The two-crystal set-up.

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The two-crystal set-up.

Before discussing the simultaneous measurement of pionic and muonic hydrogen the question is addressed whether a simultaneous measurement of a pionic hydrogen transition together with a narrow pionic line of higher Z is feasible. In this way the determination of the response function of the detector could be carried out simultaneously to the measurement. The requirement is that the calibration lines should stem from an element which can be Bragg-reflected by one crystal onto the same detector. The energy range $\Delta E$ covered is given by the detector width $\Delta x$ via the dispersion formula

$\begin{displaymath} \Delta E = \frac{E\cdot \mid \Delta x\mid } {tan(\Theta _{B})\cdot sin(\Theta _{B})\cdot R_{c}} . \end{displaymath}$

(9)

For a planned width of the detector of $48mm$ the energy window has typical values between $34eV$ for the measurement of the $\mu ^{-}H_{2\rightarrow 1}$ at $50.22^{0}$ and $84eV$ for the measurement of the $\pi ^{-}H_{3\rightarrow 1}$ transition at $40.00^{0}$ . For a spectroscopy of the $\mu ^{-}H_{3\rightarrow 1}$ transition, however, the energy width would reduce to $22eV$ for Silicon crystal at $61.58^{0}$ . The rather narrow energy interval limits the availability of calibration lines drastically. As the stop geometry should not change during calibration the calibration line should originate from a gas admixed to hydrogen. The only line in the neighbourhood of a pionic hydrogen line is the $\pi ^{-}O_{6\rightarrow 5}$ at $2876eV$ , which is only $8eV$ apart from the $\pi ^{-}H_{3\rightarrow 1}$ transition. This corresponds to a distance of only $4.6mm$ on the CCD if they are reflected by the 10.1 plane of $SiO_{2}$ . A similar situation is given for the pair $\mu ^{-}H_{3\rightarrow 1}$ and $\pi ^{-}N_{6\rightarrow 5}$ . Here the transition energies are separated by $45eV$ , which corresponds to a distance of $32.8mm$ on the CCD detector for a reflection on the 100 plane of $SiO_{2}$ . Unfortunately the oxygen and nitrogen lines exhibit a not well understood broadening because of an acceleration in the beginning of the cascade ( Coulomb explosion) typical for molecular gases. As stated above, the understanding of these processes will improve in near future as these atoms are intensively studied with high precision [86].

It can therefore be concluded that no calibration line is available near a muonic/pionic hydrogen line if one is restricted to monoatomic gases below $Z=10$ . The $\pi ^{-}O_{6\rightarrow 5}$ and the $\pi ^{-}N_{6\rightarrow 5}$ transitions remain attractive in any case as they offer the opportunity to check the pressure independence of the hydrogen transitions energies. Therefore the target will be arranged for this special measurement that allows for a simultaneous measurement of the oxygen and nitrogen lines together with the hydrogen lines. As high statistics is not needed, a measurement time of about a week (100 hours) will be sufficient to verify the pressure independence of the energies of the muonic and pionic hydrogen $3\rightarrow 1$ transitions.

As mentioned in chapter 3.1 and 3.3.2, two corresponding muonic and pionic hydrogen transitions can be measured simultaneously as the ratio of the lattice constants of the 100 and 10.1 planes in quartz almost coincides with the ratio of the reduced masses of muonic and pionic hydrogen. Therefore a 100 and a 10.1 quartz crystal will be mounted one on top of the other above and below the Rowland circle. Suitably arranged they can produce an image of the pionic and the muonic transition on the same detector at the same time. This offers the method to determine the response function of the apparatus with a similarly accelerated system $(\mu ^{-}p)$ not broadened by strong interaction. As the muon mass is known with an accuracy of $3\cdot 10^{-7}$ the muonic line also serves as a high accuracy energy standard.

The measurement with muonic hydrogen has an additional merit which is very important. Because of the small width of the transitions experimental artefacts like shifts caused by different stop distributions or absorptions in the target or target windows at different pressures can be determined. Also any cascade process or effect which up to now has escaped the attention will show up in the high resolution muonic spectra not broadened by strong interaction. To give an example, we plan to establish that the transition energy is not shifted by pressure in the density range considered. Therefore we will measure the spectra at extreme values of target pressure.

Simulated spectra of the $2\rightarrow 1$ and the $3\rightarrow 1$ transitions representing a measuring time of three weeks each is shown in Fig. 8 and Fig. 9, respectively.

Figure 8: Simulated spectra for the $2\rightarrow 1$ transitions in pionic (left) and muonic hydrogen as measured by a CCD detector of horizontal width of $48mm$ and a pixel size of $40\mu m$ . The intensity in both lines is 40000. Both lines have a common energy resolution of $221meV$ . A Lorentzian width of $950meV$ is assumed for the pionic line. The muonic line has a hyperfine splitting of $180meV$ . The distribution of the kinetic energy is as in Fig. 1 and Fig. 3, respectively. The peak/background ratio for the pionic line is 100:1.

Figure 9: Simulated spectra for the $3\rightarrow 1$ transitions in pionic (left) and muonic hydrogen as measured by a CCD detector of horizontal width of $48mm$ and a pixel size of $40\mu m$ . The intensity in both lines is 40000. Both lines have a common energy resolution of $262meV$ . A Lorentzian width of $950meV$ is assumed for the pionic line. The muonic line has a 1s hyperfine splitting of $180meV$ . The distribution of the kinetic energy is as in Fig. 2 and Fig. 4, respectively. The peak/background ratio for the pionic line is 100:1.

Table 10 gives the distance of the two lines on the CCD detector together with the dispersion for the two crystals. The distance of the two foci is close enough to fit even onto one of the chips of the CCD detector. As the two crystals can still be rotated with respect to each other there is some freedom to choose the distance $\Delta x$ freely.

Table 10: Mean Bragg angle, angle and position difference of the pionic and muonic hydrogen lines together with the dispersion of the two crystals.

Comparison		$<\Theta _{B}>$ $^{0}$	$\Delta _{\Theta _{B}}$ $^{0}$	$\Delta x$ $[mm]$	D (100) $\frac{mm}{eV}$	D(10.1) $\frac{mm}{eV}$
$\mu ^{-}H_{2\rightarrow 1}$	$\pi ^{-}H_{2\rightarrow 1}$	49.9	.45	17.95	1.45	1.14
$\mu ^{-}H_{3\rightarrow 1}$	$\pi ^{-}H_{3\rightarrow 1}$	40.2	.32	10.72	0.73	0.57

This method can be checked as there are transition energies of electronic (to be used in the preparatory phase of the experiment) and exotic atoms exhibiting a similar ratio. This is the case for the ratio of line energies listed in Table 11. Fluorescence lines of corresponding pairs can also be used. For pionic and muonic $^{4}He$ and for the comparison of $Si^{13+}\leftrightarrow S^{15+}$ it is necessary to modify the geometry of the CCD detector, which does not cause a major problem. Also the comparison $S^{15+}\leftrightarrow Ar^{17+}$ is outside the nominal range of Bragg angles which is not severe during test measurements as the shielding requirements are not as stringent as in the pionic/muonic measurement. The measurement of $\pi N_{6\rightarrow 5}$ $\leftrightarrow$ $\pi O_{6\rightarrow 5}$ should be scheduled at some time during the beginning of the beam time and will consume about one week. This period is considered very important to establish the parameters for the following $\mu ^{-}H_{3\rightarrow 1}\leftrightarrow$ $\pi ^{-}H_{3\rightarrow 1}$ and $\mu ^{-}H_{2\rightarrow 1}\leftrightarrow$ $\pi ^{-}H_{2\rightarrow 1}$ measurements.

Table 11: The angle difference and the position difference for two calibration lines is given together with the dispersion of the two crystals at the mean Bragg angle.

Comparison		$<\Theta _{B}>$ $^{0}$	$\Delta _{\Theta _{B}}$ $^{0}$	$\Delta x$ $[mm]$	D (100) $\frac{mm}{eV}$	D(10.1) $\frac{mm}{eV}$
$\mu ^{-4}He_{4\rightarrow 2}$	$\pi ^{-4}He_{4\rightarrow 2}$	44.48	1.519	55	0.99	0.78
$\pi ^{-}N_{6\rightarrow 5}$	$\pi ^{-}O_{6\rightarrow 5}$	40.77	1.254	43	0.75	0.59
$Si^{13+}$	$S^{15+}$	45.80	1.624	60	1.08	0.85
$P^{14+}$	$Cl^{16+}$	39.03	0.512	17	0.66	0.52
$S^{15+}$	$Ar^{17+}$	33.85	0.135	3.9	0.43	0.34

Next: Results of simulation tests. Up: Calibration of the spectrometer. Previous: Test experiments with pions. Contents

Pionic Hydrogen Collaboration
1998