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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 up previous contents
Next: Appendix 2: Determination of Up: Appendix 1: Useful formulae Previous: Intrinsic crystal properties.   Contents
Pionic Hydrogen Collaboration
1998