Abstract

A general formula is derived for the spectral density distribution in the far zone, produced by the diffraction of a beam of any state of spatial coherence on a medium with a spatially periodic structure. The formula may be used to determine the structure of crystals from the diffraction of partially coherent x-ray beams.

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References

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  1. A. Guinier and D. L. Dexter, X-Ray Studies of Materials (Wiley, 1963).
  2. R. W. James, The Optical Principles of the Diffraction of X-Rays (Ox Bow, 1982).
  3. The distinction between monochromaticity and spatial coherence is discussed in E. Wolf, Opt. Commun. 284, 4235 (2011).
  4. G. E. Bacon, X-ray and Neutron Diffraction (Pergamon, 1966).
  5. Diffraction of partially coherent beam on a grating was discussed in M. Dušek, Opt. Commun. 111, 203 (1994).
  6. C. Kittel, Introduction to Solid State Physics, 8th ed. (Wiley, 2005).
  7. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
  8. M. Dušek, Phys. Rev. E 52, 6833 (1995).

2011 (1)

The distinction between monochromaticity and spatial coherence is discussed in E. Wolf, Opt. Commun. 284, 4235 (2011).

1995 (1)

M. Dušek, Phys. Rev. E 52, 6833 (1995).

1994 (1)

Diffraction of partially coherent beam on a grating was discussed in M. Dušek, Opt. Commun. 111, 203 (1994).

Bacon, G. E.

G. E. Bacon, X-ray and Neutron Diffraction (Pergamon, 1966).

Dexter, D. L.

A. Guinier and D. L. Dexter, X-Ray Studies of Materials (Wiley, 1963).

Dušek, M.

M. Dušek, Phys. Rev. E 52, 6833 (1995).

Guinier, A.

A. Guinier and D. L. Dexter, X-Ray Studies of Materials (Wiley, 1963).

James, R. W.

R. W. James, The Optical Principles of the Diffraction of X-Rays (Ox Bow, 1982).

Kittel, C.

C. Kittel, Introduction to Solid State Physics, 8th ed. (Wiley, 2005).

Wolf, E.

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

Opt. Commun. (2)

The distinction between monochromaticity and spatial coherence is discussed in E. Wolf, Opt. Commun. 284, 4235 (2011).

Diffraction of partially coherent beam on a grating was discussed in M. Dušek, Opt. Commun. 111, 203 (1994).

Phys. Rev. E (1)

M. Dušek, Phys. Rev. E 52, 6833 (1995).

Other (5)

C. Kittel, Introduction to Solid State Physics, 8th ed. (Wiley, 2005).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

G. E. Bacon, X-ray and Neutron Diffraction (Pergamon, 1966).

A. Guinier and D. L. Dexter, X-Ray Studies of Materials (Wiley, 1963).

R. W. James, The Optical Principles of the Diffraction of X-Rays (Ox Bow, 1982).

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Figures (1)

Fig. 1.
Fig. 1.

S˜()(rs;ω) plotted against θarccos(s·x) for different values of σ. The wavelength of the incident radiation is taken to be λ=1010m.

Equations (19)

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F(r)=HΦ(H)exp[2πiH·r],
S()(r;ω)=S(i)(ω)×Dd3r1d3r2μ(i)(r1,r2;ω)F*(r1)F(r2)×G*(|rr1|;ω)G(|rr2|;ω),
G(|rr|;ω)=eik|rr||rr|eikrreiks·r
S()(rs;ω)=S(i)(ω)r2×Dd3r1d3r2μ(i)(r1,r2;ω)F*(r1)F(r2)×exp[iks·(r2r1)].
S()(r;ω)=S(i)(ω)r2|F˜(ks;ω)|2,
F˜(ks;ω)=Dd3rF(r)exp[iks·r]
F(r)=f(x)δ(y)δ(z),
μ(i)(ρ1,ρ2;ω)=exp[(ρ2ρ1)22σ2],
S()(rs;ω)=S(i)(ω)r2×Lf*(x1)f(x2)exp[(x2x1)22σ2]×exp[i(ks·x)(x2x1)]dx1dx2,
f(x)=n=Cnexp[i2πnx/a],
Cn=1a0af(x)exp[i2πnx/a].
S()(rs;ω)=S(i)(ω)r2n,m=Cn*CmInm(θ),
Inm(θ)=Lexp[(x2x1)22σ2]exp[i(ks·x)(x2x1)]×exp[i2πa(mx2nx1)]dx1dx2.
x+=(x1+x2)/2,x=x2x1,
Inm(θ)={Ldx+exp[i2πa(mn)x+]}×{Ldxexp[(x)22σ2]exp[i(ks·x)x]×exp[iπa(m+n)x]}.
Ldx+exp[i2πa(mn)x+]=Lδnm,
S()(rs;ω)=LS(i)(ω)r2n=|Cn|2Ldxexp[(x)22σ2]×exp[i(ks·x)x]×exp[i2πnx/a].
f(x)=12+12cos(2πx/a)
S˜()(rs;ω)(16kr22πLS(i)(ω))S()(rs;ω)=kσ{4exp[(kσs·x)2/2]+exp[(ks·x2π/a)2σ2/2]+exp[(ks·x+2π/a)2σ2/2]}.

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