Resolution of excitation energy spectra

The following factors contribute to the total resolution of the experiment:

1. HKS momentum resolution
   

   With a Monte Carlo simulation, the momentum resolution of the HKS was estimated to be 75 keV/$c$ (RMS) for a spatial resolution of 200$\mu$m of the chambers.

2. Beam momentum resolution
   

   Assuming 1 $\times 10^{-4}$, it will be 180 keV/$c$ (FWHM) for 1.8 GeV/$c$ beam.

3. Enge momentum resolution
   

   For vertical and horizontal resolutions of the chamber as 150$\mu$m, the momentum resolution of the Enge spectrometer is estimated to be 93 keV (FWHM).
   

4. Kinematical broadening due to uncertainty of the K$^+$ scattering angle
   

   The uncertainty of the K$^+$ emission angle is dominated by the multiple scattering through the materials between the target and the chambers (uncertainty due to the spatial resolution of the chambers are less than 0.2 mrad). Contribution of the following materials are taken into account: the target (100 mg/cm$^2$), vacuum windows (kevlar 0.008'', mylar 0.005''), helium bag (100 cm) and chambers (mylar 0.0045'', argon 5.08 cm @ STP).

   Total uncertainty of the K$^+$ angle was estimated to be 3.3 mrad (r.m.s.) for the carbon target. This angular uncertainty corresponds to 152 keV (FWHM) ambiguity to the $^{12}_\Lambda$B mass. Similarly, this effect to $^{28}_\Lambda$Al (Si target) will be 64 keV (FWHM).

5. Momentum loss in the target
   The momentum loss in the target was calculated for 1.2 GeV/$c$ K$^+$ assuming Vavilov distribution. The whole momentum loss in the target will contribute the mass resolution without any correction. The average momentum loss of 1.2 GeV/$c$ K$^+$ in the carbon 100 mg/cm$^2$ was 195 keV/$c$ which corresponds to the energy resolution of 180 keV(FWHM).

The results are summarized in Table 5. Present proposed experimental setup will achieve the resolution around 300 keV (FWHM).

表 5: The energy resolution of the HKS system

Item	Contribution to the resolution (keV, FWHM)
Target	C	Si	V	Y
HKS momentum	190
Beam momentum	$\le$ 180
Enge momentum	93
K$^+$ angle	152	64	36	20
Target (100 mg/cm$^2$)	$\le$ 180	$\le$ 171	$\le$ 148	$\le$ 138
Overall	$\le$ 360	$\le$ 330	$\le$ 320	$\le$ 310