Nuclear Statistics

The discriminator produces an output pulse with a fixed shape (generally square) and size when the input signal crosses a reference. Discriminators usually have multiple identical output signals. The logic pulses can be sent to a scaler that simply counts the number of pulses, to a count rate meter to monitor radiation rates or doses, and to a time-to-amplitude converter (TAC) to measure the relative times of arrival of two or more logic signals. 

The most general model to describe radioactive decay is the binomial distribution. For a process that has two outcomes (success or failure, decay or no decay), we can write for the distribution function P(x) 

TABLE 18.2 Typical Sequence of Counts of a Long-Lived Sample (170Tm)  

The binomial distribution function is cumbersome and a simplification can be made. If the probability of success p is small (p C 1) (the measurement time is very short compared with the half-life), we can approximate the binomial distribution by the Poisson distribution. The Poisson distribution is written as 

Applying these equations to the data of Table 18.2, we get a-PojS = 43.6. This illustrates the important point that these distribution functions are models, not physical laws, and when they are applied to finite data sets, their predictions may deviate from observation. 

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Finally, let us consider molecules with identical nuclei that are subject to C (n > 2) rotations. For C2v molecules in which the C2 rotation exchanges two nuclei of half-integer spin, the nuclear statistical weights of the symmetric and antisymmetric rotational levels will be one and three, respectively. For molecules where C2 exchanges two spinless nuclei, one-half of the rotational levels (odd or even J values, depending on the vibrational and electronic states)... 

The a-process Could the low [a/Fe] and low [Y/Eu] ratios in dSph stars be related by the a-process The a-process (or a-rich freeze out) occurs when a neutron-rich, a-rich gas is out of nuclear statistical equilibrium and is thought to be important in the formation of 44Ca (Woosley Weaver 1995), 48Ti (Naka-... 

Fig. 4.2. Evolution of light-element abundances with temperature, for rj io = 3.16. The dashed curves give the nuclear statistical equilibrium abundances for 4He, 3He, 3H(t) and 2D(d) respectively the dotted curve for 2D allows for the diminishing number of free neutrons. After Smith, Kawano and Malaney (1993). Courtesy Michael Smith.

As a partial summary, in normal C3 V inclusions material from the neutron-rich nuclear statistical equilibrium nucleosynthetic process is in excess relative to the average solar system composition, as well as an O-rich component. Nevertheless, the exact composition of this material is somewhat blurred by secondary processes (nebular or interstellar) as the observations show no strict interelement correlation (Jungck et al. 1984 Birck and Lugmair 1988). 

As for Allende s inclusions, variable contributions of a component produced in neutron-rich nuclear statistical equilibrium best explains the Ti- Ca data. Some parts of the solar nebula were depleted in these isotopes as deficits are also seen. There are several possibilities for explaining the variations in Ti. 1) The neutron-rich component itself may be heterogeneous and incorporate locally less neutron-rich statistical equilibrium products (Hartmann et al. 1985). 2) Ti may result from another process like explosive Si or He burning (Clayton 1988). This component would be associated with the neutron-rich component but not completely homogenized. In all cases, carriers are solid grains which may have behaved differently than the gaseous nebula during the formation of the solar system. A minimum number of components may be calculated to account for the Ca and Ti isotopic data, which number up to 3 (Ireland 1990) but to be conservative at the 5ct level, clearly resolved effects are present only on 3 isotopes ( Ca, Ti, Ti). 

Chromium. The isotopic heterogeneity is limited to this isotope which can be compared with the normal refractory inclusions of Allende. Both Cr dehcits and excesses are formd ranging from -7.6 e to +210 e (Fig. 8b). The fractions showing the highest enrichment in Cr with no correlated effects in Cr, Cr, Cr points towards a nucleosynthetic component, which is 99% pure in Cr. This component is probably the same as the component found in the CV3 inclusions, and which is produced in a neutron-rich nuclear statistical equilibrium in presupemova massive stars. 

Figure 8. Figure (a) after Clayton et al. (1976, 1977). The scales are as in Figure 1. The O isotopic compositions of the different meteorite classes are represented ordinary chondrites (H, L, LL), enstatite chondrites (EFl, EL), differentiated meteorites (eucrites, lAB irons, SNCs) and some components of the carbonaceous chondrites. As the different areas do not overlap, a classification of the meteorites can be drawn based on O isotopes. Cr (b) and Mo (c) isotope compositions obtained by stepwise dissolution of the Cl carbonaceous chondrite Orgueil (Rotaru et al. 1992 Dauphas et al. 2002), are plotted as deviations relative to the terrestrial composition in 8 units. Isotopes are labeled according to their primary nucleosynthetic sources. ExpSi is for explosive Si burning and n-eq is for neutron-rich nuclear statistical equilibrium. The open squares represent a HNOj 4 N leachate at room temperature. The filled square correspond to the dissolution of the main silicate phase in a HCl-EIF mix. The M pattern for Mo in the silicates is similar to the s-process component found in micron-size SiC presolar grains as shown in Figure 7.

Most of the Cr found in the solar system is produced in presupemova neutron-poor nuclear statistical equilibrium as Mn which then decays to Cr (Hainebach et al. 1974 Hartmann et al. 1985). The first evidence for the presence of Mn in the early solar system... 

Another important reaction chain for future phases, including the final nuclear statistical equilibrium (see below), is one which enhances the neutron content... 

Any attempt to understand the conditions in which iron and its kin were created, and identify the astrophysical site of their birth, must focus on the idea of nuclear statistical equilibrium. The situation is the exact nuclear analogy of the ionisation equilibrium occurring in hot gases. 

Fig. A3.1. Binding energy per nucleon in symmetric nuclei (Z = N) and asymmetric nuclei (0.86 < Z/N < 0.88). Ni is the most tightly bound nucleus with an equal number of protons and neutrons, whilst Fe is the strongest nucleus with Z/N = 0.87. Nuclear statistical equilibrium favours Fe if the ratio of neutrons to protons is 0.87 in the mixture undergoing nucleosynthesis. In fact nature seems to have chosen to build iron group nuclei in a crucible with Z = N.

We know today that nuclear statistical equilibrium in a neutron-poor environment (p/n = 1.01), dominated by nickel-56 rather than iron-56, gives a good overall explanation of the abundance table in the neighbourhood of the iron peak. This is a natural consequence of high-temperature combustion. The corresponding combustion times are... 

For C22H2, the nuclear spin of C12 is zero and contributes a factor of 1 to the nuclear statistical weights. The statistical weights are therefore the same as in H2. For the ground vibronic state, the even J levels are s and have nuclear statistical weight 1, corresponding to the one possible ns the odd J levels are a and have nuclear statistical weight 3. The usual selection rule (4.138) holds for collisions as well as radiative transitions, and we have ortho and para acetylene. The two forms have not been separated. 

For nonlinear molecules with identical nuclei that are interchangeable by rotation, the derivation of the nuclear statistical weights is not as simple as for linear molecules. The problem is most efficiently dealt with using group theory. We will not attempt a complete discussion, but will only point out some features of the results.16... 

For 62v molecules where the C2 rotation interchanges two spin- nuclei (e.g., the normal isotopic species of H20 and H2C=0), the function ns involves the same spin functions as for H2. Thus the a levels of H20 have a nuclear statistical weight of 3, whereas the s levels have a nuclear weight of 1 we have ortho and para water. For molecules where the C2 rotation interchanges two spin-zero nuclei (e.g., SOj6), half the rotational levels are missing, just as for C026. 

Fermi resonance in, 278-279 IR bands, table of, 264 normal modes, 242, 248, 262, 265 nuclear statistical weights for, 287-288 vibrational constants of, 275 vibrational levels of, 253 vibrations of, and group theory, 449-451,456... 

Figure 10. Thermal rate constants for capture of N2 by an ion (SACM calculation [33] with channels generating from rotational states N = 0, 1,2, accounting for nuclear statistical weights left figure positive ion right figure negative ion).

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