Accidental coincidences

Obviously, the coincidence resolving time At, shown in Fig. 4.48, has to be large enough to accept all these true coincidences. However, this finite value of At then leads to the recording of not only the desired true coincidences, but also of accidental coincidences (also called random or false coincidences). As indicated by the name, these accidental coincidences accidentally follow one another within the time At. Hence, they are due to any two electrons which match the conditions set by the experimenter for the selected double ionization process they might originate from two different double-ionization processes, two single-ionization processes, or... 

The extreme influence water can exert on the Diels-Alder reaction was rediscovered by Breslow in 1980, much by coincidence . Whale studying the effect of p-cyclodextrin on the rate of a Diels-Alder reaction in water, accidentally, the addition of the cyclodextrin was omitted, but still rate constants were observed that were one to two orders of magnitude larger than those obtained in organic solvents. The investigations that followed this remarkable observation showed that the acceleration of Diels-Alder reactions by water is a general phenomenon. Table 1.2 contains a selection from the multitude of Diels-Alder reactions in aqueous media that have been studied Note that the rate enhancements induced by water can amount up to a factor 12,800 compared to organic solvents (entry 1 in Table 1.2). 

It should explicitly be stated at this point that it is not possible, except by accidental coincidence, for an intersection between two lines i,j and k,l to be valid if either i or j is not equal to either k or 1. Thus such intersections need never be considered. 

Otherwise, it would mean that there are at least two subspaces of the space of eigenfunctions fi,..., 4>i, - , 0gn, each of them closed under the symmetry operations of G. This would mean that there are no symmetry operations connecting these two subspaces, in spite of the fact that they have the same energy This, of course, seems to be unreasonable except in the case of accidental coincidence of two energy levels. 

At low flux densities, (p < 0c. the dominant TPA occurs due to the photon entanglement. Quantum hthography is supposed to work in this regime. At higher flux densities, (p> (pc, the dominant TPA is due to the accidental coincidence between photons. In this regime benefits of quantum hthography are lost. 

The simple probabihstic model was also confirmed by quantum-mechanical analysis [72]. Quantum-mechanical analysis gave expressions for ae and a2y that are very similar to each other. This similarity indicates that TPA of entangled and accidentally coincident photons is essentially identical. Differences between both expressions arise mainly due to the fact the expression for entangled photon TPA cross-section accounts for correlations between the photons it has a constant factor l/(TeAe). Moreover, it includes complex entanglement time-dependent siunmation coefficients, which may lead to periodic modiflation in the ae(Te) dependence, or entanglement-induced two-photon transparency. 

Accidentally this relation agrees with Eq. (9.33), defining the tricritical region. This coincidence happens only in three dimensions. Indeed, our crossover diagrams ignore the three-body forces which induce an additional structure in the 0-region. This structure involves the strength of the three-body interaction as a new scale and therefore is not fixed relative to the curves shown here. 

The residence time was determined for our neutron counter by measuring the time intervals between beta start signals and neutron stop signals. With a residence half-time of 11 ms and a coincidence resolving time of 40 ms. 92 of the true coincidence events were included. The fraction of true events not detected does not influence the present results because we normalize the Pn measurements to a known Pn value measured under identical conditions. The coincidence rate was measured by a simple overlap coincidence module where the beta pulse Input was stretched to 40 ms by a gate and delay generator. To measure the accidental coincidence rate, the same beta pulse was sent to a second coincidence module and overlapped with neutron pulses which had been delayed 45 ms. After correcting each coincidence rate for deadtime effects, the difference was the true coincidence rate. 

Figure 4.48 Typical spectrum of electron-electron coincidences recorded with a TDC. The data refer to a situation in which the photon beam has no time structure. True coincidences are collected in the peak while accidental coincidences give a flat and smooth background. At indicates the coincidence resolving time and dt the time resolution of the time-measuring device. The two shaded areas represent accidental coincidences, measured on the left-hand side together with the desired true coincidences, but on the right-hand side separately (and simultaneously) in the full time spectrum.

From the relations quoted, the rate of accidental coincidences, /acc, recorded within the finite resolving time interval At (see Fig. 4.48) can be calculated from... 

An important quantity is the ratio r of true to accidental coincidences ... 

From this relation it follows that a good, i.e., large, ratio of true to accidental coincidences requires a small coincidence resolving time At and a small source strength (f J<7). However, for small values of N all counting rates, Ix, l2, and /true, are small, and therefore the ratio r is not well suited as a criterion of the quality or feasibility of coincidence experiments. Indeed, a more appropriate figure of merit follows if the relative error a of true coincidences, defined by... 

The relative error ANacc for the measurement of accidental coincidences is retained in this expression, because its value depends on the efforts undertaken for measuring JVacc. For example, from Fig. 4.48 it can be inferred that it is possible to use for the accidental coincidences a larger time interval, Afacc, than the At relevant for the true coincidences. This then helps to reduce the statistical error of ANacc, and one gets two limits for AJVacc ... 

If the accidental coincidences are calculated using equ. (4.103), ANacc- becomes negligible, because Iy and I2 are usually large numbers. However, due to the decreasing photon flux at an electron storage ring this equation would have to be applied at every instant of time. Hence, it is preferable to measure the accidental coincidences and desired total coincidences simultaneously.) With equ. (4.112)... 

Figure 4.49 The normalized collection time TcM norm. of equ. (4.115) as a function of the relative source strength N(r)/N(r = 1), and for <5 = 0. The parameter r is the ratio of true-to-accidental coincidences, and r-1 = N(r)/N(r = 1). The parameter <5 describes the error attached to the measured number of accidental coincidences (see equ. (4.112)). The collection time decreases for increasing source strength for practical applications the asymptotic limit is reached well enough for foJ(r)/M(r = 1) = 10. From [Kra94] see [VSa83].

After this detailed treatment of true and accidental coincidences, all the statements derived so far for a good performance for coincidence experiments can be summarized ... 

Figure 5.31 Time correlation spectrum between 4d5/2 photoelectrons and N5-O2 3O2 3 S0 Auger electrons in xenon, recorded with a time-to-digital converter. Note the repetition rate, 208 ns, of the circulating electron bunches in the storage ring. The large second peak contains true and accidental coincidences, and the periodic structure is due to accidental coincidences only. From [KSc93].

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