Optics and detector geometries

A 3D magnetic field calculation with OPERA3D-TOSCA was performed for each magnet, to include all higher term effects into the simulation. With this magnetic field map calculated by TOSCA (figure 23) and the optimized parameters by TRANSPORT, the GEANT Monte-Carlo simulation was carried out to estimate the HKS momentum resolution and the acceptance. Table 6 summarizes the optimized magnetic field parameters and the geometrical conditions for the HKS magnets. The parameters used in the simulation are summarized in table 7.

表 6: Summary of the HKS TRANSPORT calculation

Item	Given or Optimized values
Q1 length (cm)	84
Q1 bore radius (cm)	12
Q1 field gradient (T/m)	-5.78
Q1 integral field gradient (T/m$\cdot$m)	-5.54
Q2 length (cm)	60
Q2 bore radius (cm)	14.5
Q2 field gradient (T/m)	3.40
Q2 integral field gradient (T/m$\cdot$m)	2.53
Dipole bending angle (degrees)	70
Radius of D magnet center (cm)	270
D gap (cm)	20
Dipole Field (T)	1.44

In the simulation, the realistic shape of apertures of Q1 and Q2 (figures 18 and 19), walls of the target chamber, and walls of the dipole vacuum chamber are all taken into account.

Red lines shown in figure 4 show kaon trajectories in the horizontal plane (emission angles in horizontal plane, $\theta$ = 4.3, 7, 9.7$^\circ$) with momenta of 1.05, 1.20, 1.35 GeV/$c$, obtained with the GEANT simulation code. The horizontal focusing property of the HKS is clearly seen.

表 7: Summary of parameters used in the HKS simulation

Description	Value
Virtual target position($x$ points e-arm	$(x,y)=(-9.42 \cos(7^\circ)$ cm $, 9.42 \sin(7^\circ)$ cm)
side, $y$-axis is along Hall C beam line)
Target	Carbon 100 mg/cm$^2$
Splitter rotation angle	8.51$^\circ$ with respect to beam line
Splitter field	1.49 T
Splitter bending angle (1.2 GeV/$c$)	14.5$^\circ$
Collimator position	20 cm upstream from Q1 entrance
Distance from virtual target to Q1	105 cm
HKS rotation angle	21.5$^\circ$ with respect to beam line
Central momentum	1.2 GeV/$c$
Central scattering angle	7$^\circ$
Optics	R34=0 (vertically point to point focus)
Beam energy	1.8 GeV
Beam size	50 $\mu$m $\phi$
Kaon momentum	1.0 to 1.4 GeV/$c$
	(1.2 GeV/$c\pm$12.5% for detector size study)
Kaon angular distribution	0$^\circ$ to 15$^\circ$ uniform

Active areas of the HKS detectors are determined requiring the momentum acceptance of 1.2 GeV/$c \pm 12.5$% and maximizing the geometrical acceptance. Figure 8 shows beam profiles at each detector. The active area of each detector is indicated by a square in the figure and listed in table 15.