Rates.

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Rates.

The count rate $n_{x}$ from a crystal spectrometer set-up in Johann geometry is estimated by the relation

$\begin{displaymath} n_{x}=A_{x}\cdot \epsilon _{det}\cdot \eta , \end{displaymath}$

(29)

where $A_{x}$ denotes the activity of the source, $\epsilon _{det}$ the detection efficiency for the reflected X-rays and $\eta$ the efficiency of the parameters of the Bragg crystal. Characterising the cyclotron trap by a stop efficiency $f_{stop}$ to produce exotic atoms , the activity of the source is given by

$\begin{displaymath} A_{x}=N_{in}\cdot f_{stop}\cdot Y_{x}, \end{displaymath}$

(30)

with $N_{in}$ being the number of incoming particles and $Y_{x}$ being the yield of the X-ray transition under investigation. The quantity $\eta$ may be expressed by

$\begin{displaymath} \eta =\left( \frac{\Delta \Omega }{4\pi }\right) \cdot \frac{\Delta S}{S}\cdot P. \end{displaymath}$

(31)

$\frac{\Delta \Omega }{4\pi }$ is the fraction of solid angle of the Bragg crystal seen from the source and $P$ is the crystal's peak reflectivity. The quantity $\frac{\Delta S}{S}$ is the fraction of the source from which X-rays are reflected owing to the Bragg condition. For sources which can be regarded as uniformly radiating disks of radius $r_{s}$ the fraction is in good approximation given by

$\begin{displaymath} \frac{\Delta S}{S}=\frac{\omega \cdot R_{I}\cdot sin\Theta _... ...cdot R_{I}\cdot sin\Theta _{B}}{\pi \cdot r_{s}}\cdot \omega . \end{displaymath}$

(32)

It is assumed that the dimensions of the source are small compared to the Rowland circle radius $R_{c}$ .

Next: Appendix 2: Determination of Up: Appendix 1: Useful formulae Previous: Intrinsic crystal properties. Contents

Pionic Hydrogen Collaboration
1998