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Delayed electron lifetime

The disappearance rate of delayed electrons enters in the determination of stopping fraction (6.26), directly in the branching ratio and indirectly in the time cut efficiency . The rate was determined by fitting the time tdel - tSi, i.e., the time between the silicon $\alpha$ event in Si1 or Si2 and the first electron (telescope) event following the Si event, with a single exponential function. In order to ensure accurate determination of , cuts were made on both energy and time of the Si events.

A cut on tSi (Si time with respect to muon entrance time t0) of 0.02 < tSi < 0.5 $\mu $s was applied to select prompt fusion events from the upstream D2 moderator overlayer (where the signal-to-background ratio is most preferable), and to ensure a uniform time efficiency of the Del cuts. Since the event gate was open for a finite width ( s) after the muon entrance, the efficiency for the Del cuts for the Si events occurring late (with respect to t0) is reduced, due to the smaller time window for the detection of delayed electrons[*].

Two different energy cuts were applied. The nominal [2000, 3700] ch[*] cut (noted as Energy cut ``$\alpha$'' in Tables 6.14, 6.15) covered a good portion of the fusion $\alpha$ peak, while the lower energy cut [2000, 3000] ch (noted as ``l'' in Tables 6.14, 6.15) avoided the fusion events which occurred near the surface of the D2 overlayer. The latter cut was implemented to test a possible systematic effect which depends on the depth of the fusion event in the layer, such as or $\mu d$ escaping from the layer. The energy of the $\alpha$ is related to the event depth thanks to $\alpha$ energy loss in the layer.

Tables 6.14 and 6.15 give the fitted results of for Del-e and Del-t time spectra, respectively. Two different series of runs, (A) Runs 1671-83[*], and (B) Runs 1709-30[*], were used for the fit. While a thick hydrogen layer (500 T$\cdot l$) was present in the downstream target for Run A, there was no (Target II-13) or only a very thin (II-14) layer in the downstream target for Run B.

For the delayed electron (Table 6.14), the use of a constant background term was necessary to obtain reasonable fits. This was not the case for the delayed telescope (Table 6.15) where the background was smaller, and fits both with and without the constant term were tried to check the consistency.

Table 6.14: The lifetime of the first electrons (1st E) after the Si signal, fitted to a single exponential with a constant background. Energy cut $\alpha$ represents Si energy of [2000, 3700] ch, selecting fusion $\alpha$ events, while the cut l is for [2000, 3000] ch, avoiding fusion events occurring at the surface.
Run Energy 1st E after Si1 1st E after Si2    
  cut 1/ ($\mu $s) $\chi ^{2}$/dof cl (%) ($\mu $s) $\chi ^{2}$/dof cl (%)    


$\alpha$ 2.025(31) 1.05 34 2.094(34) 1.18 12    
  l 2.095(56) 0.99 50 2.260(63) 1.03 41    
B $\alpha$ 2.027(32) 1.03 40 1.966(34) 1.23 6.7    
  l 2.150(66) 1.09 27 2.073(49) 1.12 23    


Table 6.15: The lifetime of the first telescope (1st T) after Si signal, fitted to a single exponential with (bkgd yes) and without (bkgd no) a constant background term.
Run Energy 1st T after Si1 1st T after Si2 bkgd  
  cut 1/ ($\mu $s) $\chi ^{2}$/dof cl (%) ($\mu $s) $\chi ^{2}$/dof cl (%)    


$\alpha$ 1.978(48) 1.20 9.7 2.102(46) 0.95 62 yes  
  l 2.054(89) 1.21 9.0 2.113(78) 0.92 69    
B $\alpha$ 2.034(51) 1.13 19 1.952(44) 0.99 50    
  l 2.141(91) 0.94 64 2.077(94) 1.18 12    


$\alpha$ 2.116(31) 1.38 1.0 2.096(26) 0.94 84 no  
  l 2.102(52) 1.21 8.4 2.102(44) 0.91 73    
B $\alpha$ 2.090(30) 1.15 16 2.013(28) 1.04 37    
  l 2.187(51) 0.94 65 2.103(54) 1.17 13    

As shown in Tables 6.14 and 6.15, for Si1, Runs A and B give a consistent value of for both Del-e and Del-t, while for Si2, Run B gives a smaller value than Run A by 2 to 3 $\sigma$. If Runs A and B are averaged, however, Si1 and Si2 give consistent . The averages over Si1 and Si2 as well as over Runs A and B were thus taken and are presented in Table 6.16. We note the following. First, Del-e and Del-t are for the most part consistent with each other. Second, not including the constant background term increases the value of fitted . Though not shown in the tables, this holds true for the Del-e fits as well. Third, the energy cut l gives a that is 2-4 $\sigma$ lower than the cut $\alpha$ in all cases in Table 6.16. Our determination of is thus limited by systematic effects, which are possibly due to the finite thickness of our layer. Taking the average of the two extreme values in Table 6.16 we assign $\mu $s with the error covering the two extremes. Thus we have the time cut efficiency, , and the electron branching ratio, , which combine to give the factor for Eq. 6.26. Note that the errors are correlated.

Table 6.16: The lifetime of 1st delayed electrons, and 1st delayed telescope, averaged over Si1 and Si2, as well as Run A and Run B.
Detector bkgd ($\mu $s)
    Energy cut $\alpha$ Energy cut l
1st E yes 2.028(16) 2.145(29)
1st T yes 2.017(24) 2.096(44)
  no 2.079(14) 2.124(25)

Table: The Del cut efficiency, $\epsilon _{del}$, the absolute electron detection efficiency, $\epsilon _e \Omega _e$, and the muon stopping fraction, SFABSH determined from the absolute yield method. Run A is the same as that in Table 6.14, and Run C is similar to A but without the downstream 500 T$\cdot l$ H2 and 3 T$\cdot l$ D2 (target ID=II-7). Run D had a 6 T$\cdot l$ upstream overlayer (target ID=II-3) instead of 14 T$\cdot l$ as in others. The errors (in brackets) for $\epsilon _{del}$ and $\epsilon _e \Omega _e$ are statistical only, while those for SFABSH include the % systematic uncertainty due to . The values in bold face highlight the downstream detectors (En2 and Tel2) which are less suceptible to the Si solid angle effect.
Run Det. $\epsilon _{del}$ (%) $\epsilon _e \Omega _e$ SFABSH
    Si1 Si2 Average (%) (%)
A Ege 3.99(5) 5.26(5) 4.63(4) 5.95(5) 29.6(11)
En1 5.05(6) 3.87(7) 4.46(5) 5.74(6) 30.1(11)
En2 4.13(6) 4.25(5) 4.19(4) 5.39(5) 29.2(11)
1st E 13.1(1) 13.3(1) 13.2(1) 17.0(1) 29.8(11)
  Tel1 2.35(4) 1.99(4) 2.17(3) 2.79(4) 29.1(11)
  Tel2 1.90(4) 1.98(4) 1.94(3) 2.50(4) 28.9(11)
  1st T 4.24(6) 3.96(6) 4.10(4) 5.28(5) 29.2(11)
C Ege 3.67(13) 5.12(11) 4.40(9) 5.66(11) 31.2(13)
  En1 4.85(15) 3.78(16) 4.32(11) 5.56(14) 31.1(14)
  En2 4.07(13) 3.93(13) 4.00(9) 5.15(12) 30.6(13)
  1st E 12.54(25) 12.76(24) 12.65(17) 16.29(22) 31.1(12)
  Tel1 2.23(10) 2.07(10) 2.15(7) 2.77(9) 29.6(14)
  Tel2 1.78(9) 1.83(9) 1.81(6) 2.33(8) 30.9(15)
  1st T 4.01(13) 3.89(13) 3.95(9) 5.09(12) 30.3(13)
D Ege 4.17(23) 4.44(22) 4.31(16) 5.55(21) 31.8(17)
  En1 4.63(25) 3.80(27) 4.22(18) 5.43(23) 31.8(18)
  En2 4.11(23) 3.81(24) 3.96(17) 5.10(22) 30.9(17)
  1st E 12.86(44) 11.93(44) 12.40(31) 15.96(40) 31.8(14)


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Next: Delayed electron cuts efficiency Up: Absolute amplitude method Previous: Absolute amplitude method