Let be the normalized kinetic energy distribution of the exotic atom, then the cumulative energy distribution is defined by the formula
with the normalization condition having the form
(34) 
where is the maximum kinetic energy.
The neutron time of flight corresponding to the reaction at rest is where is the neutron flight path and cm/ns is the neutron velocity. The Doppler broadening of the neutron timeofflight spectrum , where is the difference between the measured time and , is related to the kinetic energy distribution by
where is speed of light and is the mass.
Given the Doppler profile , one can determine the cumulative energy distribution using the following formula which is derived straightforwardly from Eqs. (3740):
Equation (41) can be used for the determination of the kinetic energy distribution from the measured neutron ToF spectrum in a model independent way as illustrated in Figs. 11, 12 and 13.

Figure 11: The calculated kinetic energy distribution of the atom at the instant of strong interaction in liquid hydrogen. 
The theoretical kinetic energy distribution shown in Fig 11 corresponds to the nToF distribution plotted in Fig. 12. Applying Eq. (41) to the Monte Carlo generated spectrum with events we obtain perfect reconstruction of the original kinetic energy distribution as shown in Fig. 13.
The Doppler broadening of the Xray lines is described by formulas similar to Eqs. (39,40):
where is the transition energy. Therefore the same method can in principle be applied for the kinetic energy distribution of the muonic hydrogen from the Doppler profile of the Xray lines. In this case, however, the limits of the final energy resolution are significant and proper corrections must be applied before using the formulas mentioned above. The extension to the case of pionic hydrogen, where the nuclear reaction widths should be taken into account, is straightforward.

Figure 12: The calculated neutron ToF spectrum for the reaction in liquid hydrogen for a neutron flight path of 5m. 

Figure 13: The cumulative energy distribution W(E) reconstructed from the nToF spectrum with events (dotted line) in comparison with the exact result corresponding to Fig 11. 