Nagano, N. Zettsu, S. Maeo, S. Shimada, T. Utaka, and K. Taniguchi, “Fabrication of damage-free Johansson-type doubly curved crystal spectrometer substrate by numerically controlled local wet etching,” Nucl. Instrum. Methods Phys. Res. A 616, 281–284 (2010).
[CrossRef]

Plasma Devices Oper. (1)

M. M. Stepanenko, “A spectral resolution of Johann-type x-ray spectrometers,” Plasma Devices Oper. 17, 191–200 (2009).
[CrossRef]

Rev. Sci. Instrum. (2)

N. Gao and Z. Chen, “A microbeam wavelength-dispersive x-ray fluorescence system and its application for thin-film analysis,” Rev. Sci. Instrum. 76, 123104 (2005).
[CrossRef]

D. B. Berreman, “Single quartz crystal point focusing x-ray monochromator,” Rev. Sci. Instrum. 26, 1048–1052 (1955).
[CrossRef]

Spectrochim. Acta B (2)

Z. W. Chen, T. N. Mailb, F. Z. Weia, C. A. MacDonald, and W. M. Gibson, “Focused beam total reflection x-ray fluorescence with low power sources coupled to doubly curved crystal optics,” Spectrochim. Acta B 60, 471–478 (2005).
[CrossRef]

M. Krämer, K. Kuzushita, S. Maeo, T. Utaka, and K. Taniguchi, “Design of a doubly-curved crystal to improve X-ray fluorescence analysis of aerosol particles,” Spectrochim. Acta B 63, 1408–1414 (2008).
[CrossRef]

Tech. Phys. Lett. (1)

E. M. Latush and M. I. Mazuritsky, “A focusing x-ray diffractor: effect of the crystal bending parameters on the spectral resolution,” Tech. Phys. Lett. 28, 142–144 (2002).
[CrossRef]

Other (5)

D. M. Golijanin and D. B. Wittry, “Microprobe x-ray fluorescence: new developments in an old technique,” in Microbeam Analysis 1988: Proceedings of the 23rd Conference of the Microbeam Analysis Society, D.E.Newbury, ed. (San Francisco Press, 1988), p. 397–402.

D. B. Wittry, “Scanning monochrometer crystal and method of formation,” U.S. Patent No. 4,807,268 (21 February 1989).

Z. W. Chen, W. M. Gibson, and H. Huang, “High definition x-ray fluorescence: principles and techniques,” ID 318171, X-Ray Optics and Instrumentation, Inc., 15 Tech Valley Drive, East Greenbush, New York 12061, USA (2008).

Z. W. Chen, F. Wei, and D. Gibson, “Advance in detection of low sulfur content by wavelenght dispersive XRF,” X-Ray Optical Systems Inc., 30 Corporate Circle, Albany, New York 12203 (2003).

E. P. Bertin, “Principles and Practice of X-Ray Spectrometric Analysis,” 2nd ed. (Plenum, 1975), p. 200.

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Figures (14)

Fig. 1
Fig. 1

Diagram of a crystal with a doubly curved surface and doubly curved planes. To simplify the illustration, the first quadrant of the x z plane is shown.

Fig. 2
Fig. 2

Relationship between the toroidally curvilinear system running through the crystal midpoint, the Cartesian coordinate system, and the Rowland circle. To simplify the illustration, the first quadrant of the x z plane is shown.

Fig. 3
Fig. 3

Cross-sectional view of the considered systems on a Rowland surface.

Fig. 4
Fig. 4

Illustration of the two considered toroidal planes and surface in Fig. 1 in two different directions in detail.

Fig. 5
Fig. 5

Between the two toroidal atomic planes and the toroidal crystal surface passing through M tangentially, there are several other toroidal diffracting surfaces whose origins are located at the origin of the Cartesian system, but their radii differ from each other (these origins are not shown here).

Fig. 6
Fig. 6

Arbitrary point P is considered to be on the toroidal diffracting plane passing through M.

Fig. 7
Fig. 7

Effective scattering area on the crystal surface of case 6 (spherical surface–toroidal planes or 45 ° point focusing crystals) using Δ θ = 0 , ± 10 4   rad for Bragg angles of 15 ° , 40 ° , 46 ° , and 75 ° . The solid, dotted, and dashed lines indicate the results for Δ θ = 0 and the negative and the positive Δ θ , respectively. (a) Wttry’s results, (b) this paper’s results.

Fig. 8
Fig. 8

(a) Wittry’s [3, 4] and (b) the present obtained effective scattering areas on the crystal surface of case 7 (toroidal surface–toroidal planes or general point focusing case) using Δ θ = 0 , ± 10 4   rad for Bragg angles of 15 ° , 30 ° , 34 ° , and 45 ° for a typical value of θ B 0 = 35 ° . The solid, dotted, and dashed lines indicate the results for Δ θ = 0 and the negative and the positive Δ θ , respectively. As predicted with the newly obtained analytical expression Δ θ , there are significant differences in diagrams, especially for Bragg angles in the vicinity of θ B 0 = 35 ° . Therefore, this geometry with the given radii by Wittry [3, 4] cannot be a general point focusing configuration.

Fig. 9
Fig. 9

Effective scattering area on the crystal surface of case 8 (Berreman’s case) using Δ θ = 0 , ± 10 4   rad for Bragg angles of 30 ° , 34 ° , 35 ° , and 45 ° for a typical value of θ B 0 = 35 ° . The solid, dotted, and dashed lines indicate the results for Δ θ = 0 and the negative and the positive Δ θ , respectively. (a) Wittry’s results [3, 4], (b)  the obtained results. There are some differences in the diagrams of effective scattering area for Bragg angles in θ B 0 = 35 ° .

Fig. 10
Fig. 10

Effective scattering area on the crystal surface of case 9 (Johann point focusing case) using Δ θ = 0 , ± 10 4   rad for a typical value θ B 0 = 15 ° and Bragg angles of 11 ° , 13 ° , 15 ° , and 17 ° . (a) Wttry’s reported results [3, 4], (b) Wittry’s results using R 1 = 1 , R 2 = R 2 = sin 2 θ B 0 and the exact normal vector obtained in this work. It can be seen that Wittry's results are significantly affected by using the exact normal vector.

Fig. 11
Fig. 11

Effective scattering area on the crystal surface of (a) case 7 (Johansson general point focusing scheme) for a typical value of θ B 0 = 35 ° and (b) case 9 (Johann point focusing) for a typical value of θ B 0 = 15 ° using Δ θ = 0 , ± 10 3   rad with different Bragg angles. It can be seen that the effective areas are behaving in approximately the same way as with Figs. 8b, 10b, but they are expanded using a bigger rocking curve.

Fig. 12
Fig. 12

Variation of the ratio of the vertical crystal surface to vertical atomic plane radii, R 2 / R 2 , versus the Bragg angle in a point focusing system.

Fig. 13
Fig. 13

Effective scattering area of (a) case 6 and (b) case 7 for θ B 0 = 35 ° , at different Bragg angles, using the derived point focusing configuration. The broadening effect of effective area is seen from the figures in the vicinity of θ B 0 = 45 ° and θ B 0 = 35 ° , namely, θ B = 46 ° and θ B = 34 ° , respectively.

Fig. 14
Fig. 14

Effective scattering area of (a) case 8 for θ B 0 = 35 ° and (b) case 9 for θ B 0 = 15 ° , at different Bragg angles, using the derived point focusing configuration (viz. R 1 = 1 / 2 , R 2 = sin θ B 0 , R 2 = sin θ B 0 ( 1 + sin θ B 0 ) / 2 ). It can be seen that the newly designed crystal geometry shows the approximate point focusing characteristics of both Berreman’s and Johann’s geometries.

Tables (4)

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Table 1 Nine Considered Conventional Cases of Singly and Doubly Curved X-Ray Crystal Configurations

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Table 2 Derived and Wittry’s Results for Four Nonzero Coefficients of Δ θ ( x , z )

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Table 3 Summary of Derived and Wittry’s Results for Δ θ of Cases 6, 7, 8, and 9

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Table 4 Angle Deviation, Δ θ , for Cases 6, 7, 8, and 9 in the Designed Point Focusing Scheme

Equations (16)

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n · PS | n | | PS | = cos ( π 2 θ i ) = sin θ i = sin ( θ B + Δ θ ) .
Δ θ ( x P , y P , z P ) 1 cos θ B ( e ^ p ( x P , y P , z P ) · PS | PS | ) tan θ B ,
S ( R 1 sin θ B cos θ B , R 1 cos 2 θ B , 0 ) and PS = PS ( R 1 sin θ B cos θ B x P , R 1 cos 2 θ B y P , z P ) .
x P = ( R 1 R 2 + r cos θ ) cos φ , y P = ( R 1 R 2 + r cos θ ) sin φ , z P = r sin θ .
e ^ ζ = X P / h ζ q ζ ; ζ = r , θ , φ ,
e ^ p ( x P , y P , z P ) = X P h r r = cos θ cos φ i ^ + cos θ sin φ j ^ + sin θ k ^ .
sin φ = y P / ( x P 2 + y P 2 ) 1 / 2 = y P / ( x P 2 + y P 2 ) 1 / 2 , cos φ = x P / ( x P 2 + y P 2 ) 1 / 2 = x P / ( x P 2 + y P 2 ) 1 / 2 , sin θ = z P / R 2 = z P / ( ( R 1 R 2 ( x P 2 + y P 2 ) 1 / 2 ) 2 + z P 2 ) 1 / 2 ,
x O P = O O P cos φ = ( R 1 R 2 ) cos φ , y O P = O O P sin φ = ( R 1 R 2 ) sin φ .
y P = ( ( R 1 R 2 + ( R 2 2 z P 2 ) 1 / 2 ) 2 x P 2 ) 1 / 2 + R 1 R 1 ,
y P R 1 x P / 2 R 1 z P / 2 R 2 .
e ^ p ( x P , z P ) 1 R 2 + x P 2 2 ( 1 R 1 1 R 1 ) + z P 2 2 ( 1 R 2 1 R 2 ) [ ( 1 R 1 R 2 R 1 + x P 2 2 ( 1 R 1 1 R 1 ) z P 2 2 R 2 ) x P i ^ + ( 1 R 1 R 2 R 1 + x P 2 2 ( 1 R 1 1 R 1 ) z P 2 2 R 2 ) ( R 1 x P 2 2 R 1 z P 2 2 R 2 ) j ^ + z P k ^ ] .
Δ θ = f ( x , z ) = f ( 0 , 0 ) + 1 1 ! ( x f x | x , z = 0 + z f z | x . z = 0 ) + 1 2 ! ( x 2 2 f x 2 | x , z = 0 + 2 x z 2 f x z | x , z = 0 + z 2 2 f z 2 | x , z = 0 ) + 1 3 ! ( x 3 3 f x 3 | x , z = 0 + 3 x 2 z 3 f x 2 z | x , z = 0 + 3 x z 2 3 f x z 2 | x , z = 0 + 3 f z 3 | x , z = 0 ) .
Δ θ ( x , z ) = a 1 + a 2 x + a 3 z + a 4 x 2 + a 5 x z + a 6 z 2 + a 7 x 3 + a 8 x 2 z + a 9 x z 2 + a 10 z 3 ,
{ cot θ B ( 1 1 2 R 1 ) = 0 tan θ B 2 [ 1 R 2 1 R 2 ( 1 R 2 ) R 2 2 + ( 2 R 2 1 R 2 1 ) 1 sin 2 θ B ] = 0 cot 2 θ B ( 1 1 2 R 1 ) = 0 ( 1 R 2 ) 2 R 2 2 + 1 2 [ 1 R 2 ( 1 R 2 ) R 2 2 + ( 2 R 2 1 R 2 2 ) 1 sin 2 θ B ] = 0 .
R 1 = 1 2 , R 2 = 1 , R 2 = 1 ,
R 1 = 1 2 , R 2 = sin θ B 0 , R 2 = sin θ B 0 2 ( 1 + sin θ B 0 ) .

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