Figure 3 shows the pion punch through probabilities as a function of
depth in iron for pions of different momenta. Table 1 tabulates the normal
depth of iron which has been penetrated by a particle in reaching the indicated
gap. From Figure 3, the punch through probabilities of pions reaching the 5th
gap (150 cm of Fe) and the 6th gap (170 cm of Fe) as a function of momentum can
be determined. The results of such a tabulation is shown in Figure 4. Figure 5
shows the ratio of the two. From Figure 5, it is seen that the 
/
rejection ratio will be degraded by a factor
of about 2.5 for momentum below 10 GeV/c if the Muon Identifier is instrumented
only to a depth of gap 5.
The above conclusions, which are all based upon Figure 3, have also been verified by Monte Carlo studies as reported by Naohito Saito (as shown in the E-mail attached as Appendix A). In the Monte Carlo study it was shown that if the 6th gap were not instrumented, the probability of misidentifying a pion as a muon increased by about a factor of 1.5. The difference between the Monte Carlo and these data can be accounted for by the fact that the Monte Carlo includes the contribution of the Cu nose cones which have been neglected here.
Based upon the information shown in these analysis of the performance of the
Muon Identifier, one would want to instrument the 6th gap. Then the question
comes as to whether to instrument the 5th or the 4th gap. If one were to
attempt to build a rangefinder which had a constant 
E/E, it would be logrithimically
longitudinally instrumented. In the case of the PHENIX detector, an
approximation to the idea could be achieved by the instrumentation of gap 4 and
skipping gap 5. Such a scheme should result in almost no difference in
performance from an identifier which were completely instrumented in all six
gaps. The Monte Carlo studies indicate that that is indeed the case.
Figure 3: Punch through probabilities as a function of depth of iron penetrated for pions of various momenta. This figure is taken from "PHENIX Muon Detector Review" (June 4-5, 1993, PHENIX Note PN-84).
Figure 4: The punch through probability of pions as a function of pion momentum for two different thicknesses of iron absorber. The two different thicknesses of iron correspond to pions reaching gaps 5 (150 cm of Fe) and 6 (170 cm of Fe) in the North Muon Identifier.
Figure 5: The ratio of punch through probabilities of pions penetrating to gap 5 vs penetrating to gap 6 in the North Muon Identifier as a function of momentum.