Relative efficiency

Some papers

File:Nn correlation extraction.pdf

File:Ynn vs Dnn.pdf

Rletive efficiency obtained from the 2n opening angle w/ uncorrelated neutrons

The result of simulation of 2n openeing angle obtained by different neutron detectors (each neutron hit different detector) having y-resolution is presented below. The source of neutrons was isotropic.

Empty target, [math]10^8[/math] events sampled	DU target, [math]10^8[/math] events sampled
Point isotropic n-source w/o material	Volume isotropic n-source + DU material

The result of simulation of 2n openeing angle obtained by the same neutron detectors (2 neutrons hit same detector) having y-resolution is presented below:

In the experiment our expereimental setup did not have y-resolution across the surface of the neutron detectors. So it was necessary to simulate the 2n opening angle for the neutron detectors w/o y-resolution. The result of the simulation of 2n opening angle (the 2n angles detected by all detectors are superimposed) for the case where thre was no y-resolution is going to be noted as [math]D_{nn}[/math]. The experimental data obtained from run 4172 (DU target) on 2n opening angle for uncorrelated neutrons (i.e. in the case of isotropic source of neutrons) from different pulses is going to be noted as [math]Y_{nn}[/math]. The relative efficiency is defined as [math]\epsilon=Y_{nn}/D_{nn}[/math] and it is plotted below as a result of bib-by-bin division of [math]Y_{nn}[/math] over [math]D_{nn}[/math].

Relative efficiency of the detecting system obtained using experimental data only (all runs)

In this part the experimantal data are presented for the cases when (1) the 2n opening angle was measured for neutrons correlated and (2) the 2n opening angle was measured for neutrons uncorrelated (neutrons from separate pulses). In the figure below the experimental data on the 2n opening angle measured for the case of correlated neutrons is presented where the statistical error bar width is equal to [math]\pm \sqrt{N_{bin}}[/math] along y-axis and 4.5 deg along x-axis (the bin width). The total number of triggering pulses during the experiment was Nexp_pulses=2.469256E+7.

The experimental data on the 2n opening angle measured for the case of uncorrelated neutrons is presented in the next figure. 80 neutrons from different pulses were taken uniformly to reproduce 2n opening angle for the case of uncorrelated neutrons. The total number of uncorrelated neutron pairs was Npair_uncorr = 3.608397E+7. The width of the error bars is smaller then the size of the markers.

The result of the division of the two histograms ([math]Y_{nn}^{correlated}/Y_{nn}^{uncorrelated}[/math]) obtained for the described two cases is shown below. In order to obtain the resulting histogram as a division of the previous two histograms we need to do the division on the bin-by-bin basis as [math]\frac{a \pm \delta a}{b \pm \delta b}[/math], where [math]a[/math] and [math]b[/math] are the contents of the corresponding bins and [math]\delta a[/math] and [math]\delta b[/math] are the coresponding statistical errors for the bin contents [math]a[/math] and [math]b[/math]. The error bars were obtained by the propagation in the following way: [math]\delta Err = \frac{a}{b}\sqrt{(\frac{\delta a}{a})^2+(\frac{\delta b}{b})^2}[/math]. The resulting histogram wa normalized to the ratio of the total number of pulses to the total number of uncorrelated neutron pairs considered Norm = Nexp_pulses/Npair_uncorr = 2.469256E+7/3.608397E+7 . It should be noted that the total number of uncorrelated neutron pairs can be varied by changing the number of corresponding combinations and the number of pulses involved into the 2n opening angle reproduction, so the normalization may change.

Normalization

Data table 1.

run#	Total # of pulses per run
run4118	1668482
run4119	2135901
run4133	1704331
run4134	1732490
run4135	1603604
run4137	800699
run4144	1645667
run4145	1544605
run4146	510826
run4153	1750657
run4154	1631436
run4155	1062048
run4166	214600
run4169	191592
run4170	429974
run4171	1141113
run4172	650734
run4173	354992
run4174	953831
run4175	299086
run4176	144774
run4177	225877
run4203	659050
run4204	1636192
totals	24692561

Normalization factor is 24692561.

Data table 2. Uncorrelated neutron pairs.

run#	80 puls pairs	all nonZ pairs	total # of pairs
run4118	1949796	35168980	58460510939894
run4119	2470399	57115623	95803579611944
run4133	1741760	30145143	60999663129029
run4134	1832885	33384324	63031990015343
run4135	1727993	28580648	54002495269034
run4137	833189	7099684	13463513489481
run4144	1770693	26603021	56872651957830
run4145	1592222	20562899	50101929186175
run4146	491705	2157292	5479817982802
run4153	8753312	177560268	64360835399566
run4154	1695297	26345381	55893286150871
run4155	1182325	12257834	23686887361551
run4166	174390	226636	967120869210
run4169	175558	235498	770861409690
run4170	319678	1008712	3882439488990
run4171	1415700	19159461	27344940440490
run4172	623978	3412116	8892563188751
run4173	395573	1327163	2646413182232
run4174	894440	5765727	19105685150123
run4175	255450	619705	1878507428108
run4176	110058	119818	440152774689
run4177	976415	1721476	1071433552715
run4203	3003073	23564022	9121298818219
run4204	1682843	28829022	56219643867161
totals	36068732	542970453	734498220663898

In the case if all possible non zero 2n pairs are considered the shape then the unncorrelated 2n opening angle distribution normalized looks like (the normalization factor is the total number of all possible n-pairs 734498220663898):

The division of normalized experimental data fir the case of correlared 2n pairs and uncorrelated 2n pairs (all non zero n pairs are taken into account) gives the following 2n angle distribution:

If the normalization factor of uncorrelated neutron pairs increased by 4 then the 2n opening angle distribution looks like:

Y(80 n-pairs)/Y(tot # of n-pairs)

80 uncorrelated neutron pairs uniformly distributed	All possible uncorrelated n-pairs
80 n-pairs algorithm	All possible combinations

The result of division of Y(80 n-pairs) over Y(tot # of n-pairs) is presented below

Relative efficiency by random pairing neutrons from different pulses

2n opening angle distribution obtained by random sampling of neutron pairs (without repetition of a neutrons taken once) is presented below:

The relative efficiency obtained as a result of division of 2n opening angle distribution in the case of correlated neutron pairs by 2n opening angle distribution obtained by random sampling of uncorrelated neutron pairs is presented below:

Relative efficiency by pairing neutrons from different pulses (ascending event #)

Another algorithm of neutron pairing was used to obtain the relative efficiency of the detecting system. Every run data file was sorted such that the event number order was ascending (see [1], second column in the table) and two neighboring neutrons from different detectors and different events were used to make a pair and the opening angle was calculated.

The normalization procedure of the opening angle distribution in the case of different pulses was the following

File:Run summary norm.pdf

After going through the data one more time there were found some suspicious events. They were filtered out from the data. Now the range of 2n opening angles matches the range of the angles obtained via geant simulation for the current experimental setup (see the top section on the current page).

The fit on the picture below shows one of the possible trends which the experimental data may follow and is needed to be experimentally proved.

Over the y-axis it is plotted normalized ratio [math](Y^{corr}+Y^{uncorr})/Y^{uncorr} = Y^{tot}/Y^{uncorr}[/math].

The ratio of the integrated yields

The ratio of the normalized integrated yields of correlated + uncorrelated neutrons per pulse [math]Y^{\text{tot}}=2.2E-5[/math] and calculated yield of uncorrelated neutrons per pulse [math]Y^{\text{2n/fisson}}=6.49E-6[/math] gives the following number [math]\frac{Y^{\text{tot}}}{Y^{\text{2 fisson}}}=3.4[/math]. If we want to get the ratio of pure yield of correlated neutrons [math]Y^{\text{1n/fission}}[/math] over the yield of uncorrelated neutrons [math]Y^{\text{2n/fisson}}[/math] we should take the difference [math]Y^{\text{1n/fission}}=Y^{\text{tot}}-Y^{\text{2n/fisson}}[/math] and then take the ratio [math]\frac{Y^{\text{1n/fission}}}{Y^{\text{2n/fisson}}} = \frac{(2.2E-5)-(6.49E-6)}{6.49E-6}=2.4[/math]

Poisson distribution [math]P(n)=\frac{\bar{n}^n e^{-\bar{n}}}{n!}=\frac{(2.5)^ne^{-2.5}}{n!}[/math] of the number of fission neutrons per fission event if the average number of neutrons emitted in fission event is 2.5 neutrons.

Bin-by-bin subtraction

Now let's plot the ratio [math](Y^{tot}-Y^{uncorr})/Y^{uncorr}=Y^{corr}/Y^{uncorr}[/math]. The error propagation is going to be the following:

[math]\frac{Y^{tot}-Y^{uncorr}}{Y^{uncorr}} = \frac{(a \pm \delta a) - (b \pm \delta b)}{(b \pm \delta b)}[/math]

[math]=\frac{(a-b) \pm (\sqrt{\delta a^2 + \delta b^2})}{b \pm \delta b}[/math]

[math]=\frac{a-b}{b} \pm Err(\frac{Y^{tot}-Y^{uncorr}}{Y^{uncorr}})[/math]

[math]Err(\frac{Y^{tot}-Y^{uncorr}}{Y^{uncorr}}) = \frac{a-b}{b} \sqrt{\frac{\delta a^2 + \delta b^2}{(a-b)^2}+(\frac{\delta b}{b})^2}[/math]

The plot of the ratio of yield of pure correlated neutrons over the yield of pure uncorrelated neutrons looks like