Position of exit X-rays from rotated-inclined double-crystal monochromators

Kashihara, Y.; Yamazaki, H.; Tamasaku, K.; Ishikawa, T.

doi:10.1107/S0909049597015355

short communications

JOURNAL OF
SYNCHROTRON
RADIATION

ISSN: 1600-5775

Volume 5| Part 3| May 1998| Pages 679-681

https://doi.org/10.1107/S0909049597015355

Position of exit X-rays from rotated-inclined double-crystal monochromators

Yasuharu Kashihara,^a ^* Hiroshi Yamazaki,^a Kenji Tamasaku ^b and Tetsuya Ishikawa ^c

^aJapan Synchrotron Radiation Research Institute, Kamigouri, Ako-gun, Hyogo 678-12, Japan, ^bMicrowave Laboratory, RIKEN, Kamigouri, Ako-gun, Hyogo 678-12, Japan, and ^cJAERI-RIKEN SPring-8 Project Team, Kamigouri, Ako-gun, Hyogo 678-12, Japan
^*Correspondence e-mail: kashi@haru01.spring8.or.jp

(Received 4 August 1997; accepted 3 November 1997)

The rotated-inclined double-crystal monochromator (RIDCM) has been adopted to reduce the heat load from third-generation undulator radiation. The position of the exit X-rays from RIDCM has been calculated as a function of X-ray energy on the basis of diffraction theory including refraction effects. The results show that the positions of the exit X-rays vary over a wide range due to asymmetric reflection. Methods of fixing the exit position in RIDCM are also discussed.

Keywords: double-crystal monochromators; rotated-inclined double-crystal monochromators; exit X-rays; undulator monochromators; SPring-8.

1. Introduction

The European Synchrotron Radiation Facility (ESRF), the Advanced Photon Source (APS) and SPring-8 are third-generation synchrotron facilities operated at 6, 7 and 8 GeV, respectively, in which X-ray undulators will be the main light sources. The heat-load problem with undulator monochromators arises from the power density rather than the total power. In order to overcome the heat problem, several methods, including geometrical reduction of the power density on the first crystal of a double-crystal monochromator, have been adopted. For example, the inclined double-crystal monochromator (IDCM) (Blasdell et al., 1994 ; Hrdy, 1992 ; Khounsary, 1992 ; Lee et al., 1992 ; Lee & Mills, 1994 ; Macrander et al., 1992 ) with liquid-Ga cooling and the rotated-inclined double-crystal monochromator (RIDCM) (Uruga et al., 1995 ) with pin-post water cooling have been developed at APS and SPring-8, respectively.

The vertical and horizontal positions of the exit X-rays from IDCM and RIDCM can be easily fixed for various X-ray energies, if the diffraction of X-rays follows the kinematical diffraction theory. Actual X-rays refract at the entrance and exit surfaces of a crystal so that the exit X-rays from the monochromator deviate from the position calculated using the kinematical diffraction theory. We therefore examine the exit X-ray position by using diffraction theory including refraction effects and discuss ways to fix the exit X-ray position.

2. Optical arrangements of RIDCM

The optical arrangement of RIDCM is illustrated in Fig. 1. K_O and K_H are the wave vectors of the incident and reflected X-rays on the first crystal, respectively. K_H′ is the wavevector of the exit X-ray from the second crystal. γ_O and γ_H (or γ_O′ and γ_H′) are the polar angles between the incident and reflected X-rays and the normal to the crystal surface n (or n′), respectively, and are defined as shown in Fig. 1. The inclination angle between the normal n (or n′) and the active reciprocal lattice vector H (or H′) of the crystal is denoted as β. The RIDCM is set so that the surfaces of two crystals are parallel to each other and H′ is parallel to H. Then the exit X-ray K_H′ from the second crystal is always parallel to the incident X-ray K_O.

Figure 1
Schematic diagram of RIDCM. The O₁-XYZ coordinate system is fixed in an experimental hall. IS is the image screen perpendicular to the Z axis. O₁-X′Y′Z′ is fixed on the first crystal surface (see Fig. 2

Let us consider an orthogonal coordinate system, O₁-XYZ, fixed in an experimental hall. We take the Z axis in the direction of the incident X-ray K_O, the Y axis in the vertical plane and the X axis in the horizontal plane. Let the intersection of the incident X-ray K_O and the surface of the first crystal be the origin O₁ of the coordinate system. The exit X-ray K_H′ from the double-crystal monochromator intersects the image screen perpendicular to the Z axis at the point (X, Y).

To realize the fixed exit we put the second crystal at O₂ with the vector $[\overline{O_1O_2}]$ in O₁-XYZ written as

$[\overline{O_1O_2}=(0,g, g/\tan2\theta_{\rm B}),\eqno(1)]$

where g is a nominal offset value and θ_B is the kinematic Bragg angle. In the kinematical diffraction theory the point (X, Y) on the image screen is represented by (0, g) since H and H′ are usually in the Y-Z plane; the real image point, however, is different from the point (0, g) because of refraction.

3. Diffraction condition

To examine the diffraction condition of X-rays on RIDCM we define an orthogonal coordinate system, O₁-X′Y′Z′, fixed on the first crystal surface as shown in Fig. 2. We take the Z′ axis in the n direction and the Y′ axis perpendicular to the plane which is spanned by the vectors n and H. The X′ axis is perpendicular to both the Y′ and Z′ axes. Let an azimuthal angle of the incident X-ray wavevector K_O around the Z′ axis be ψ_O with respect to the X′ axis. Similarly, let the azimuthal angle for the diffracted X-ray wavevector K_H be ψ_H.

Figure 2
Schematic diagram of the scattering geometry on the first crystal surface.

In the kinematical X-ray diffraction theory, the incident X-ray can be reflected from the crystal when the incident angles, polar angle γ_KO and azimuthal angle ψ_KO satisfy

$[\sin\gamma_{KO}\cos\psi_{KO} \sin\beta+\cos\gamma_{KO} \cos\beta=H/2K, \eqno(2)]$

where H and K are magnitudes of the reciprocal lattice vector H and the incident X-ray wavevector K_O, respectively. The diffracted X-ray calculated in the kinematical X-ray diffraction theory have exit angles, polar angle γ_KH and azimuthal angle ψ_KH satisfying

$[\cos\gamma_{KH}=-\cos\gamma _{KO}+(H/K)\cos\beta,\eqno(3)]$

$[\sin\psi_{KH}=-\sin\psi_{KO} \sin\gamma_{KO}/\sin \gamma_{KH},\eqno(4)]$

$[\cos\psi_{KH}=[-\cos\psi_{KO} \sin\gamma_{KO}+(H/K)\sin \beta]/\sin\gamma_{KH}.\eqno(5)]$

However, the X-ray actually refracts through the entrance and exit surfaces of a crystal and so, for diffraction, γ_O, ψ_O, γ_H and ψ_H must be different from those calculated in the kinematical diffraction theory. If their deviations are represented by Δγ_O, Δψ_O, Δγ_H and Δψ_H, then the real angles can be written

$[\gamma_O=\gamma_{KO}+\Delta \gamma_O,\eqno(6)]$

$[\psi_O=\psi_{KO} +\Delta\psi_O,\eqno(7)]$

$[\gamma_H=\gamma_{KH}+\Delta \gamma_H,\eqno(8)]$

$[\psi_H=\psi_{KH} +\Delta\psi_H.\eqno(9)]$

From both the Laue equation at a Lorentz point used in Ewald–Von Laue's dynamical diffraction theory (Batterman & Cole, 1964 ; Colella, 1974 ; Zachariasen, 1967 $[Zachariasen, W. H. (1967). Theory of X-ray Diffraction in Crystals. New York: Dover.]$ ) and boundary conditions of the wavevector on the entrance surface, Δγ_O and Δψ_O satisfy the following relation,

$[\xi\Delta\gamma_O+\eta\Delta \psi_O = (\chi_O^r/2)(1+\cos \gamma_{KH}/\cos\gamma_ {KO}),\eqno(10)]$

where $[\chi_O^r]$ is the real part of the complex electric susceptibility. ξ and η are defined as

$[\eqalignno{\xi={}&2(\cos\gamma _{KO}\cos\beta+\sin\gamma_{KO} \cos\psi_{KO}\sin\beta)\cr &\times(\sin\gamma _{KO}\cos\beta-\cos\gamma_{KO} \cos\psi_{KO}\sin\beta),&(11)}]$

$[\eqalignno{\eta={}&2(\cos\gamma _{KO}\cos\beta+\sin\gamma_{KO} \cos\psi_{KO}\sin\beta)\cr &\times\sin\gamma _{KO}\sin\psi_{KO}\sin\beta.&(12)}]$

From the boundary conditions of the wavevector on the exit surface, Δγ_H and Δψ_H are given by

$[\eqalignno{\Delta\gamma_H={}& \left[-\cos\gamma_{KO}\cos (\psi_{KO}-\psi_{KH})\Delta \gamma_O\right.\cr &\left.+\sin \gamma_{KO}\sin(\psi_{KO} -\psi_{KH})\Delta\psi_O \right]/\cos\gamma_{KH}, \cr&&(13)}]$

$[\eqalignno{\Delta\psi_H={}& \left[-\cos\gamma_{KO}\sin (\psi_{KO}-\psi_{KH})\Delta \gamma_O\right.\cr &\left.-\sin \gamma_{KO}\cos(\psi_{KO} -\psi_{KH})\Delta\psi_O \right]/\sin\gamma_{KH}. \cr&&(14)}]$

We cannot determine the angles of γ_O and ψ_O independently from (2) and (10). If we want to determine them independently, we need further geometrical relations between them.

For the RIDCM in which the incident polar angle γ_O is kept constant, the following two relations can be obtained,

$[\gamma_{KO}=\gamma={\rm constant},\eqno(15)]$

$[\Delta\gamma_O=0.\eqno(16)]$

By substituting (15) into (2),

$[\cos\psi_{KO}=[(H/2K)-\cos \gamma_{KO}\cos\beta]/\sin\gamma_{KO}\sin\beta, \eqno(17)]$

is given. Substituting (16) into (10) yields

$[\Delta\psi_O = (\chi_O^r/2) (1+\cos\gamma_{KH}/ \cos\gamma_{KO}) /\eta.\eqno(18)]$

Therefore, for diffraction to occur, the crystal must be rotated about the normal n by −Δψ_O .

4. Results and discussion

Before we investigate the real position of the exit X-ray from the RIDCM, we examine the coordinate transformation between O₁-XYZ and O₁-X′Y′Z′ (see Figs. 1 and 2). The position vector, (X′, Y′, Z′) in O₁-X′Y′Z′, is represented by (X, Y, Z) in O₁-XYZ through

$[\left| \matrix{X^\prime \cr Y^\prime \cr Z^\prime \cr}\right|={\bf R}\left|\matrix{X \cr Y \cr Z \cr}\right|,\eqno(19)]$

where the transformation matrix, R, is written as

$[\eqalignno{&{\bf R}=\cr&\left|\sevenpoint{\matrix{\eqalign{&\sin\gamma_O\sin\psi_O\cos\beta\cr&\,\,\,/(1-\alpha^2)^{1/2}} & \eqalign{&(-\alpha\sin\gamma_O\cos\psi_O\cr&\,+\sin\beta)/(1-\alpha^2)^{1/2}} & -\sin\gamma_O\cos\psi_O\cr\cr\eqalign{&(-\sin\gamma_O\cos\psi_O\cos\beta\cr&\,\,\,+\cos\gamma_O\sin\beta)\cr&\,\,\,\,\,/(1-\alpha^2)^{1/2}} & \eqalign{&-\alpha\sin\gamma_O\sin\psi_O\cr&\,\,\,\,\,/(1-\alpha^2)^{1/2}} & -\sin\gamma_O\sin\psi_O\cr\cr\cr \eqalign{-&\sin\gamma_O\sin\psi_O\sin\beta\cr&\,\,\,/(1-\alpha^2)^{1/2}} & \eqalign{&(-\alpha\cos\gamma_O+\cos\beta)\cr&\,\,\,/(1-\alpha^2)^{1/2}} & -\cos\gamma_O\cr}}\right|\cr&&(20)}]$

Here, α is defined by

$[\alpha=\sin\gamma_O\cos \psi_O\sin\beta+\cos\gamma _O\cos\beta.\eqno(21)]$

The second crystal surface through the point O₂, whose coordinate in O₁-XYZ is given by (1), is represented by

$[Z^\prime=gr_{32}+gr_{33}/\tan 2\theta_{\rm B},\eqno(22)]$

in O₁-X′Y′Z′, where r_ij is an element of the matrix R. The diffracted X-ray from the first crystal, therefore, intersects the second crystal surface at the point

$[X^\prime=Z^\prime\sin\gamma_H \cos\psi_H/\cos\gamma_H, \eqno(23)]$

$[Y^\prime=Z^\prime\sin\gamma_H \sin\psi_H/\cos\gamma_H, \eqno(24)]$

in O₁-X′Y′Z′. The point shown in (22), (23) and (24) can be represented in O₁-XYZ as

$[\left|\matrix{X\cr Y\cr Z\cr}\right|={\bf R}^{-1} \left|\matrix{X^\prime \cr Y^\prime\cr Z^\prime\cr}\right|,\eqno(25)]$

where matrix R⁻¹ is the inverse of R. Because the exit X-ray from the second crystal is parallel to the Z axis in O₁-XYZ, it intersects the image screen shown in Fig. 1 at the point (X, Y) given by (25).

The positions on the image screen of the exit X-ray from Si(111)-RIDCM, whose inclination angle (β) is 80°, at X-ray energies of 5, 10, 15, 20, 25 and 30 keV are shown in Fig. 3, where we assumed that the incident polar angle γ_O in (15) was 89° and that the nominal offset value g was 30.0 mm. The exit X-ray calculated in the kinematical theory intersects the image screen at the point (0, 30.0) mm. From Fig. 3 it is apparent that deviations of the exit positions for RIDCM are fairly great. The widths of deviation along X and Y directions are about 0.03 mm and 0.4 mm, respectively. Above and below an energy of about 20 keV, the deviations in RIDCM become larger. At about 20 keV the geometry of RIDCM is almost symmetric (γ_O ≃ γ_H). RIDCM takes on the grazing-incident geometry (γ_O > γ_H) at energies below 20 keV and the grazing-exit geometry (γ_O < γ_H) at higher energies. Because of the asymmetric geometry, the positions of the exit X-rays from RIDCM vary over a wide range.

Figure 3
Position of exit X-rays from RIDCM at an X-ray energy 5, 10, 15, 20, 25 and 30 keV.

We now discuss how to suppress this deviation of exit X-ray position. The reciprocal lattice vector H does not lie in the plane of incidence containing both the incident wavevector K_O and the diffracted one K_H because of refraction. Therefore, when we put H in the Y-Z plane, we cannot obtain the exit X-ray at X = 0. We introduce the scattering vector H_S, defined as

$[{\bf H}_S={\bf K}_O-{\bf K}_H.\eqno(26)]$

From the boundary conditions of the wave vectors on both the incident and exit crystal surfaces, we can obtain the important result that H_S always lies in the plane spanned with H and n (Colella, 1974). We define β′ to be the angle between H_S and n and express it as

$[\beta^\prime=\beta+ \Delta\beta.\eqno(27)]$

Δβ can be approximately written from the boundary conditions of the wave vectors as

$[\Delta\beta=-(K/H)\zeta \sin\beta,\eqno(28)]$

where

$[\zeta=-\sin\gamma_{KO}\Delta \gamma_O-\sin\gamma_{KH} \Delta\gamma_H.\eqno(29)]$

We find from these equations that Δβ depends on both the X-ray energy and the geometry of the monochromator.

It is required from (26) that we always put H_S in the Y-Z plane to keep the exit X-ray in the Y-Z plane. The actual scattering angle of the X-ray diffracted by the crystal is different from 2θ_B as calculated by the kinematical diffraction theory, and is given by

$[\eqalignno{\cos2\theta={}& -\sin\gamma_O\cos\psi_O\sin \gamma_H\cos\psi_H\cr& -\sin\gamma_O\sin\psi_O\sin\gamma_H\sin\psi_H-\cos\gamma_O\cos\gamma_H.\cr&&(30)}]$

We finally translate the second crystal from O₁ to O₂, where the translation vector $[\overline {O_1O_2}]$ is given by (1) but we replace 2θ_B in (1) with 2θ in (30). Then, we can fix the exit position at the point (X, Y) = (0, g) on the image screen shown in Fig. 1.

References

Batterman, B. W. & Cole, H. (1964). Rev. Mod. Phys. 36, 681–717. CrossRef CAS Web of Science
Blasdell, R. C., Macrander, A. T. & Lee, W. K. (1994). Nucl. Instrum. Methods, A347, 327–330. CrossRef Web of Science
Colella, R. (1974). Acta Cryst. A30, 413–422. CrossRef CAS IUCr Journals Web of Science
Hrdy, V. (1992). Rev. Sci. Instrum. 63, 459–460.
Khounsary, A. M. (1992). Rev. Sci. Instrum. 63, 461–464. CrossRef Web of Science
Lee, W. K., Macrander, A. T., Mills, D. M., Rogers, C. S., Smither, R. K. & Berman, L. E. (1992). Nucl. Instrum. Methods, A320, 381–387. CrossRef CAS Web of Science
Lee, W. K. & Mills, D. M. (1994). Nucl. Instrum. Methods, A347, 618–626. CrossRef Web of Science
Macrander, A. T., Lee, W. K., Smither, R. K., Mills, D. M., Rogers, C. S. & Khounsary, A. M. (1992). Nucl. Instrum. Methods, A319, 188–196. CrossRef CAS Web of Science
Uruga, T., Kimura, H., Kohmura, Y., Kuroda, M., Nagasawa, H., Ohtomo, K., Yamaoka, H., Ishikawa, T., Ueki, T., Iwasaki, H., Hashimoto, S., Kashihara, Y. & Okui, K. (1995). Rev. Sci. Instrum. 66, 2254–2256. CrossRef CAS Web of Science
Zachariasen, W. H. (1967). Theory of X-ray Diffraction in Crystals. New York: Dover.

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JOURNAL OF
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RADIATION

ISSN: 1600-5775

Volume 5| Part 3| May 1998| Pages 679-681

https://doi.org/10.1107/S0909049597015355

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short communications\(\def\hfill{\hskip 5em}\def\hfil{\hskip 3em}\def\eqno#1{\hfil {#1}}\)

Position of exit X-rays from rotated-inclined double-crystal monochromators

1. Introduction

2. Optical arrangements of RIDCM

3. Diffraction condition

4. Results and discussion

References

short communications