The uncertainty of the half-life

As the activity in a radioactive sample changes with time, it is appropriate to associate a measured activity with a reference time t₀, which does not necessarily coincide with the start or stop time of measurement (t₁, t₂). Rescaling of a measured amount of decays N of a radionuclide with decay constant λ involves correction factors for decay between the reference time and the start of the measurement t_d = t₁ − t₀ and during the measurement t_m = t₂ − t_1.

$$\begin{eqnarray}&&A\left({{t}_{0}}\right)=\frac{N}{{{t}_{\text{m}}}}{{\text{e}}^{\lambda {{t}_{\text{d}}}}}\frac{\lambda {{t}_{\text{m}}}}{\left(1-{{\text{e}}^{-\lambda {{t}_{\text{m}}}}}\right)}\end{eqnarray} \tag{ 1 }$$

The counting factor, $C=\left(1-{{\text{e}}^{-\lambda {{t}_{\text{m}}}}}\right)/\lambda {{t}_{\text{m}}}$, accounts for decay during the measurement and is significant (>x%) only if t_m is not negligibly small compared to the half-life T_1/2 = ln2/λ or, using a first order series expansion $C=1-\lambda {{t}_{\text{m}}}/2+O\left({{\lambda}^{2}}t_{\text{m}}^{2}\right),$ if ${{t}_{\text{m}}}/{{T}_{1/2}}>2/\ln 2x\%\approx 3x\%$.

The standard uncertainty of the half-life, ${{\sigma}_{{{T}_{1/2}}}}$, propagates through the decay correction factors as follows

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{\left(\frac{{{\sigma}_{A}}}{A}\right)}^{2}}&={{\left(\frac{\partial A}{\partial \lambda}\right)}^{2}}{{\left(\frac{\lambda}{A}\right)}^{2}}{{\left(\frac{{{\sigma}_{\lambda}}}{\lambda}\right)}^{2}} \\ &={{\left(\lambda {{t}_{\text{d}}}+1-\frac{\lambda {{t}_{\text{m}}}}{{{\text{e}}^{\lambda {{t}_{\text{m}}}}}-1}\right)}^{2}}{{\left(\frac{{{\sigma}_{\lambda}}}{\lambda}\right)}^{2}} \\ &={{\left(\lambda {{t}_{\text{d}}}+\lambda {{t}_{\text{m}}}-\frac{1-C}{C}\right)}^{2}}{{\left(\frac{{{\sigma}_{{{T}_{1/2}}}}}{{{T}_{1/2}}}\right)}^{2}} \\ &={{\left(\lambda {{t}_{\text{d}}}+\frac{\lambda {{t}_{\text{m}}}}{2}+O\left({{\lambda}^{2}}t_{\text{m}}^{2}\right)\right)}^{2}}{{\left(\frac{{{\sigma}_{{{T}_{1/2}}}}}{{{T}_{1/2}}}\right)}^{2}} \\ \end{array}\end{eqnarray} \tag{ 2 }$$

The uncertainty on the corrected activity increases linearly in time (via t_d and t_m, if t_m < T_1/2) and the propagation factor is significant for time periods of comparable magnitude as the half-life. With every 0.1% uncertainty on the half-life, the uncertainty of the activity increases by 0.07% per half-life.

Routine laboratories calibrate their activity detectors by means of secondary standards; calibrated sources with traceable activities of certain radionuclides. These standards are kept and reused for a practical period of time. Consider for example 10 year-old ²²Na (2.6029 (8) a), ⁵⁴Mn (312.13 (3) d), ⁵⁵Fe (2.747 (8) a), ⁵⁷Co (271.80 (5) d), ⁶⁵Zn (244.01 (9) d), ¹⁰⁹Cd (461.4 (12) d), ¹²⁴Sb (60.208 (11) d) and ¹³⁴Cs (2.0644 (14) a [21]) sources being used for calibration in an x- and γ -ray spectrometry laboratory. The standard uncertainty on the calibration factor due to uncertainty on the half-life is then respectively 0.08% (²²Na), 0.08% (⁵⁴Mn), 0.7% (⁵⁵Fe), 0.17% (⁵⁷Co), 0.4% (⁶⁵Zn), 1.4% (¹⁰⁹Cd), 0.8% (¹²⁴Sb) and 0.23% (¹³⁴Cs), which is significant for some of these nuclides. Larger errors can occur for radionuclides with shorter or less reliably known half-life.

More intricate rescaling is applied in the field of neutron activation analysis, where the link between elemental concentrations and measured activity of decay products is established through complex activation and decay formulas obtained as solutions of sets of linear first-order differential equations [22]. They typically contain a saturation factor $S=\left(1-{{\text{e}}^{-\lambda {{t}_{\text{irr}}}}}\right)$ for the activation during a period t_irr, a decay factor $D={{\text{e}}^{-\lambda {{t}_{\text{d}}}}}$ and a counting factor C as above. The relative uncertainty of a measured count rate via such factors SDC is

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{\left(\frac{{{\sigma}_{\text{SDC}}}}{SDC}\right)}^{2}}={{\left(\frac{\lambda}{S}\frac{\partial S}{\partial \lambda}+\frac{\lambda}{D}\frac{\partial D}{\partial \lambda}+\frac{\lambda}{C}\frac{\partial C}{\partial \lambda}\right)}^{2}}{{\left(\frac{{{\sigma}_{\lambda}}}{\lambda}\right)}^{2}} \\ ={{\left(\lambda \left({{t}_{\text{d}}}+{{t}_{\text{m}}}-{{t}_{\text{irr}}}\frac{1-S}{S}\right)-\frac{1-C}{C}\right)}^{2}}{{\left(\frac{{{\sigma}_{{{T}_{1/2}}}}}{{{T}_{1/2}}}\right)}^{2}} \\ ={{\left(\lambda \left({{t}_{\text{d}}}+\frac{{{t}_{\text{m}}}}{2}+\frac{{{t}_{\text{irr}}}}{2}\right)-1+O\left({{\lambda}^{2}}t_{\text{irr}}^{2}\right)+O\left({{\lambda}^{2}}t_{\text{m}}^{2}\right)\right)}^{2}}{{\left(\frac{{{\sigma}_{{{T}_{1/2}}}}}{{{T}_{1/2}}}\right)}^{2}} \\ \end{array}\end{eqnarray} \tag{ 3 }$$

Assuming that the time periods involved—t_irr, t_d and t_m—are of similar magnitude, the propagation factor is about 1 for long-lived nuclides (λt << 1, λt_irr(1 − S)/S ≈ 1, C ≈ 1) and λt_d + 1 for extremely short-lived nuclides (λt >> 1, λt_irr(1 − S)/S ≈ 0, λt_m − (1 − C)/C ≈ 1). For intermediate half-lives (λt ≈ 0.1–1), the propagation factor can be <1 or even cancel out completely, because the activation and decay terms have opposite signs.

Radiometric dating methods are based on the exponential decay law describing the decreasing number of radioactive atoms in a material, and/or the related ingrowth of daughter nuclides [23]. One way of age dating is based on known atomic concentrations P(t) of the parent nuclide at different times, via

$$\begin{eqnarray}&&\text{Age}=t=\frac{1}{\lambda}\ln \left[\frac{P(0)}{P(t)}\right]\end{eqnarray} \tag{ 4 }$$

A typical example is ¹⁴C dating (5700 (30) a).

If the daughter atoms are stable and their initial concentration is zero, D(0) = 0, the sum of parent and daughter concentrations is constant, D(t) = P(0) − P(t) or P(0) = P(t) + D(t), and the age can be obtained from

$$\begin{eqnarray}&&\text{Age}=\frac{1}{\lambda}\ln \left[1+\frac{D(t)}{P(t)}\right]\end{eqnarray} \tag{ 5 }$$

This scheme can be used in uranium-lead dating for systems that remained closed and with both decay routes ²³⁸U/²⁰⁶Pb and ²³⁵U/²⁰⁷Pb yielding concordant ages.

Corrections are required if D(0) ≠ 0 or if the concentrations have varied in the sample through physicochemical processes. Isochron dating methods use the concentration S of a stable isotope of the daughter element as a reference. Since (P(0) + D(0))/S = (P(t) + D(t))/S, the relative concentration of the daughter isotopes D(t)/S is the sum of the initial isotopic ratio D(0)/S and a fraction originating from the decay of the parent nuclide:

$$\begin{eqnarray}&&\frac{D(t)}{S}=\frac{P(t)}{S}\left[{{\text{e}}^{\lambda t}}-1\right]+\frac{D(0)}{S}\end{eqnarray} \tag{ 6 }$$

As P(t)/S can vary freely within material samples of the same age, a consistent set of data is characterised by a linear relationship, called isochron. The age follows from the slope (e^λt − 1)

$$\begin{eqnarray}&&\text{Age}=\frac{1}{\lambda}\ln \left[1+\frac{\Delta \left(D(t)/S\right)}{\Delta \left(P(t)/S\right)}\right]\end{eqnarray} \tag{ 7 }$$

This procedure is routinely used in rubidium-strontium dating, in which the parent ⁸⁷Rb and daughter ⁸⁷Sr are compared to stable ⁸⁶Sr. Ratios are used instead of absolute concentrations because they are conveniently measured with mass spectrometers. Another application is dating 'young' groundwater, i.e. assessing when it became isolated from the atmosphere, through the decay of tritium ³H to the rare stable daughter ³He.

The isochron method can also be applied in uranium-lead dating, using ²⁰⁴Pb as the non-radiogenic isotope. Combining the isochron equations for the ²³⁸U/²⁰⁶Pb and ²³⁵U/²⁰⁷Pb systems, one can derive the mathematical basis of 'lead-lead dating' based merely on measurement of the lead isotopic ratios R_6,4(t) = ²⁰⁶Pb/²⁰⁴Pb and R_7,4(t) = ²⁰⁷Pb/²⁰⁴Pb

$$\begin{eqnarray}&&\frac{{{R}_{7,4}}(t)-{{R}_{7,4}}(0)}{{{R}_{6,4}}(t)-{{R}_{6,4}}(0)}=\frac{1}{137.88}\left(\frac{{{e}^{{{\lambda}_{235}}t}}-1}{{{e}^{{{\lambda}_{238}}t}}-1}\right)\end{eqnarray} \tag{ 8 }$$

where the factor of 137.88 is the present-day ²³⁸U/²³⁵U ratio. The initial Pb isotopic ratios (extracted from meteorites) and age of the system are the two factors determining the present day Pb isotopic ratios. In a closed system, equation (7) represents a linear relationship in which the slope depends on the age. The age of Earth, 4.54 (5) 10⁹ a, was thus obtained (see e.g. [24]).

The dating equations (4), (5) and (7) are linearly proportional to the half-life of the parent (and equation (8) roughly on a ratio of half-lives), which means that the relative uncertainty of the half-life constitutes an upper limit to the attainable accuracy on the age. For several applications, this has become the bottleneck [25].

Another dating technique, ²¹⁰Pb sediment radiochronology [26], is frequently applied to reconstruct past environmental conditions of ecosystems. In systems that have been closed for a sufficiently long time (>150 a), the supported²¹⁰Pb is in equilibrium with its parent radionuclide ²²⁶Ra. Due to various transport processes, e.g. exhalation of ²²²Rn and subsequent deposition of daughter products, excess or unsupported²¹⁰Pb can accumulate in bottom sediments. The amount of excess ²¹⁰Pb in various layers of a sediment core is an indicator of the accumulation rate. For this technique, the uncertainty on the half-life is of minor importance compared to the modelling issues and the measurement uncertainty on the difference between ²¹⁰Pb and ²²⁶Ra/²¹⁴Pb activities.

Radioactive disequilibrium between parent and daughter arises when both are separated by physicochemical processes, e.g. by applying radiochemical separation techniques. Following this separation, the daughter starts to grow in again. Information on the time of separation may be obtained from measuring the ratio of parent to daughter atoms. This property is applied in nuclear forensics, aiming at fingerprinting of nuclear materials [23, 27, 28]. The relative amounts of parent and decay products can be used to identify the age and source of the material. Nuclides of interest are actinides (Th, U, Np, Pu, Am) with potential applications in improvised nuclear devices or other nuclides that can be applied in radiological dispersion devices (Co, Sr, Cs). A sample consisting of mixed U-Pu isotopes provides as many as a dozen chronometers, including ²³⁴U/²³⁰Th, ²³⁵U/²³¹Pa, ²³⁸Pu/²³⁴U, ²³⁹Pu/²³⁵U, ²⁴⁰Pu/²³⁶U and ²⁴¹Pu/²⁴¹Am. The time scale of interest is in the 1–50 year range, which is much shorter than in geology. Age dating of U is more difficult than Pu dating, because their long half-lives lead to minute amounts of ingrowing daughter nuclides. Besides the quality of the separation, the current uncertainties on the relevant half-lives are a major limiting factor on the attainable accuracy.

Parent and daughter both being unstable, the ratio of their atom concentrations at a time t after complete separation (D(0) = 0) is calculated from [29]

$$\begin{eqnarray}&&R(t)=\frac{D(t)}{P(t)}=\frac{{{\lambda}_{\text{P}}}}{{{\lambda}_{\text{D}}}-{{\lambda}_{\text{P}}}}\left(1-{{e}^{-\left({{\lambda}_{\text{D}}}-{{\lambda}_{\text{P}}}\right)t}}\right)\end{eqnarray} \tag{ 9 }$$

and the 'age' t is solved from

$$\begin{eqnarray}&&\text{Age}=\frac{1}{{{\lambda}_{\text{P}}}-{{\lambda}_{\text{D}}}}\ln \left(1-\frac{D(t)}{P(t)}\frac{{{\lambda}_{\text{D}}}-{{\lambda}_{\text{P}}}}{{{\lambda}_{\text{P}}}}\right)\end{eqnarray} \tag{ 10 }$$

Linear propagation of uncertainty on the atom ratio R(t) and the half-lives λ_P and λ_D results to [29]

$$\begin{eqnarray}\begin{array}{*{35}{l}}{{\left(\frac{{{\sigma}_{t}}}{t}\right)}^{2}}&={{\left(\frac{{{\lambda}_{\text{D}}}}{{{\lambda}_{\text{P}}}-{{\lambda}_{\text{D}}}}\left(\frac{T}{t}-\frac{{{\lambda}_{\text{P}}}}{{{\lambda}_{\text{D}}}}\right)\right)}^{2}}{{\left(\frac{{{\sigma}_{{{\lambda}_{\text{P}}}}}}{{{\lambda}_{\text{P}}}}\right)}^{2}}\\ &+{{\left(\frac{-{{\lambda}_{\text{D}}}}{{{\lambda}_{\text{P}}}-{{\lambda}_{\text{D}}}}\left(\frac{T}{t}-1\right)\right)}^{2}}{{\left(\frac{{{\sigma}_{{{\lambda}_{\text{D}}}}}}{{{\lambda}_{\text{D}}}}\right)}^{2}}+{{\left(\frac{T}{t}\right)}^{2}}{{\left(\frac{{{\sigma}_{R}}}{R}\right)}^{2}}\end{array}\end{eqnarray} \tag{ 11 }$$

in which the variable T is defined through

$$\begin{eqnarray}&&\frac{T}{t}=\frac{{{\text{e}}^{\left({{\lambda}_{\text{D}}}-{{\lambda}_{\text{P}}}\right)t}}-1}{\left({{\lambda}_{\text{D}}}-{{\lambda}_{\text{P}}}\right)t}\end{eqnarray} \tag{ 12 }$$

For material with a relatively young age compared to the half-lives involved, λ_Pt << 1 and λ_Dt << 1, the propagation factors for λ_P and R are unity, while the factor for λ_D is insignificantly small. Consequently, age determinations based on atom ratios are more sensitive to the parent half-life than to the daughter half-life. This is compatible with the fact that R(t) ≈ λ_Pt for small values of t, i.e. linear with λ_P and independent of λ_D.

Only for old material, |λ_D − λ_P|t >> 1, does the uncertainty propagation of the daughter half-life become important, but under these conditions, the over-all accuracy of the method is relatively poor as the propagation factors increase almost exponentially with λ_Dt.

Similar equations can be derived for cases in which the activity ratio R_A of parent and daughter nuclide is measured instead of the atom ratio, with a subtle change in a multiplication factor

$$\begin{eqnarray}&&\text{Age}=\frac{1}{{{\lambda}_{\text{P}}}-{{\lambda}_{\text{D}}}}\ln \left(1-\frac{{{A}_{\text{D}}}(t)}{{{A}_{\text{P}}}(t)}\frac{{{\lambda}_{\text{D}}}-{{\lambda}_{\text{P}}}}{{{\lambda}_{\text{D}}}}\right)\end{eqnarray} \tag{ 13 }$$

The uncertainty propagation of the concentration ratio R_A(t) and the half-lives λ_P and λ_D to the age is given by [29]

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{\left(\frac{{{\sigma}_{t}}}{t}\right)}^{2}}&={{\left(\frac{{{\lambda}_{\text{P}}}}{{{\lambda}_{\text{P}}}-{{\lambda}_{\text{D}}}}\left(\frac{T}{t}-1\right)\right)}^{2}}{{\left(\frac{{{\sigma}_{{{\lambda}_{\text{P}}}}}}{{{\lambda}_{\text{P}}}}\right)}^{2}} \\ &+{{\left(\frac{-{{\lambda}_{\text{P}}}}{{{\lambda}_{\text{P}}}-{{\lambda}_{\text{D}}}}\left(\frac{T}{t}-\frac{{{\lambda}_{\text{D}}}}{{{\lambda}_{\text{P}}}}\right)\right)}^{2}}{{\left(\frac{{{\sigma}_{{{\lambda}_{\text{D}}}}}}{{{\lambda}_{\text{D}}}}\right)}^{2}} \\ &+{{\left(\frac{T}{t}\right)}^{2}}{{\left(\frac{{{\sigma}_{{{R}_{\text{A}}}}}}{{{R}_{\text{A}}}}\right)}^{2}} \\ \end{array}\end{eqnarray} \tag{ 14 }$$

For material with a young age compared to the half-lives involved, λ_Pt << 1 and λ_Dt << 1, the propagation factors for λ_D and R_A are unity, while the factor for λ_P is insignificantly small. Consequently, age determinations based on activity ratios are more sensitive to the daughter half-life than to the parent half-life, which is the opposite of the effect noticed for atom ratios (equation (11)).

Activity ratio measurements enhance the signal of the short-lived nuclide, and may be a good alternative to atom ratio measurements in cases where the atom concentration of the short-lived nuclide is particularly low. The relative uncertainty of the half-lives of the parent (via R) or the daughter nuclide (via R_A) is of equal importance as the relative uncertainty of the measured ratio (R or R_A). In order to achieve a time resolution of 50 d (k = 1) over a period of 1–50 years, the half-life uncertainties need to be lower than 0.25%, which has not yet been achieved for several important nuclides.

The principles of radiometric dating can also be applied to fission products created in a nuclear explosion. Radionuclides may attach to aerosols or be released as noble gases and get collected in air samplers at a remote location. One can distinguish isobaric and non-isobaric clocks [30]. Non-isobaric clocks start from theoretical (cumulative) fission yields for the calculation of initial activity ratios and use the current activity ratio of fission products with different half-lives to estimate the time elapsed since the nuclear event. Isobaric clocks are based on parent–daughter pairs of which the daughter nuclide is not directly produced in the fission reaction. The equations are the same as the equations (9)–(14) applied in nuclear forensics. For example, they are directly applicable to the ¹⁴⁰Ba-¹⁴⁰La clock [31], based on progeny of the short-lived noble gas ¹⁴⁰Xe (13.6 s). The aerosol-bound ⁹⁵Zr-⁹⁵Nb chronometer is another important clock that requires a more elaborate mathematical treatment due the presence of meta-stable ^95mNb in the decay scheme [32]. Specific equations for time-zero and uncertainty calculation are available to perform bias-free ⁹⁵Zr-⁹⁵Nb dating of a nuclear event over a time range of more than a year [33].

Radionuclides introduced as nuclear medicine into the human body deliver a dose that depends on their effective half-life T_eff = ln2/λ_eff, a combination of radioactive decay and biological excretion:

$$\begin{eqnarray}&&{{\lambda}_{\text{eff}}}=\lambda +{{\lambda}_{\text{bio}}}\end{eqnarray} \tag{ 15 }$$

Starting from an initial activity A, the accumulated dose is proportional to the number of atoms that decay in the body:

$$\begin{eqnarray}&&\text{dose}\propto \underset{0}{\overset{T\to \infty}{\int}} A{{e}^{-{{\lambda}_{\text{eff}}}t}}=\underset{T\to \infty}{{\text{lim}}} \frac{A}{{{\lambda}_{\text{eff}}}}\left(1-{{e}^{-{{\lambda}_{\text{eff}}}T}}\right)\end{eqnarray} \tag{ 16 }$$

The relative uncertainty of the effective half-life propagates linearly to the dose. The biological half-life cannot be determined as precisely as the physical half-life, and the dominance of either rate differs from one radionuclide to another. For many long-lived nuclides, such as ³H, ¹⁴C, ²²Na, ³⁶Cl, ⁶⁰Co, ¹³⁷Cs and ²³⁵U, the much shorter biological half-life (10–70 d) is dominant [34]. The inverse is true for short-lived nuclides, such as ³²P, ⁵⁹Fe, ^99mTc, ^123,131I, ¹⁴⁰Ba and ¹⁹⁸Au. The latter are the only cases in which the dose calculation depends significantly on the accuracy of the physical half-life. The bone seekers ⁹⁰Sr, ²²⁶Ra and ²³⁹Pu have extremely long effective half-lives (18–197 a [34]), which reduces their uncertainty propagation towards the accumulated dose during a person's lifetime.

Short lifetimes of excited nuclear levels ranging between about 1 ps to several ns have been measured by the Recoil Distance Doppler-Shift (RDDS) method [35, 36] for a large variety of nuclei all over the nuclear chart. In this method, nuclei produced by a beam-induced nuclear reaction in a thin target are allowed to fly freely in vacuum over variable distances from the target to a stopper ('plunger'). Gamma-spectroscopy measurements are performed with a single off-axis HPGe detector, preferably at extreme forward or backward angles, or by a cluster of detectors arranged at the same polar angle to increase sensitivity. Gamma rays emitted in flight are distinguished from those emitted after the nucleus has come to rest in the stopper, due to their Doppler shift. The mean life of a given level (and the accumulated time delay from higher lying states that feed it) can be deduced from the relative intensities of the shifted and unshifted peaks at different target-to-stopper distances.

In principle all the effects which should be considered when applying the RDDS method were known right from the beginning when the method was developed, but from time to time some of these effects were underestimated or invalid assumptions were made. In some cases the reasons for failures could be identified later but for other cases the situation remained unclear. Typical sources of uncertainty mentioned in the review paper by Dewald et al [35] are:

• fit of decay curve: the decay curve comprises of a single exponential if only one excited level is involved, but usually a more complex level-feeding scheme needs to be solved with many free parameters, such as level lifetimes, branching ratios and initial populations. Assumptions have to be made for time behaviour of side-feeding, i.e. level feeding for which no discrete transitions are observed. Advanced code was developed to deal with the complexity of the non-linear fitting problem, but in many cases the solution is not unique. In general, the function to be fitted is not exactly known since the feeding states are not completely known.
• distances and solid angles: the standard method relies on absolute distances, incl. the position of the plunger. Solid angle effects show up in the shifted peak when the target–stopper distance becomes comparable to the stopper–detector distance. In this case, only the unshifted component is analysed or a fixed stopper and moving target is used instead.
• detector efficiencies: the shifted and unshifted peaks are registered with different detector efficiencies, depending also on the angle of observation with respect to the recoil velocity.
• relativistic effects: high recoil velocities affect the lifetime of the excited nuclear state (time dilation), the emission angle (aberration) and the solid angle (Lorentz boost).
• line shapes: knowledge of the line shapes of the shifted and unshifted peaks is needed when both are not well separated. The recoil velocity distribution is significantly broadened and asymmetric due to the finite target thickness. Moreover, the faster recoils reach the stopper in a shorter time and therefore contribute less to the area of the flight peak. Also during slowing-down time in the stopper, the ion velocity changes magnitude and direction, leading to a tailed γ-line shape that prevents determination of comparably small lifetimes.
• deorientation: the angular distribution of the γ-ray emissions becomes more and more isotropic with increasing flight time, due to a decay of the spin orientation during the relaxation period. Even though this effect has been known for a long time, it was often underestimated, resulting in wrong half-lives. The error may reach 20% when using only the unshifted peak or 6% when using both.

These problems stimulated efforts to improve the technique with respect to higher reliability and to make it more transparent so that the presence of possible systematic errors is revealed more easily. Some of these improvements are [35, 36]:

• γ–γ coincidence measurements: by gating on specific feeding transitions, the problem of side-feeding and unknown feeding intensities and times can be circumvented. The drawback is the need for longer beam time and many Ge-detectors. Gating as a means of reducing background is beneficial to the statistical precision only if it improves the peak-to-background ratio by a large factor.
• the differential decay curve method and new alternatives: powerful new analysis techniques help solve problems of the standard method, relying more on experimentally accessible data, requiring only relative flight times/target-to-stopper distances and taking into account the angular and velocity spread of recoiling nuclei due to energy loss and multiple scattering in the target.

Nano- and microsecond nuclear states are mostly measured electronically in delayed coincidence experiments or alternatively by time interval analysis.

3.2.1. Delayed coincidence counting.

The radiation yielding the state of interest is detected to provide a start pulse for an electronic clock, and the radiation by which the state decays is detected in the same or another device to provide a stop pulse. The half-life is derived from the analysis of the time difference between both signals. Using ultrafast scintillators (plastic scintillator, LaBr₃(Ce), BaF₂), the applicability range of the method has been extended down to picoseconds [37]. Also states of tens of milliseconds (or even seconds) are within reach, provided that the count rate is sufficiently low. Conventional experimental set-ups contain electronic modules, such as delay line amplifiers, timing single channel analysers (TSCA), time-to-amplitude converters (TAC) and multi-channel analysers (ADC), but they can also be replaced by programmable processors (FPGA) or software performing off-line data analysis of digitally acquired data saved in list files [38].

Common features of the delayed coincidence method are that time differences are stored in a spectrum, that the decay constant is derived from the slope of an exponential function fitted to a part of the spectrum and that account has to be taken for data not due to the intended parent–daughter pairs. Random coincidences may be recorded due to background signals, decays from other states or closely spaced but unrelated parent and daughter events. The first two interferences can be significantly reduced if the experiment allows for energy selectivity, e.g. by selecting peaks in γ- or α-spectra, and the third by keeping the count rate well below the decay constant. By means of energy selection, multiple half-lives can be derived from one experiment.

There exist different variants of how the data are recorded, each requiring a specific mathematical basis for the functional shape of the time spectrum. Two options are discussed below, based on different manners to collect and interpret time differences.

• Single delayed coincidence

In a classical set-up, a timer is started whenever a parent decay is detected and stopped when a daughter decay is detected. Then the system is made sensitive again for parent decays. In doing so, all the parent events arriving before the first daughter event are ignored. In this type of experiment, the time spectrum theoretically takes the following shape: [39]

$$\begin{eqnarray}&&N(t)\cdot \text{d}t=A\cdot {{\text{e}}^{-\left({{R}_{2}}+\lambda \right)t}}\text{d}t+B\cdot {{\text{e}}^{-{{R}_{2}}t}}\text{d}t\end{eqnarray} \tag{ 17 }$$

in which A and B are multiplication factors related to efficiency and count rate, λ is the decay constant of the state of interest and R₂ is the count rate of the daughter decay. If the count rate R₂ is kept small (R₂ << λ), this formula can be approximated by:

$$\begin{eqnarray}&&N(t)\cdot \text{d}t=\left[A\cdot {{\text{e}}^{-\lambda t}}+B\right]\cdot \text{d}t\end{eqnarray} \tag{ 18 }$$

which is commonly used for the determination of a half-life from experimental data.

• Multiple delayed coincidences

If the activity of the source is high − i.e. R₂ is not negligible compared to λ − the true parent–daughter decays are being interfered by uncorrelated events, leading to a distortion of the single-coincidence time spectrum. In that case, systematic errors are to be expected from using equation (18). This can be solved by means of an alternative time analysing circuit in which a parent decay starts the timer and not a single but multiple delayed coincidences are added to the time spectrum, one time value for every recorded daughter decay over a long period of time [40]. Theoretical analysis [39] predicts that time spectra obtained by this approach are represented by the sum of a constant term and an exponential function with the time constant of the radioactive decay of the daughter, exactly as in equation (18). Using multiple delayed coincidences eliminates the spectral distortion effect, but at the price of an increased number of random coincidences.

3.2.2. Time-interval distribution analysis method.

Time-interval distribution analysis is a well-established method for measuring the activity of radiation sources [41–43]. For a simple decay, the probability density of the time interval between two successive disintegrations is given by the formula:

$$\begin{eqnarray}&&P(t)\text{d}t=\lambda N{{\text{e}}^{-\lambda Nt}}\text{d}t\end{eqnarray} \tag{ 19 }$$

in which λN denotes the activity of the source. This formula can also be applied to a good approximation in short time intervals for a mixture of long-lived radionuclides − decaying independently or in radioactive equilibrium in a decay chain − but is incorrect if the decay chain also contains short-lived members. Lindeman and Rosen [42] deduced the general formula for the time-interval distribution of decays from a long-lived parent in equilibrium with n − 1 short-lived daughters. For a large number of parent atoms N₁, the probability of a given time interval between two successive disintegrations is given by:

$$\begin{eqnarray}&&P(t)=\frac{1}{n}{{\text{e}}^{-{{\lambda}_{1}}{{N}_{1}}t-{\sum\limits_i} {{N}_{i}}\left(1-{{e}^{-{{\lambda}_{i}}t}}\right)}}\left[{{\lambda}_{1}}{{N}_{1}}{{\left(1+\underset{i}{\sum} {{\text{e}}^{-{{\lambda}_{i}}t}}\right)}^{2}}+\underset{i}{\sum} {{\lambda}_{i}}{{\text{e}}^{-{{\lambda}_{i}}t}}\right]\end{eqnarray} \tag{ 20 }$$

It shows that, at low time intervals, the shape of the curve has a term that depends not on the activity of the daughter nuclide, but on its half-life. The time-interval spectrum is obtained by binning the time differences between all events. It is possible to focus on one of the transitions separately by selecting regions in the energy spectrum mainly belonging to the parent-daughter sequence of interest. The resulting curves have the characteristics of equation (20) applied to a single daughter nuclide (i = 2), but need the inclusion of a weighting factor C to balance both parts of the curve:

$$\begin{eqnarray}&&P(t)=\frac{1}{2}{{\text{e}}^{-{{\lambda}_{1}}{{N}_{1}}t-{{N}_{2}}\left(1-{{\text{e}}^{-{{\lambda}_{2}}t}}\right)}}\left[{{\lambda}_{1}}{{N}_{1}}{{\left(1+{{\text{e}}^{-{{\lambda}_{2}}t}}\right)}^{2}}+C{{\lambda}_{2}}{{\text{e}}^{-{{\lambda}_{2}}t}}\right]\end{eqnarray} \tag{ 21 }$$

in which C is included as a free parameter in the fit. For a source in equilibrium, the number of short-lived daughter atoms N₂ = N₁ λ₁/λ₂ will be low compared to N₁ and thus the second term in the first exponential is small. Time-interval distribution analysis yields comparable results to the delayed coincidence method [38, 43].

3.2.3. Data analysis.

The half-life of the nuclide is usually derived from a least-squares fit of an exponential function to the measured time spectra. However, some complications have to be taken into account. Least squares fitting procedures imply the assignment of proper weighting factors to the stochastically distributed data involved. In the case of a Poisson distribution, the obvious choice of setting the weighting factor equal to the inverse of the measured value is prone to bias towards low values. Alternatively, using the inverse of the fitted value may turn out to be biased towards higher values, depending on the procedure followed. Additional problems arise when also the possibility of zero counts has to be taken into account. An overview of the problems and possible solutions can be found in [44, 45] and references therein.

In fact, it is possible to perform unbiased least-squares fitting with Pearson's chi-square i.e. using the fitted value Y as weighting factor,

$$\begin{eqnarray}&&\chi _{\text{P}}^{2}=\underset{i=1}{\overset{n}{\sum}} \frac{{{\left({{y}_{i}}-Y\right)}^{2}}}{Y}\end{eqnarray} \tag{ 22 }$$

on condition that the weighting factor is kept fixed during each iteration of the fit. After each optimisation, it is set equal to the fit value Y obtained in the last iteration and a new fit is performed until convergence is reached [45]. There exist alternative estimators that allow using existing software for least-squares in a less biased way, with a minimal adaptation to the procedure. For example, one can minimise [45]

$$\begin{eqnarray}&&{{T}^{2}}=\underset{i=1}{\overset{n}{\sum}} \frac{{{\left({{y}_{i}}-Y-0.5\right)}^{2}}}{Y}\end{eqnarray} \tag{ 23 }$$

freely without the need for an iterative procedure, and obtain an unbiased result if the data are not too close to zero.

A recommended solution to the problem of bias is to replace the χ² estimator by a maximum likelihood ratio statistic (MLE) for Poisson distributions [44]:

$$\begin{eqnarray}&&\chi _{\lambda}^{2}=2\underset{i=1}{\overset{n}{\sum}} \left[Y-{{y}_{i}}-{{y}_{i}}\ln \left(\frac{Y}{{{y}_{i}}}\right)\right]\end{eqnarray} \tag{ 24 }$$

in which y_i is the observed number of events in time bin i, Y is the fitted value and the last term is set equal to zero for y_i = 0.

To decay curves with a (quasi) purely exponential shape, another excellent method to derive an unbiased value for the decay constant is applying 'moment analysis'. It is known that the 'first moment' (i.e. the expectation value for the decay time t) and the 'second central moment' of the exponential distribution are equal to λ⁻¹ and λ⁻², respectively. If the moment analysis of the decay curve is restricted to a time interval (t₁,t₂), the following relationship can be derived between the half-life and the first moment [45]:

$$\begin{eqnarray}&&{{T}_{1/2}}=\ln (2)\left({{\mu}_{1}}\left({{t}_{1}},{{t}_{2}}\right)-{{t}_{1}}+{{C}_{1}}\right)\end{eqnarray} \tag{ 25 }$$

and with the second central moment [45]:

$$\begin{eqnarray}&&{{T}_{1/2}}=\ln (2){{\left(\mu _{2}^{'}\left({{t}_{1}},{{t}_{2}}\right)+{{C}_{2}}\right)}^{1/2}}\end{eqnarray} \tag{ 26 }$$

in which the correction factors C_i are calculated from:

$$\begin{eqnarray}&&{{C}_{i}}=\frac{{{\left({{t}_{2}}-{{t}_{1}}\right)}^{i}}{{e}^{-\lambda \left({{t}_{2}}-{{t}_{1}}\right)}}}{{{\left(1-{{e}^{-\lambda \left({{t}_{2}}-{{t}_{1}}\right)}}\right)}^{i}}}i=1,2\end{eqnarray} \tag{ 27 }$$

The statistical uncertainty associated with equation (25) is:

$$\begin{eqnarray}&&\frac{\sigma \left({{T}_{1/2}}\right)}{{{T}_{1/2}}}=\frac{\sigma (\lambda )}{\lambda}=1/\sqrt{{{N}_{\left({{t}_{1}},{{t}_{2}}\right)}}\left(1-{{\lambda}^{2}}{{C}_{2}}\right)}\end{eqnarray} \tag{ 28 }$$

in which ${{N}_{\left({{t}_{1}},{{t}_{2}}\right)}}$ represents the number of events observed within the time interval (t₁,t₂). The relative uncertainty associated with T_1/2 of equation (26) is about a factor √2 higher, i.e. proportional to $\sqrt{2/{{N}_{\left({{t}_{1}},{{t}_{2}}\right)}}}$. Equation (26) confirms the observation by others [46, 47] that the best attainable precision on a decay constant is equal to 1/√N.

The first moment, which is not a central moment, is sensitive to the time value assigned to each time bin. The data being distributed into time bins of equal duration Δt, the weighted mean time value of the first time bin, going from t = 0 to Δt, is calculated from:

$$\begin{eqnarray}&&\langle t\rangle =\frac{1}{\lambda}+\frac{\Delta t}{1-{{e}^{\lambda \Delta t}}}\end{eqnarray} \tag{ 29 }$$

The time values for the other bins are found by adding to equation (29) a multiple of Δt. Central moments are insensitive to a linear translation of the time origin. Moment analysis should be performed only in a region (t₁,t₂) of the spectrum in which the random coincidence part (B in equation (18)) or contributions of other nuclides with a significantly shorter or longer half-life are negligible.

3.2.4. Uncertainties.

Many radionuclides with practical implications and applications have half-lives varying between seconds and about 100 years. This is a range of half-lives that can be measured directly by repeated activity measurements of a source. The simplicity of the measurement principle has incited many authors to publish thus obtained half-life data, unsuspecting of hidden processes that inflate the measurement errors far beyond their uncertainty estimates. A common scenario is that an exponential decay curve is fitted to various activity values measured as a function of time and that the uncertainty on the decay constant is obtained from the least-squares minimisation algorithm. This procedure is often faulty [48], as real measurement data may deviate in a subtle but systematic way from the theoretically assumed decay curve. The fitted decay constant is the value leading to the smallest residuals, which explains why an erroneous result seldom raises suspicion in the mind of the experimenter [14, 15]. Also other methods can be applied to extract the half-life from the data, such as moment analysis (section 3.2.3) and statistical sampling [14, 49], but they are equally sensitive to systematic errors.

The sum of an exponential function superposed on a constant background is often assumed to model the temporal dependence of the measured activity:

$$\begin{eqnarray}&&\hat{A}(t)={{A}_{0}}\cdot {{\text{e}}^{-\lambda t}}+B\end{eqnarray} \tag{ 30 }$$

The fitted value of the parameter λ is believed to be the most probably value of the decay constant and the uncertainty is derived from the covariance matrix provided by the fitting algorithm, which implicitly assumes that the measurement data rigorously follow the prescribed decay function, the residuals being purely stochastic. In reality, there is a less than perfectly linear relationship between the activity and the measured signal (count rate or current) in a detector. Various processes make measurement data deviate from the ideal decay curve, long-term instabilities being at the same time the most influential but least visible sources of error. Due to differences in uncertainty propagation, it is convenient to subdivide these process according to the frequency at which they occur [14, 15]:

• High-frequency deviations: these sources of uncertainties occur at a rate that is higher than or comparable to the measurement duration of one data point. A typical example would be counting statistics: Poisson processes are characterised by an exponential distribution of the interval times between successive events. By extending the measurement, one improves the statistical uncertainty on the actual count rate. Also ultra-high frequency instabilities, such as e.g. electronic noise, are included in this category. They are partly cancelled, though, by the 'integrating' effect of the duration of the measurement. Normal statistical treatments (including least squares fitting) apply because of the preservation of randomness of the data.
• Medium-frequency deviations: these instabilities show up (at least partly) as a trend in the residuals and can also be identified by means of an autocorrelation plot [14]. They include the so-called 'seasonal effects' (e.g. day-night effects or changes with the seasons of the year), but also other variations occurring at a rate that can be observed over the measurement campaign. Their effect on the fitted half-life is greatly underestimated if treated as just a random residual. Moreover, a fit tends to minimise the residuals and partly covers up the true size of the medium frequency effects. Typical examples are the activity interference by radioactive impurities, reproducibility of the source-detector geometry, noticeable changes in the detector efficiency (e.g. through temperature, pressure, humidity, electronic flaws, scale changes in an electrometer) and modelling errors (e.g. neglecting a decay correction during measurement when progressively adjusting the measurement duration with time, inadequate separation between photopeak and underlying Compton continuum by gamma-ray spectrometry).
• Low-frequency deviations: these sources of uncertainties occur at a rate that is lower than or comparable to the duration of the whole measurement campaign. They remain practically invisible in the residuals, as the fit will compensate for this trend, hence erase it erroneously. Common problems in this range are a non-linear detector response to activity (charge recombination in ionisation chamber, discharge of a capacitor, non-linearity in an electrometer, pile-up and dead time in counter), systematic errors in the background subtraction, under- or over-compensation of count loss by live-time systems (double count loss through summing-out of piled-up events, hidden dead time due to unexpected behaviour of detector and electronic modules, pulse undershoot and overshoot, cascade effects of pile-up and dead time, effect of variation of dead time during measurement of short-lived nuclides), long-term drift of the counting efficiency and source degradation (e.g. oxidation of the source, mechanical wear, precipitation of active material in a solution).

In figure 1, hypothetical residuals of a fitted decay curve to data with high, medium and low frequency instabilities are shown. The dark dots represent the deviations of the data from the 'true' decay curve, the light circles show the deviations of the data from the fitted decay curve and the solid curve is the difference between the fitted and true decay curve. The observer that performs the fit perceives only the light points. Considering that a slope equal to zero corresponds to the true half-life, the medium and low frequency deviations in figure 1 lead to a biased result. As the residuals give insufficient indication of these errors, one has to evaluate an exhaustive list of potential uncertainty components and combine them in a squared sum in the uncertainty budget.

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In a simplified scheme, the decay constant in equation (30) can be determined from two activity measurements of a source: A₀ at t = 0 and A at time t. Generally, corrections have to be made to transform the measured signals  into the activity A of the radionuclide, which can be represented by a subtraction of a value B (e.g. background signals) and multiplication by a factor f (e.g. dead-time correction). The estimated value of the decay constant is

$$\begin{eqnarray}\begin{array}{*{35}{l}} \hat{\lambda}&=\frac{1}{{\hat{t}}}\ln \left(\frac{{{{\hat{A}}}_{0}}}{{\hat{A}}}\right) \\ &=\frac{1}{{\hat{t}}}\left[\ln \left(\frac{{{A}_{0}}{{f}_{0}}+B-{{{\hat{B}}}_{0}}}{Af+B-\hat{B}}\right)\right] \\ &\approx \frac{1}{t}\left[\ln \left(\frac{{{A}_{0}}}{A}\right)+\ln \left(\frac{{{f}_{0}}}{f}\right)+\ln \left(1+\frac{\Delta B}{A}\right)\right] \\ \end{array}\end{eqnarray} \tag{ 31 }$$

Errors ΔB and Δf in the corrections propagate into an error in the decay constant:

$$\begin{eqnarray}&&\frac{\hat{\lambda}-\lambda}{\lambda}\approx \frac{1}{\lambda t}\left[{{\left(\frac{\Delta f}{f}\right)}_{0}}-\left(\frac{\Delta f}{f}\right)+\left(\frac{\Delta B}{A}\right)-{{\left(\frac{\Delta B}{A}\right)}_{0}}\right]\end{eqnarray} \tag{ 32 }$$

Some conclusions can be derived from equation (32):

• Assuming the errors to be constant, the uncertainty on the half-life is inversely proportional with time via the factor 1/λt. Consequently, in order to reduce the uncertainty by a factor of two, one needs to double the decay time. This entails a linear relationship on a log-log scale, therefore less and less is gained by continuing the experiment at great length. Moreover, the gain in accuracy can be counteracted by a long-term increase of ΔB/A and Δf/f with time.
• A systematic relative error Δf/f = (Δf/f)₀ does not alter the half-life result; e.g. there is no need to determine the detection efficiency of the detector, as long as it is constant in time. On the other hand, a systematic error in the characteristic dead time leads to non-constant relative error in the count-loss correction factor. Some authors, even in recent publications, make the error of emphasising efficiency calibration over linearity checks.
• The impact of systematic errors ΔB in background (or impurity) subtraction are dependent on their size relative to the net signal A. Energy-selectivity is a useful tool to separate the source activity from interfering signals and to be insensitive to variations in threshold settings and instrumental noise. Therefore it is easier to measure half-lives through (high-energy) gamma rays and alpha particles than through (low-energy) x-ray, electron and beta emissions. The latter decay types are more sensitive to medium-frequency instabilities, which explains the speculations about the non-constancy of half-lives involved in, e.g. electron capture decay.
• The ideal time interval for a measurement campaign is typically chosen in function of non-linearity effects at the start (e.g. dead-time, short-lived impurities) and poor signal quality at the end (e.g. counting statistics, background subtraction, long-lived impurities, instability).

The uncertainty propagation formula for a half-life T_1/2 = T ln2/lnR determined from the activity ratio R = A/A₀ at time t = 0 and t = T is

$$\begin{eqnarray}&&\frac{\sigma \left({{T}_{1/2}}\right)}{{{T}_{1/2}}}=\frac{1}{\ln R}\frac{\sigma (R)}{R}=\frac{1}{\lambda T}\sqrt{\frac{{{\sigma}^{2}}\left({{A}_{0}}\right)}{A_{0}^{2}}+\frac{{{\sigma}^{2}}(A)}{{{A}^{2}}}}\end{eqnarray} \tag{ 33 }$$

In practice, measurements are repeated at regular time intervals during a measurement campaign over a time interval [0,T], with the intention to reduce the uncertainty on the half-life. Assuming a hypothetical situation in which the relative uncertainty on the activity measurement σ(A)/A is identical for each data point, the uncertainty propagation can be calculated from [50]:

$$\begin{eqnarray}&&\frac{{{\sigma}_{{{T}_{1/2}}}}}{{{T}_{1/2}}}\approx \frac{2}{\lambda T}\sqrt{\frac{3(n-1)}{n(n+1)}}\frac{{{\sigma}_{A}}}{A}\end{eqnarray} \tag{ 34 }$$

The formula implicitly assumes that the data are independent; therefore, it is applicable only to statistical (short-term) variations and not to auto-correlated (medium and long-term) variations. When performing three measurements, the middle data point neither affects the half-life nor its uncertainty. It makes a fitted decay curve shift as a whole in the vertical direction, but does not alter its slope.

A similar approximating formula was derived independently [14, 15] from the assumption that the relative impact of a measurement on the fitted half-life is proportional to the time difference with the middle of the campaign, i.e. ~│t − T/2│/(T/2). Under the aforementioned conditions, and data points being spread somewhat evenly in time over the period T, the uncertainty on T_1/2 is approximated by:

$$\begin{eqnarray}&&\frac{\sigma \left({{T}_{1/2}}\right)}{{{T}_{1/2}}}\approx \frac{2}{\lambda T}{{\left(\underset{i=1}{\overset{n}{\sum}} \left|\frac{2{{t}_{i}}}{T}-1\right|\right)}^{-1/2}}\frac{\sigma (A)}{A}\end{eqnarray} \tag{ 35 }$$

and the reduction factor (for n > 2) is roughly equal to

$$\begin{eqnarray}&&\sqrt{\underset{i=1}{\overset{n}{\sum}} \left|\frac{2{{t}_{i}}}{T}-1\right|}=\sqrt{\underset{i=0}{\overset{n-1}{\sum}} \left|\frac{2i}{n-1}-1\right|}\approx \sqrt{\frac{n+1}{2}}\end{eqnarray} \tag{ 36 }$$

resulting in

$$\begin{eqnarray}&&\frac{\sigma \left({{T}_{1/2}}\right)}{{{T}_{1/2}}}\approx \frac{2}{\lambda T}\sqrt{\frac{2}{n+1}}\frac{\sigma (A)}{A}\end{eqnarray} \tag{ 37 }$$

Using equations (34) or (37), an evaluator as well as an experimenter can make a quick estimate of the contribution of, e.g. counting statistics to the half-life if three variables are available: the number of activity measurements, n, the duration of the campaign, T, and a measure of the uncertainty for a typical activity measurement, σ(A). Equation (34) is probably more rigorous for large n, but the difference is less than 20% and significant only in the rare case that high-frequency uncertainty components are dominant. If an uncertainty component σ(A)/A is not constant with time, an average is taken of its value at time t = 0 and t = T.

It is important to apply independent uncertainty propagation according to the 'frequency' of the uncertainty components. When pertaining to medium and low-frequency instabilities, the parameter n in equations (34) and (37) receives a new meaning. For (cyclic) medium-frequency components, the value of n is set equal to the number of periods covered by the measurement campaign. In the case of, e.g. yearly repeating effects, n would correspond to the number of years covered. For low-frequency effects, the value n = 1 is always selected and the propagation factor is 2/λT. For example, a hypothetical 1% change in detection efficiency during the measurement campaign leads to an average uncertainty at t = 0 and t = T of < σ(A)/A > = (0% + 1%)/2 = 0.5% and is propagated to an uncertainty of σ(T_1/2)/T_1/2 = 1%/ λT, in agreement with equation (33).

Whereas two activity measurements in principle suffice for a half-life determination, repeated measurements have the advantage of reducing the propagation of random uncertainties, as well as allowing quantification of medium frequency components if they are not insignificant compared to the random variations. For this purpose, it is important to measure with good statistical precision. As time progresses, the main weight of the uncertainty budget typically shifts from random variations to cyclic instabilities and systematic errors [14]. The latter are the most difficult to quantify, so the most precise half-lives have the most uncertain confidence intervals. Also the analysis method itself may contribute to unexpected errors, as already discussed in section 3.2.3.

4.4.1. Reproducibility in ionisation chamber (IC).

4.4.2. Impurity and background correction.

An ionisation chamber is used to measure a source that is not perfectly mono-nuclidic due to the presence of an impurity, possibly a radionuclide occurring in the same decay chain. Gamma-ray spectrometry may be used to determine the relative activity of both nuclides and to calculate the impurity signal in the IC to be subtracted. Errors in the γ-ray emission probabilities and response of the IC and spectrometer can influence the half-life result. As an alternative, the signals of both nuclides can be fitted simultaneously to the decay curve. Another possible source of error is neglecting the decay rate change during measurements of variable duration, which requires a different correction factor ${{C}_{i}}\left({{t}_{\text{m}}}\right) = \left(1-{{e}^{-{{\lambda}_{i}}{{t}_{\text{m}}}}}\right)/{{\lambda}_{i}}{{t}_{\text{m}}}$ (see also section 2.1) for the fractions of the signal associated with both decay constants. The following function can be fitted to the measured data (average current over a measurement time interval [55]):

$$\begin{eqnarray}&&I\left(t,t+{{t}_{\text{m}}}\right)=a{{C}_{1}}\left({{t}_{\text{m}}}\right){{\text{e}}^{-{{\lambda}_{1}}t}}+b{{C}_{2}}\left({{t}_{\text{m}}}\right){{\text{e}}^{-{{\lambda}_{2}}t}}+B\end{eqnarray} \tag{ 38 }$$

in which a and b are constants and B represents the background signal. For measurements of constant duration t_m in real time, the factors C_i are invariable and can be integrated in the factors a and b. Assuming that these corrections have been taken into account, the activity ratio at two different moments in time t = t₁ and t = t₂ is calculated from:

$$\begin{eqnarray}&&R=\frac{\hat{A}\left({{t}_{1}}\right)}{\hat{A}\left({{t}_{2}}\right)}=\frac{I\left({{t}_{1}}\right)-\hat{b}{{e}^{-{{{\hat{\lambda}}}_{2}}{{t}_{1}}}}-\hat{B}}{I\left({{t}_{2}}\right)-\hat{b}{{e}^{-{{{\hat{\lambda}}}_{2}}{{t}_{2}}}}-\hat{B}}\end{eqnarray} \tag{ 39 }$$

in which I(t) is measured IC current, $\hat{b}$ is a fitted or calculated impurity signal and $\hat{B}$ is the measured or fitted background signal. The most straightforward way to investigate the uncertainty on the half-life T_1/2 = (t₂ − t₁)ln2/lnR due to potential errors on background and impurity subtraction is to compare the result of hypothetical half-life calculations using equation (38) with different values of $\hat{B}$, $\hat{b}$ and ${{\hat{\lambda}}_{2}}$. Alternatively, one can calculate the uncertainty from a first order series expansion [55] and equation (33):

$$\begin{eqnarray}&&\frac{{{\sigma}_{R}}}{R}\approx \frac{1+\left(\frac{{{\sigma}_{b}}}{a}+\frac{b}{a}{{\sigma}_{{{\lambda}_{2}}}}{{t}_{1}}\right){{\text{e}}^{-\left({{\lambda}_{2}}-{{\lambda}_{1}}\right){{t}_{1}}}}+\frac{{{\sigma}_{B}}}{a}{{\text{e}}^{{{\lambda}_{1}}{{t}_{1}}}}}{1+\left(\frac{{{\sigma}_{b}}}{a}+\frac{b}{a}{{\sigma}_{{{\lambda}_{2}}}}{{t}_{2}}\right){{\text{e}}^{-\left({{\lambda}_{2}}-{{\lambda}_{1}}\right){{t}_{2}}}}+\frac{{{\sigma}_{B}}}{a}{{\text{e}}^{{{\lambda}_{1}}{{t}_{2}}}}}-1\end{eqnarray} \tag{ 40 }$$

The fitted values of background $\hat{B}$ and impurity signal $\hat{b}$ may be anti-correlated as they compete in describing the part of the decay curve not following the exponential decay of the main nuclide. This should be taken into account when combining the different uncertainty components. It is important to take into account that the number of background counts may show statistical fluctuations higher than expected from a Poisson process. It is good practice to include a fraction of $\hat{B}$ also as a hypothetical systematic uncertainty component, e.g. because the counting conditions for a blank and a source may differ.

The above reasoning and equations (38)–(40) are also formally applicable to the case of a parent and daughter half-life being derived simultaneously from a fit to an ingrowth-and-decay curve obtained with an IC or calorimeter. The resulting half-lives can be highly correlated and their uncertainty underestimated. For example, this could be the case with ²²⁷Th where applying a fixed value for the half-life of the ²²³Ra daughter has led to a significant but questionable reduction of the uncertainty estimate (see discussion in [56]).

4.4.3. Counting with dead time and pileup.

The activity is registered as a pulse counting process, as opposed to current measurements in non-selective methods as ionisation chambers and calorimeters. The weakness of counting methods lies in the finite time resolution of the detector and associated pulse processing electronics [57, 58]. The linearity of the detector response is affected by count loss due to pulse pileup and system dead time. This prohibits the use of high amounts of activity at the start and therefore shortens the duration of the measurement campaign, since it is also limited at the end by poor counting statistics and significant background subtraction.

Live-timing techniques can compensate for count loss, by accounting for the characteristic dead time τ associated with each recorded event. If the dead time is of the extending type, the activity ratio R of the activities A₁ and A₂ at a time t₁ and t₂, respectively, is obtained from the measured count rates r₁ and r₂ and an exponential correction factor [57–59] to estimate the true input rates ρ₁ and ρ₂:

$$\begin{eqnarray}&&R=\frac{{{\rho}_{1}}}{{{\rho}_{2}}}=\frac{{{r}_{1}}}{{{r}_{2}}}{{e}^{\left({{\rho}_{1}}-{{\rho}_{2}}\right)\tau}}\end{eqnarray} \tag{ 41 }$$

The relative error on R produced by applying a slightly incorrect characteristic dead time τ + Δ is

$$\begin{eqnarray}\begin{array}{*{35}{l}} \frac{{{\sigma}_{R}}}{R}&={{e}^{\left({{\rho}_{1}}-{{\rho}_{2}}\right)\Delta}}-1\approx \left({{\rho}_{1}}-{{\rho}_{2}}\right)\Delta \\ &\approx \frac{\Delta}{\tau}\left\{\ln {{\left[\frac{{{T}_{\text{R}}}}{{{T}_{\text{L}}}}\right]}_{1}}-\ln {{\left[\frac{{{T}_{\text{R}}}}{{{T}_{\text{L}}}}\right]}_{2}}\right\} \\ \end{array}\end{eqnarray} \tag{ 42 }$$

in which T_R and T_L represent the system real time and live time, respectively, and the ratio of both corresponds to the dead time correction factor used in the respective measurements. The error grows exponentially with the count rate ρ₁, which overrules the 1/λT reduction factor in the uncertainty propagation to T_1/2 (via equation (33)):

$$\begin{eqnarray}&&\frac{{{\sigma}_{{{T}_{1/2}}}}}{{{T}_{1/2}}}\approx \left({{\rho}_{2}}\tau \right)\left(\frac{\Delta}{\tau}\right)\frac{{{e}^{\lambda T}}-1}{\lambda T}\end{eqnarray} \tag{ 43 }$$

For a fixed end point of the measurement campaign with count rate ρ₂, the uncertainty due to extending dead time or pileup increases by a factor of 1.5 or more for every half-life that the measurement campaign is started earlier. Therefore, there is no gain in accuracy by increasing the initial activity if dead time or pileup is the main uncertainty component.

In the case of non-extending dead time, the uncertainty on the activity ratio is

$$\begin{eqnarray}&&\frac{\sigma (R)}{R}=\frac{1+{{\rho}_{1}}\Delta /\left(1+{{\rho}_{1}}\tau \right)}{1+{{\rho}_{2}}\Delta /\left(1+{{\rho}_{2}}\tau \right)}-1\end{eqnarray} \tag{ 44 }$$

and propagates to T_1/2 via equation (33). Equations (42) and (43) apply by approximation only for moderate count rates, i.e. ρ₁τ < 0.2.

The influence of dead time on the activity measurement is even more complicated for radionuclides with a short half-life, such that the count rate and corresponding dead time varies during one measurement. Mathematical equations are available to convert the integrated count rate to a momentary value [57]. In the field of primary standardisation of activity, counting is usually performed with a single-channel system and an artificial dead time of selectable length and type (extending or non-extending) imposed on every counted event. However, this live-time technique may also suffer from residual cascade effects of pileup with the imposed dead time, a process which can be corrected for through confirmed theoretical models [60, 61].

4.4.4. Time interval analysis.

A formalism has been developed to replace histogram analysis of the decay curve by time interval analysis of valid events [62, 63]. The events are saved in list mode and subsequently analysed, first by assigning a fixed extendable dead time τ to each event and eliminating all events falling within a dead time period, and second by calculating a likelihood function for each surviving event i from a total of n:

$$\begin{eqnarray}&&L=\underset{i=1}{\overset{n}{\prod}} \rho \left({{t}_{i}}\right)\text{exp}\left(-\underset{{{t}_{i}}-{{T}_{\text{L}}}(i)}{\overset{{{t}_{i}}}{\int}} \rho \left(t'\right)\text{d}t'\right)\end{eqnarray} \tag{ 45 }$$

in which T_L(i) is the 'live time' that preceded the event and the count rate is made slightly variable with time (cf non-homogenous Poisson process). Rate-related count loss is mostly compensated for through the imposed extending dead time, but similar issues remain regarding time interval distortions and cascade effects with pulse-pileup [59–61], impurity and background subtraction, etc.

4.4.5. Spectrometric method.

4.4.6. Mass spectrometry.