TABLE I.

Variables and parameters used within the theoretical model of coaxial prompt γ-ray detection, cf. fig. 1.

z	beam axis coordinate	[mm]
z0	front face position of water target	0 mm
zd	front face position of the detector	152 mm
L	length of the water phantom	150 mm
Rw	proton range in water	[mm]
r	detector radius	½”
t	detector thickness	1½”
Nγ	γ-rays emitted per incident proton	[ ]
${\overline{g}}_{\lambda }$	prompt γ-ray yield per unit length	6·10−4 mm−1
$\overline{f}$	distal falloff slope	0.3 mm−1
$\tau$	transmission probability	[ ]
${\overline{\mu }}_{\text{w}}$	average absorption coefficient of water	10−4 mm−1
$\mathrm{\Omega }/4\pi$	relative solid angle subtended	[ ]
$\eta$	probability of interaction in detector	[ ]
${\overline{\mu }}_{\text{d}}$	average attenuation coefficient of detector	2·10−2 mm−1
td	decay time of the scintillation light pulse	16 ns
Np	number of protons within one spot cluster	108
${\widehat{N}}_{\text{d}}$	prompt γ-rays detected per proton (theory)	[ ]
Nd	prompt γ-rays detected per spot (theory	[ ]
Md	prompt γ-rays measured per spot cluster	[ ]
${\overline{k}}_{\gamma }$	ratio between all counts and the prompt γ-rays	2.5
${\dot{N}}_{\text{d},\text{all}}$	overall detector count rate	[Mcps]
Ip	the proton beam (peak) current at the target	2 nA
e	elementary charge	1.6·10−19 C
${\delta }_R$	absolute error in the proton range Rw	1 mm
${\delta }_z$	absolute error in the phantom position z0	1 mm
κ	thickness of a cavity within the water target	2 mm
zκ	front face position of a cavity within the target	[mm]
ΔN	relative count variation for a range error δR	[ ]
${\mathrm{\Delta }}_{\mathrm{\Omega }}$	relative count variation for a target error δz	[ ]
${\mathrm{\Delta }}_{\gamma }$	γ-ray emission variation for a range error δR	[ ]
${\mathrm{\Delta }}_{\kappa }$	relative count variation due to a cavity κ	[ ]
Δs	statistical measurement uncertainty	[ ]
$\chi$	background fraction separated by energy filters	0.5
${\mathrm{\Sigma }}_{\text{t}}$	proton bunch time spread (FWHM)	2 ns
Tp	proton bunch period	9.4 ns