and
where: A= a GL, B= a GR, and k is the inverse of the attenuation length. The value a is proportional to the true signal size (i.e. the number of MeV deposited in the detector). The values GL and GR are the respective gain factors for the left and right side photomultiplier tubes. The geometric mean is defined as:
and is proportional to the energy lost regardless of where along the stave the particle passed through the stave. Now, consider the quantity:
or
where the value
is constant. A linear fit of the scatter plot of EL / Egm versus x (as obtained from the MWPCs) can be performed with the resulting slope being k. Two examples of such scatter plots can be seen in Figure 4.6. The value k is the inverse of the effective attenuation length of the detector. By using the ratio of EL to Egm, the events are normalized to
Figure 4.6: Scatter plots used to determine the effective attenuation length of a PV stave.
account for differences in energy deposition due to different pathlengths taken by cosmic rays striking the detectors at various angles. The events used in the scatter plots were restricted to those in which the cosmic ray entered at an angle of less that 20o from the normal to the stave. The inverse attenuation lengths for each of the three staves in the tomography chamber were found in this matter and the results are summarized in Table 4.2.
The reciprocal of the average yields a value for the effective attenuation length of 39.6 ± 1.3 cm. This is considerably shorter than the nominal value of 160 cm given by Bicron for their BC-400 type scintillator noted in Table 4.1. This is to be expected, however, since the Bicron number is based on data taken from a 1 × 20 × 200 cm3 sheet of material. A PV scintillator is only 3.175 mm thick so any light reaching the phototube is likely to have undergone many reflections. The geometry then works to degrade the signal in two ways: First, by effectively increasing the amount of material the light must traverse
| Phototube
|
k
(cm-1)
|
| 1
|
-(2.18
± 0.08) × 10-3
|
| 2
|
-(2.53
± 0.09) × 10-3
|
| 3
|
-(2.10
± 0.06) × 10-3
|
| 4
|
(3.16
± 0.09) × 10-3
|
| 5
|
(2.60
± 0.09) × 10-3
|
| 6
|
(2.56
± 0.09) × 10-3
|
| Average
|
(2.52
± 0.08) × 10-3
|
before reaching the PMT, and, secondly, by reducing the signal at each reflection by an amount dependent upon the smoothness of the surface and the wrapping used.