where d is the lattice spacing of crystallographic planes relevant for the reflection, QB is the Bragg angle, which is defined as the angle between the incident beam and the lattice planes, l is the wavelength and n denotes the order of the reflection.
With the derivative of Bragg's law the monochromatization of the radiation can be expressed as a function of
With an energy band of
are therefore necessary for NRS experiments:
The figure above indicates the beam trace through the crystal monochromator systems: The high heat load monochromator (left side) selects an energy band of approximately 3 eV from the incident 300 eV broad energy band. The beam is further monochromatized to 6 meV or less by a high resolution monochromator.
The design of the high heat-load monochromator
is governed by the following demands:
(mean deviation of
about 1 µrad per month).
The rotation stage for the first crystal is mounted to a horizontal translation, whereas the rotation stage of the second crystal (which is more stable because of less heat load) is rigidly attached to a base plate. This improved the
(mean deviation of
about 0.3 eV per month)
of the monochromatized beam. Once the energy is adjusted with the second crystal, which is the more stable one, the first crystal is optimized for maximum intensity coming through the setup.
If perfect crystals are used, the energy resolution in the case of Bragg diffraction can be approximated by DE / E » Dq cot qB, where Dq is the divergence of the incident beam (typically
One therefore attempts to achieve high energy resolution using reflections with large Bragg angles (i.e. large values of the Miller indices), for in this case cotqB becomes small.
|(1 1 1)||7.88||3.72|
|(4 2 2)||22.83||1.17|
|(12 2 2)||77.53||0.40|
|(9 7 5)||80.40||0.35|
This choice has nevertheless a drawback:
For a symmetrically cut crystal, the angular acceptance of a reflection corresponds to its Darwin width, which becomes smaller as the Bragg angle increases (only when qB goes above ~ 80° it starts to increase again), so that the beam divergence that is accepted by the reflection is very small (typically < 1 mrad). The problem can be overcome if the beam divergence is previously reduced. This can be achieved by using an asymmetrically cut crystal (for details see below).
So far nested monochromators have been used for NFS experiments with four successive reflections on two nested channel-cut crystals. This geometry provides
The outer channel-cut
Both channel-cut crystals are mounted on a goniometer with two co-axial axes one sitting on the other. By this means one assured a compact design and reached high temperature and mechanical stability over the crystals. Furthermore, we benefit from the fact that only one relative angular measurement with high resolution and accuracy is necessary.
In order to ensure a good energy calibration of the system a linear encoder with a resolution of 10 nm which corresponds to 0.015 arcsec is used. The corresponding encoder resolutions are tabulated below:
|encoder res.||[meV / step]||0.2108||0.1646||0.08814|
For nuclear forward scattering (NFS) the desired energy band is in the range of 10-6 eV... 10-9 eV, depending on the magnitude of the hyperfine splitting. Since the monochromatization 'only' serves as background reduction, the nested monochromators are appropriate for this kind of experiment.
The high energy resolution on the other hand was found to be necessary also for recording the energy dependence of inelastic excitations (NIS) in the sample. Those experiments demand even higher energy resolutions which have been achieved by cuts of larger asymmetry:
The whole setup is shrunk to a two-bounce monochromator adopting two asymmetric
Furthermore the two
About 15 different HRMs for various nuclear transitions and applications were built up.
The most frequently used cases are summarized in the following table.
|Iso-tope||Reson. Energy [keV]||Reflections||Asymm. param. b (outer refl.)||Energy resol. [meV]||Flux [phot./s] ISR=90mA||CRL|
|57Fe||14.413||(4 2 2)||(12 2 2)||-0.100||6.4||1.8*109|
|57Fe||14.413||(4 2 2)||(9 7 5)||-0.097||4.4||0.6*109|
|151Eu||21.542||(4 4 0)||(12 12 8)||-0.064||1.6||7.5*107||with|
|151Eu||21.542||(8 0 0)||(12 4 4)||-0.071||7.3||1.0*108|
|119Sn||23.880||(2 4 6)||-0.043||0.65||1.8*107||with|
|119Sn||23.880||(4 4 4)||(8 8 0)||-0.058||12.0||4.0*108||with|
|161Dy||25.651||(4 4 4)||(18 12 6)||-0.100||1.1||5.0*106||with|
|57Fe||14.413||(9 7 5)||(9 7 5)||
b = - (sinQ1) / (sinQ2)
is the angle
Parameters for an asymmetric reflection with Bragg angle QB. Qcut is the angle between the scattering crystal-planes and the surface, DQi the angular acceptance/emittance of the crystal and hi the vertical size of the beam.
That means that for a small b
while the beamsize becomes larger.