Bragg spectroscopy and available crystals.

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Bragg spectroscopy and available crystals.

The energy $E$ of X-rays emitted from the target region is analyzed by Bragg reflection from spherically bent single crystals. These are mounted in the so-called Johann geometry [70]. The energy analysis relies on the Bragg reflection condition

$\begin{displaymath} n\cdot \lambda =2\cdot d\cdot sin\Theta _{B} \end{displaymath}$

(7)

with

$\begin{displaymath} \lambda =2\pi \frac{\hbar c}{E}. \end{displaymath}$

(8)

The Bragg condition corresponds to the coherent superposition of X-rays of wavelength $\lambda$ reflected from lattice planes with a spacing of $d$ into a direction given by the reflection angle $\Theta _{B}$ , $n$ is the order of reflection. In the following we only consider Bragg reflection of the first order.

The angular width of reflected monoenergetic X-rays determines the intrinsic energy resolution. It depends on the crystal properties for a given wavelength. More information about the imaging properties of bent crystals and the facts limiting the resolution of bent crystal spectroscopy are given in Appendix 1.

The single crystals with best known properties for our energy region are $Si$ and $SiO_{2}$ (quartz) crystals. For the energy region below $3.3keV$ , which is about the series limit for the pionic hydrogen $K$ transitions, only the 111 lattice plane for $Si$ and the 100 and 10.1 lattice planes for quartz can be used. Some properties of these crystals are listed in Table 4.

Table 4: Physical properties of the Bragg crystals available for the experiment.

Crystal	plane	2d[ $nm$ ] at $22.5^{0}C$	Energy[ $eV$ ]		Temp. coeff.
			$\Theta _{B}=90^{0}$	$\Theta _{B}=65^{0}$
$SiO_{2}$	100	0.85110(4) [71]	1457	1607	$1.4\cdot 10^{-5}/K$
$SiO_{2}$	10.1	0.66862(4) [71]	1854	2045	$1.4\cdot 10^{-5}/K$
$Si$	111	0.62712016(18) [72]	1977	2181	$2.6\cdot 10^{-6}/K$

With quartz 100 it is also possible to measure the muonic $K\alpha$ transition. All these crystals cut to the appropiate planes are at the disposal of the experiment. Their geometrical form is a circular disk with a thickness of $0.3mm$ and a diameter of $100mm$ . In order to achieve spatial focussing they are spherically bent by mounting them on a concave glass lens. The radius of curvature is $R_{c}=2985.4mm$ for three available $Si$ (111) crystals and three $SiO_{2}$ (100) crystals. The two available $SiO_{2}$ (10.1) crystals are presently still mounted with $R_{c}=2660mm$ which is foreseen to be changed. As the measurement aims at a final accuracy in the energy determination of about $10^{-6}$ the Bragg crystals have to be temperature stabilized.

Next: Transition energies of pionic Up: The crystal spectrometer. Previous: The crystal spectrometer. Contents

Pionic Hydrogen Collaboration
1998