clever design for a “fixed-exit” beam monochromator was developed by
Golevchenko and coworkers at Bell Labs [121]. By incorporating a rigid right
angle rotating midway between the two crystals, with appropriate linkages, the
horizontal position of the second crystal is adjusted to maintain a constant beam
height. Instead of mechanical linkages, most beamline monochromators now accomplish a fixed-exit beam with computerized motion control.
For a 2-bounce monochromator in the symmetric or “non-dispersive” geometry,
the resolution is not improved by the second crystal. This can be seen by looking at
the overlap of slices in a Dumond diagram—when the two crystals are brought into
alignment, their transmission functions are parallel and perfectly overlap (Fig. 4.18).
Typical resolutions are 1.4 Â 10
À4 with Si(1 1 1) crystals and 3 Â 10
À5 with Si
(3 1 1). The resolution can be improved by changing to a dispersive geometry,
yielding a smaller overlap of transmission functions. However, with only two
crystals, the angular deviation problem would return. All is solved by using two
pairs of crystals, for a “4-bounce” monochromator, with the second and third crystals
in the dispersive geometry, as shown in Fig. 4.18. In this geometry the resolution
with Si(3 1 1) crystals can approach 10
À5 .
Although 4-bounce designs with conventional crystals can achieve an impressive
0.1 eV resolution, some experiments such as IXS (Chap. 8) and NRVS (Chap. 10)
require at least two orders of magnitude better resolution. This is normally achieved
either with asymmetrically cut crystals or by operating in an extreme backscattering
geometry.
4.6.2.1 Extreme Backscattering
The extreme backscattering approach exploits the fact that the angular Darwin width
acceptance ω D becomes quite large when the Bragg angle is close to 90
. It is given
by:
ω D θ $ 90
À
Á ffi 2√ j χ j
ð 4:29Þ
where the susceptibility |χ| is on the order of 10
À6 . The acceptance can thus become
on the order of milliradians (Table 4.2) [122]. By choosing a high-order reflection to
yield such a Bragg angle, a resolution of ~1 meV can be obtained. For example, with
the Si(11 11 11) reflection, an intrinsic resolution of 0.8 meV can be obtained at
21.75 keV—for a ΔE/E of 3.6 Â 10
À8 !
One complication is that close to backscattering, the diffracted wavelength
λ ffi 2d, and the energy changes very little with angle. So, in extreme backscattering,
the only practical way to scan such a monochromator is to change the d-spacing.
This is done by changing the crystal temperature, using:
Δλ=λ ¼ ΔE=E ¼ Δd=d ¼ α ΔT
ð4:30Þ
where α is the coefficient of thermal expansion. At room temperature, the lattice
spacing for Si changes by 2.6 Â 10
À6 per degree Kelvin, so for the above reflection,
4.6 Diffraction: Crystals and Multilayers
89
Golevchenko and coworkers at Bell Labs [121]. By incorporating a rigid right
angle rotating midway between the two crystals, with appropriate linkages, the
horizontal position of the second crystal is adjusted to maintain a constant beam
height. Instead of mechanical linkages, most beamline monochromators now accomplish a fixed-exit beam with computerized motion control.
For a 2-bounce monochromator in the symmetric or “non-dispersive” geometry,
the resolution is not improved by the second crystal. This can be seen by looking at
the overlap of slices in a Dumond diagram—when the two crystals are brought into
alignment, their transmission functions are parallel and perfectly overlap (Fig. 4.18).
Typical resolutions are 1.4 Â 10
À4 with Si(1 1 1) crystals and 3 Â 10
À5 with Si
(3 1 1). The resolution can be improved by changing to a dispersive geometry,
yielding a smaller overlap of transmission functions. However, with only two
crystals, the angular deviation problem would return. All is solved by using two
pairs of crystals, for a “4-bounce” monochromator, with the second and third crystals
in the dispersive geometry, as shown in Fig. 4.18. In this geometry the resolution
with Si(3 1 1) crystals can approach 10
À5 .
Although 4-bounce designs with conventional crystals can achieve an impressive
0.1 eV resolution, some experiments such as IXS (Chap. 8) and NRVS (Chap. 10)
require at least two orders of magnitude better resolution. This is normally achieved
either with asymmetrically cut crystals or by operating in an extreme backscattering
geometry.
4.6.2.1 Extreme Backscattering
The extreme backscattering approach exploits the fact that the angular Darwin width
acceptance ω D becomes quite large when the Bragg angle is close to 90
. It is given
by:
ω D θ $ 90
À
Á ffi 2√ j χ j
ð 4:29Þ
where the susceptibility |χ| is on the order of 10
À6 . The acceptance can thus become
on the order of milliradians (Table 4.2) [122]. By choosing a high-order reflection to
yield such a Bragg angle, a resolution of ~1 meV can be obtained. For example, with
the Si(11 11 11) reflection, an intrinsic resolution of 0.8 meV can be obtained at
21.75 keV—for a ΔE/E of 3.6 Â 10
À8 !
One complication is that close to backscattering, the diffracted wavelength
λ ffi 2d, and the energy changes very little with angle. So, in extreme backscattering,
the only practical way to scan such a monochromator is to change the d-spacing.
This is done by changing the crystal temperature, using:
Δλ=λ ¼ ΔE=E ¼ Δd=d ¼ α ΔT
ð4:30Þ
where α is the coefficient of thermal expansion. At room temperature, the lattice
spacing for Si changes by 2.6 Â 10
À6 per degree Kelvin, so for the above reflection,
4.6 Diffraction: Crystals and Multilayers
89