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short communications
Position of exit X-rays from rotated-inclined double-crystal monochromators
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
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
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
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. KO and KH are the wave vectors of the incident and reflected X-rays on the first crystal, respectively. KH′ is the wavevector of the exit X-ray from the second crystal. γO and γH (or γO′ and γH′) 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 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 KH′ from the second crystal is always parallel to the incident X-ray KO.
![]() | 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 KO, the Y axis in the vertical plane and the X axis in the horizontal plane. Let the intersection of the incident X-ray KO and the surface of the first crystal be the origin O1 of the coordinate system. The exit X-ray KH′ 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 in O1-XYZ written as
where g is a nominal offset value and θB is the kinematic 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 KO around the Z′ axis be ψO with respect to the X′ axis. Similarly, let the azimuthal angle for the diffracted X-ray wavevector KH be ψH.
![]() | 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 γKO and azimuthal angle ψKO satisfy
where H and K are magnitudes of the vector H and the incident X-ray wavevector KO, respectively. The diffracted X-ray calculated in the kinematical X-ray diffraction theory have exit angles, polar angle γKH and azimuthal angle ψKH satisfying
However, the X-ray actually refracts through the entrance and exit surfaces of a crystal and so, for diffraction, γO, ψO, γH and ψH must be different from those calculated in the kinematical diffraction theory. If their deviations are represented by ΔγO, ΔψO, ΔγH and ΔψH, then the real angles can be written
From both the Laue equation at a Lorentz point used in Ewald–Von Laue's ; Colella, 1974
; Zachariasen, 1967
) and boundary conditions of the wavevector on the entrance surface, ΔγO and ΔψO satisfy the following relation,
where is the real part of the complex electric susceptibility. ξ and η are defined as
From the boundary conditions of the wavevector on the exit surface, ΔγH and ΔψH are given by
We cannot determine the angles of γO and ψO 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 γO is kept constant, the following two relations can be obtained,
By substituting (15) into (2)
,
is given. Substituting (16) into (10)
yields
Therefore, for diffraction to occur, the crystal must be rotated about the normal n by −ΔψO .
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
where the transformation matrix, R, is written as
Here, α is defined by
The second crystal surface through the point O2, whose coordinate in O1-XYZ is given by (1), is represented by
in O1-X′Y′Z′, where rij is an element of the matrix R. The diffracted X-ray from the first crystal, therefore, intersects the second crystal surface at the point
in O1-X′Y′Z′. The point shown in (22), (23)
and (24)
can be represented in O1-XYZ as
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 γO in (15)
was 89° and that the nominal offset value g was 30.0 mm. The exit X-ray calculated in the 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 (γO ≃ γH). RIDCM takes on the grazing-incident geometry (γO > γH) at energies below 20 keV and the grazing-exit geometry (γO < γH) 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 H does not lie in the plane of incidence containing both the incident wavevector KO and the diffracted one KH 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 HS, defined as
vectorFrom the boundary conditions of the wave vectors on both the incident and exit crystal surfaces, we can obtain the important result that HS always lies in the plane spanned with H and n (Colella, 1974). We define β′ to be the angle between HS and n and express it as
Δβ can be approximately written from the boundary conditions of the wave vectors as
where
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 HS 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 2θB as calculated by the kinematical diffraction theory, and is given by
We finally translate the second crystal from O1 to O2, where the translation vector is given by (1)
but we replace 2θB 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
.
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