Abstract

The geometrical optics of symmetrical point-focusing x-ray monochromators made with elastically bent ideal monocrystalline wafers in the geometry employed by Despujols and Berreman are reinvestigated. Sharpness of focus is limited mainly by accuracy of curvature of the crystal wafer and by thermal diffuse scattering. Monochromaticity is limited by these factors and by source and crystal dimensions in the plane of reflection and rarely by the inherent width of the reflection band of the curved crystal. Two existing monochromators reflect about 2–3% of the incident CuKα1 radiation.

© 1977 Optical Society of America

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References

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  1. D. W. Berreman, “A Method for Monochromatization and Precise Point Focusing of X-rays and its Application to Low Angle Diffraction Studies” (Thesis, California Institute of Technology, 1955).
  2. J. W. M. Du Mond, P. Kirkpatrick, Rev. Sci. Instrum. 1, 88 (1930).
    [CrossRef]
  3. T. Johansson, Naturwissenschaften 20, 758 (1932); Z. Phys. 82, 507 (1933).
    [CrossRef]
  4. A. H. Compton, S. K. Allison, X-rays in Theory and Experiment (Van Nostrand, New York, 1935), p. 750ff.
  5. J. Despujols, C. R. Acad. Sci. 235, 716 (1952).
  6. D. W. Berreman, J. W. M. Du Mond, P. E. Marmier, Rev. Sci. Instrum. 25, 1220 (1954).
    [CrossRef]
  7. D. W. Berreman, Rev. Sci. Instrum. 26, 1048 (1955).
    [CrossRef]
  8. D. W. Berreman, Phys. Rev. B 15, 4313 (1976).
    [CrossRef]
  9. J. Hartmann, Z. Instrumentenk. 24, 1, 33, 97 (1904).
  10. J. Strong, Concepts of Classical Optics (Freeman, San Francisco, 1958), pp. 354–355.
  11. See, for example, R. W. James, The Optical Principles of the Diffraction of X-rays (Bell, London, 1958), Chap. 5.
  12. L. G. Parratt, Phys. Rev. 46, 749 (1934).
    [CrossRef]

1976

D. W. Berreman, Phys. Rev. B 15, 4313 (1976).
[CrossRef]

1955

D. W. Berreman, Rev. Sci. Instrum. 26, 1048 (1955).
[CrossRef]

1954

D. W. Berreman, J. W. M. Du Mond, P. E. Marmier, Rev. Sci. Instrum. 25, 1220 (1954).
[CrossRef]

1952

J. Despujols, C. R. Acad. Sci. 235, 716 (1952).

1934

L. G. Parratt, Phys. Rev. 46, 749 (1934).
[CrossRef]

1932

T. Johansson, Naturwissenschaften 20, 758 (1932); Z. Phys. 82, 507 (1933).
[CrossRef]

1930

J. W. M. Du Mond, P. Kirkpatrick, Rev. Sci. Instrum. 1, 88 (1930).
[CrossRef]

1904

J. Hartmann, Z. Instrumentenk. 24, 1, 33, 97 (1904).

Allison, S. K.

A. H. Compton, S. K. Allison, X-rays in Theory and Experiment (Van Nostrand, New York, 1935), p. 750ff.

Berreman, D. W.

D. W. Berreman, Phys. Rev. B 15, 4313 (1976).
[CrossRef]

D. W. Berreman, Rev. Sci. Instrum. 26, 1048 (1955).
[CrossRef]

D. W. Berreman, J. W. M. Du Mond, P. E. Marmier, Rev. Sci. Instrum. 25, 1220 (1954).
[CrossRef]

D. W. Berreman, “A Method for Monochromatization and Precise Point Focusing of X-rays and its Application to Low Angle Diffraction Studies” (Thesis, California Institute of Technology, 1955).

Compton, A. H.

A. H. Compton, S. K. Allison, X-rays in Theory and Experiment (Van Nostrand, New York, 1935), p. 750ff.

Despujols, J.

J. Despujols, C. R. Acad. Sci. 235, 716 (1952).

Du Mond, J. W. M.

D. W. Berreman, J. W. M. Du Mond, P. E. Marmier, Rev. Sci. Instrum. 25, 1220 (1954).
[CrossRef]

J. W. M. Du Mond, P. Kirkpatrick, Rev. Sci. Instrum. 1, 88 (1930).
[CrossRef]

Hartmann, J.

J. Hartmann, Z. Instrumentenk. 24, 1, 33, 97 (1904).

James, R. W.

See, for example, R. W. James, The Optical Principles of the Diffraction of X-rays (Bell, London, 1958), Chap. 5.

Johansson, T.

T. Johansson, Naturwissenschaften 20, 758 (1932); Z. Phys. 82, 507 (1933).
[CrossRef]

Kirkpatrick, P.

J. W. M. Du Mond, P. Kirkpatrick, Rev. Sci. Instrum. 1, 88 (1930).
[CrossRef]

Marmier, P. E.

D. W. Berreman, J. W. M. Du Mond, P. E. Marmier, Rev. Sci. Instrum. 25, 1220 (1954).
[CrossRef]

Parratt, L. G.

L. G. Parratt, Phys. Rev. 46, 749 (1934).
[CrossRef]

Strong, J.

J. Strong, Concepts of Classical Optics (Freeman, San Francisco, 1958), pp. 354–355.

C. R. Acad. Sci.

J. Despujols, C. R. Acad. Sci. 235, 716 (1952).

Naturwissenschaften

T. Johansson, Naturwissenschaften 20, 758 (1932); Z. Phys. 82, 507 (1933).
[CrossRef]

Phys. Rev.

L. G. Parratt, Phys. Rev. 46, 749 (1934).
[CrossRef]

Phys. Rev. B

D. W. Berreman, Phys. Rev. B 15, 4313 (1976).
[CrossRef]

Rev. Sci. Instrum.

J. W. M. Du Mond, P. Kirkpatrick, Rev. Sci. Instrum. 1, 88 (1930).
[CrossRef]

D. W. Berreman, J. W. M. Du Mond, P. E. Marmier, Rev. Sci. Instrum. 25, 1220 (1954).
[CrossRef]

D. W. Berreman, Rev. Sci. Instrum. 26, 1048 (1955).
[CrossRef]

Z. Instrumentenk.

J. Hartmann, Z. Instrumentenk. 24, 1, 33, 97 (1904).

Other

J. Strong, Concepts of Classical Optics (Freeman, San Francisco, 1958), pp. 354–355.

See, for example, R. W. James, The Optical Principles of the Diffraction of X-rays (Bell, London, 1958), Chap. 5.

D. W. Berreman, “A Method for Monochromatization and Precise Point Focusing of X-rays and its Application to Low Angle Diffraction Studies” (Thesis, California Institute of Technology, 1955).

A. H. Compton, S. K. Allison, X-rays in Theory and Experiment (Van Nostrand, New York, 1935), p. 750ff.

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Figures (4)

Fig. 1
Fig. 1

Confocal ellipsoids of equal optical path difference to form a perfect point focusing monochromator everywhere in space. Along Rowland circles, Bragg spacing is constant.

Fig. 2
Fig. 2

(A) Crystalline wafer in unstrained cylindrical form; (B) strained to flat form so that planes curve one way; (C) strained around transverse axis to form a point-focusing x-ray monochromator.

Fig. 3
Fig. 3

Cross section of point-focusing x-ray monochromfltor illustrating variables used.

Fig. 4
Fig. 4

Hartmann test with monochromatic x rays of a quartz crystal point-focusing monochromator showing disorientation about dust particles under wafer and at edges and some over-all distortion.

Tables (1)

Tables Icon

Table I Design Parameters, Aberrations, and Reflectance in Two Point-Focusing Monochromators of Quartz for CuKα1 Radiation

Equations (19)

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γ 1 ( z , x ) arctan [ ( R 2 - x ) / ( Z 0 + z ) ] ;
γ 2 ( z , x ) arctan { ( R 2 - x ) / [ Z ( z , x ) - z ] } ;
γ 2 ( z , x ) γ 1 ( z , x ) + 2 α ( z , x ) .
α ( z , x ) = arctan [ ( R 1 + x R 1 ) tan ( z R 1 ) ] = z R 1 + x R 1 sin z R 1 cos z R 1 + O ( x R 1 ) 2 .
d ( z , x ) = d 0 cos α ( z , x ) / cos ( z / R 1 ) = d 0 [ 1 - x R 1 sin 2 ( z R 1 ) + O ( x R 1 ) 2 ] .
Z ( z , x ) = z + ( R 2 - x ) cos [ γ 1 ( z , x ) + 2 α ( z , x ) ] .
Z ( z , x ) = Z 0 + Z z z + Z z z z 2 + Z z z z z 3 + Z x z x z + .
Z z = 2 - [ 2 R 2 / ( R 1 sin 2 θ ) ] ,
Z z z = - 2 Z z / ( R 1 tan θ ) ,
Z z z z = tan 2 θ 3 R 2 2 [ 3 - 7 sin 2 θ + 4 sin 4 θ + 6 cos 2 θ ( 2 R 2 R 1 sin 2 θ - 1 ) + cos 2 θ ( 2 sin 2 θ - 3 ) ( 2 R 2 R 1 sin 2 θ - 1 ) 3 ] ,
Z x z = 2 [ sin 2 θ - cos 2 θ - ( R 2 / R 1 ) ] / ( R 1 sin 2 θ ) .
R 2 = R 1 sin 2 θ ,
Z z z z = ( 6 - 8 sin 2 θ + 2 sin 4 θ ) tan 2 θ / ( 3 R 2 2 )
Z x z = - 2 cos 2 θ / R 2 .
θ ( z , x ) = γ 1 ( z , x ) - α ( z , x ) θ z z + θ z z z 2 + θ x x .
θ z = θ + 1 R 1 - sin 2 θ R 2 ;
θ z z = sin 3 θ cos θ / R 2 2 ;
θ x = - sin θ cos θ / R 2 .
R π , σ = R π , σ ( λ ) d λ ,

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