Experimental determination of k ₀ , Q ₀ , $\bar{E}_{r}$ factors and neutron cross-sections for 41 isotopes of interest in Neutron Activation Analysis

Abstract

The k ₀ -literature has been reviewed every decade but some of its nuclear data is still more than 30 years old. Sometimes Q ₀ values were adopted from the nuclear data at that time or were experimentally determined by only 1 laboratory. Other isotopes were listed with accurate pairs of (k ₀ , Q ₀ ) values but were also quoted as candidates for redetermination for different reasons (i.e. imprecise cadmium transmission factors, half-lives). In this work we aim at the experimental re-determination of k ₀ and Q ₀ values for 41 isotopes of analytical interest while introducing a methodology for simultaneous $\bar{E}_{r}$ and Q ₀ determination employing N irradiation channels. In order to satisfy the metrological level required, up to 12 repeats per standard were irradiated in up to 4 irradiation channels of the Belgian Reactor 1 (BR1, SCK∙CEN) having a wide spread in neutron characteristics. Our relative percentile differences to the literature values were usually ≤10 % for Q ₀ factors, ≥25 % for $\bar{E}_{r}$ values and ≤4 % for k ₀ values. Our precision and accuracy are discussed thoroughly.

Introduction

The k ₀ -standardization of Neutron Activation Analysis (k ₀ -NAA) provides a user-friendly framework for the mapping of reactor channels and the multi-elemental quantification of up to 70 elements, without the need of extensive standard preparation. Its success depends on nuclear constants known as k ₀ and Q ₀ factors. These factors were experimentally determined mostly during the eighties, by the joint collaboration of 2 specialized laboratories and with the occasional participation of a third one [1–6]. Since then, the method has been successfully applied and extended worldwide but the experimental redetermination of these factors has been carried out for a few cases only. Some of its current literature was either adopted from nuclear data at that time or is correlated to fundamental parameters that have changed since then (half-lives, decay branching fractions). Among others cases, a few factors were reported by only 1 laboratory or were quoted as candidates for redetermination [5].

In this work we aimed at the experimental determination of k ₀ nuclear data for 41 radioisotopes of analytical interest through the cadmium subtraction technique employed over 3 irradiation channels of the BR1 reactor (SCK∙CEN), having considerable spread in neutron characteristics. Occasionally, a 4th channel having a pure thermal neutron spectrum (the Cavity) was employed. A method for simultaneous Q ₀ and $\bar{E}_{r}$ determination through N irradiation channels is proposed here (the α-vector method; N ≥ 2). The method was tested on 25 investigated isotopes and for N ≤ 3. We refer to [1–6] for a full description of the k ₀ -method. De Corte [7] contains a survey of the main aspects. The results from this work are compiled and provided as electronic supplementary materials (Online Resources).

Experimental

Isotopes under investigation and chosen convention

For all the determinations use was made of the standards shown in Table 1. These materials, mostly foils and wires, are especially suited because of the minimum preparation required, negligible chances of mass losses, homogeneity, stability and low uncertainty in the certified content. Furthermore, several of the diluted Al alloys or spiked paper filters can be regarded as free from strong neutron self-shielding effects during irradiations. The thermal (index t) and epithermal (index e) self-shielding correction factors (G _t and G _e ; both ≤1) were obtained through the MATSSF and the Sigmoid calculation methods [8–10].

Table 1 Materials employed in this work: provider, matrix, certified elemental content, average thickness (D) and their calculated self-shielding factors. The target (T) and formed isotopes (FI) and the reason for their investigation

The use of the much simpler Høgdahl convention [11] over the Westcott formalism [12] was chosen since from Holden [13] it can be seen that only the rhenium isotopes deviate slightly from the 1/v expected behavior. Their deviations are <0.8 % at 60 °C, while the temperature of our channels is estimated to be 30 °C.

Q ₀ determination

For Q ₀ determination, the standards (index s) were positioned within a 2 cm distance to the nearest comparator (index c) along the axis of a cylindrical polyethylene container (PE rabbit) following a sandwich pattern. Up to 10 samples per rabbit were irradiated on each opportunity. Table 2 summarizes their neutron characteristics, the typical irradiation times and rabbit sizes employed. These channel parameters are in agreement with our latest reported values [14]. From that work, the choice of thin plastic bags (0.05 mm thick) for packing of the samples, over typical PE vials (1 mm thick), was clarified from the statistical significance observed when comparing the calibration curves obtained under both scenarios.

Table 2 Neutron parameters (f, α) for the irradiation channels employed in this work (Y4, S84, X26 and the Cavity) and their confidence intervals (2s). Radius (r) and length (l) of these channels and irradiation containers (in mm; in that order) and typical irradiation times (t _irr ) applied

Replicate samples were positioned inside 1 mm thick cadmium covers (index Cd) and were irradiated on the same irradiation positions. These covers filter the thermal component from the total capture reaction rate. Hence, the ratio of the induced specific gamma activity (A) between a bare and its Cd-covered replicate is an indicator of the thermal-to-epithermal sensitivity attainable for a given isotope on an irradiation channel (cadmium ratio; R _Cd  = A/A _Cd ). In this work we define a normalized cadmium ratio (r _Cd ) in a threefold way as:

$$ r_{Cd} \equiv \frac{{G_{e} }}{{G_{t} }}\left( {R_{Cd} F_{Cd} - 1} \right) \equiv \frac{f}{{Q_{\alpha } }} \equiv \frac{{\varphi_{t} }}{{\varphi_{e} }}\frac{{\sigma_{0} }}{{I_{\alpha } }}, $$

(1)

where F _Cd is the cadmium transmission factor for the isotope of interest, f is the ratio between the thermal and the epithermal conventional neutron flux (φ; in n cm⁻² s⁻¹), σ ₀ is the thermal neutron cross-section and I _α is the resonance integral (both in barns). The subscript α corresponds to a channel-specific dimensionless parameter necessary for an accurate mapping of the ~1/E ^1+α distribution of epithermal neutrons observed in reactor channels. If α = 0, the r _Cd factor is just inversely proportional to the Q ₀ factor.

By writing eq. (1) for the comparator as well, is possible to avoid f:

$$ Q_{\alpha } = \frac{{r_{Cd,c} }}{{r_{Cd} }}Q_{\alpha ,c} $$

(2)

and shift the problem to the Q _α  → Q ₀ conversion. This is done by means of a theoretical expression that introduces the concept of a hypothetical single-resonance energy (or effective resonance energy; $\bar{E}_{r}$). This single-resonance would give the same resonance activation rate as all the actual resonances combined. The conversion function proposed in [15, 16] is:

$$ Q_{\alpha } = (Q_{0} - C_{0} )\bar{E}_{r}^{ - \alpha } + C_{\alpha } $$

(3)

where $C_{\alpha } = C_{0} \left( {2\alpha + 1} \right)^{ - 1} \left( {E_{Cd} } \right)^{ - \alpha }$, C ₀  = 0.429 and E _Cd  = 0.55 eV is the cadmium cut-off energy for 1 mm Cd-covers. The α value of a channel can be found experimentally by bare and Cd-covered irradiations of other standards with well-known nuclear data, after computing the slope (−α) on the resulting Log–Log plot of ($\bar{E}_{r,i}^{ - \alpha } /r_{Cd,i}$) versus $\bar{E}_{r,i}$, when expressed for i = 1, …, n isotopes. The parameter f is related to the intercept in this kind of plot [17].

Finally, the Q ₀ problem is classically solved by writing eq. (3) for the standard and the comparator and their substitutions on eq. (2), assuming that the $\bar{E}_{r}$ values are well-known. While this is obviously true for the chosen comparator, there is still doubt about the choice of $\bar{E}_{r}$ values for certain isotopes, since these have been officially adopted from early calculations.

The α-vector method: simultaneous Q ₀ and $\bar{E}_{r}$ determination with N irradiation channels

The typical approach for $\bar{E}_{r}$ determination is either: (a) to calculate it from weighted resonance data, assuming that it is independent of α (at least to a first order approximation) or, (b) to determine it experimentally from standard vs. comparator Cd-ratios obtained from 1 irradiation channel, assuming their Q ₀ factors are well-known [18]. Clearly, the latter option would lead to a vicious circle when performing Q ₀ determination.

The Q ₀ factors obtained from largely different irradiation channels (spread in α values) can only be consistent if this simplified $\bar{E}_{r}$ model is accurate irrespectively of the irradiation channel employed. By defining an experimental parameter Y _α for a given isotope and a given irradiation channel (index j) as:

$$ Y_{{\alpha_{j} }} \equiv \ln \left\{ {\left( {Q_{0} - C_{0} } \right)\bar{E}_{r}^{{ - \alpha_{j} }} } \right\} \equiv \ln \left\{ {\left( {{{r_{Cd,c} } \mathord{\left/ {\vphantom {{r_{Cd,c} } {r_{Cd} }}} \right. \kern-\nulldelimiterspace} {r_{Cd} }}} \right)_{j} \left( {Q_{0,c} - C_{0} } \right)\bar{E}_{r,c}^{{ - \alpha_{j} }} + \left[ {\left( {{{r_{Cd,c} } \mathord{\left/ {\vphantom {{r_{Cd,c} } {r_{Cd} }}} \right. \kern-\nulldelimiterspace} {r_{Cd} }}} \right)_{j} - 1} \right]C_{{\alpha_{j} }} } \right\} $$

(4)

it is possible to express the problem as a transcendental vector system:

$$ \vec{Y}_{{\vec{\alpha }}} = b - \vec{\alpha }m $$

(5)

and to determine $\bar{E}_{r} = \exp \left( m \right)$ and $Q_{0} = \exp \left( b \right) + C_{0}$ from the scalars b and m after solving the system through substitution of the α-vector from the N channels. Alternatively, a graphical solution can be obtained from a linear regression on a Y _α versus α plot, since b corresponds to the intercept and m to the slope. The parameter b changes sign for Q ₀  < 1.429, while m does so for $\bar{E}_{r} < 1\,{\text{eV}}$. An interesting case is ⁴⁵Sc, since it is the best example of a Q ₀  ≈ C ₀ situation.

For a few cases the effective resonance energy is still a function of α, so the expression $\bar{E}_{r,\alpha } = \bar{E}_{r} \exp ( - pa)$ should be used instead of $\bar{E}_{r}$. These p values are reported in [5] but not in the latest official compilation [6]. For ⁶⁴Zn, ⁹⁸Mo and ¹⁸⁷Re, the p values are 1.17, 1.82 and 0.95 (respectively). In this case, the α-vector method would aim at solving the polynomial vector system:

$$ \vec{Y}_{{\vec{\alpha }}} = b - m\vec{\alpha } + p\left| {\vec{\alpha }} \right|^{2} $$

(6)

k ₀ determination

From the ratio of A values between the standard and the comparator, a k ₀ factor can be calculated assuming that f, α, Q ₀ and $\bar{E}_{r}$ are known. Alternatively, in terms of specific activities from bare and Cd-covered irradiations, a k ₀ can be computed from:

$$ k_{0,s,c} = \frac{{\left( {A - {{A_{Cd} } \mathord{\left/ {\vphantom {{A_{Cd} } {F_{Cd} }}} \right. \kern-\nulldelimiterspace} {F_{Cd} }}} \right)_{s} }}{{\left( {A - {{A_{Cd} } \mathord{\left/ {\vphantom {{A_{Cd} } {F_{Cd} }}} \right. \kern-\nulldelimiterspace} {F_{Cd} }}} \right)_{c} }}\frac{{G_{t,c} }}{{G_{t,s} }} $$

(7)

This avoids the introduction of f, α, Q _o and $\bar{E}_{r}$ values and their uncertainties into the analytical result, which are of greater magnitude than the ones arising from counting statistics. For the experiments performed in the Cavity a k ₀ factor was obtained from eq. (7) by setting A _Cd  = 0, that is, the factor is just proportional to the ratio between activities and thermal self-shielding correction factors (no epithermal component). In this sense, the Cavity is a unique tool, but its low neutron flux limits our sensitivity (7 × 10⁸ n cm⁻²/s).

On the traceability, accuracy and precision of the determinations

When computing a Q ₀ factor with aid of f estimated from a calibration curve, the correlation will be weighted in favor of the average neutron flux perceived by the calibration isotopes and to their nuclear data. A Q ₀ can be computed in this way without the need of the comparator data at all (if it is not part of this calibration set). On the other hand, the introduction of the f parameter for Q ₀ and/or k ₀ determination is inevitable when the Cd-ratio was not determined, for instance, when adopting a Q ₀ or when assuming that f >> Q _α . Since the application of k ₀ -NAA relies on the specific activities from a comparator that was co-irradiated next to the sample, it follows that its Cd-ratio would be a better indicator of the neutron flux in that specific region. This alternative bears the lowest uncertainty and maintains the correlation in favor of the comparator results and its nuclear data. In this work all of our reported k ₀ values were determined experimentally through the use of Cd-subtraction (or ratio) technique. These values are therefore independent from the chosen f, α, Q ₀ and $\bar{E}_{r}$ values.

The precision of our measurement setup was tested by taking on average up to 20 measurements per standard on 5 calibrated source-detector distances (0, 3, 15, 20 and 27 cm) on 5 HPGe detectors equipped with LFC modules (for dead-time, pulse pile-up corrections). The measurements aimed at ≤0.3 % uncertainty in counting statistics when possible. The photo-peaks deconvolutions were done with the software HyperLab [19]. For Q ₀ determination the results from all detector positions available were employed while for k ₀ determination only the results from the 2 farthest positions were considered. These positions pose negligible coincidence gamma (COI) and secondary unwanted effects (i.e. peak distortions, higher dead-times, pulse pile-up) as compared to closer ones. The COI and the efficiency calculations were performed by a software adaptation of the efficiency transfer code SOLCOI [20] that employs mass attenuation coefficients from the NIST X-COM online database [21] for any specific gamma energy of interest.

A computer code was developed in Structured Query Language (SQL) 2008 and C# 4.0 programming languages, comprising the latest official k ₀ -NAA literature and formulae. With aid of SQL constrains and parent/child data-table relationships it was possible to treat great amounts of data points while avoiding systematic errors from human input and data redundancy.

The uncertainty from the laboratory setup on a specific activity and/or Cd-ratio can be estimated from the individual uncertainty contributions listed in Table 3. A Cd-ratio has the uncertainty arising from the counting statistics of 2 samples but the COI, efficiency and elemental content contributions are cancelled by employing the same detectors and measurement positions (total of 1 %, 1s). The uncertainty contribution from F _Cd is usually low (<0.5 %; except for ¹⁸⁶W, ¹⁸⁵Re, ⁶⁵Cu and ¹⁶⁵Ho), so the accuracy on a determined k ₀ factor should not be worse than 4 % (at 2s). Thus, a k ₀ factor was recommended when the standard deviation from 4–6 repeats per standard (per irradiation channel) was lower than this threshold value.

Table 3 Maximum expected uncertainty on an induced gamma specific activity (A) and a cadmium ratio (R _Cd ) at 1s confidence level. A cadmium ratio uncertainty corresponds to the quadratic summation of twice (×2) the uncertainty reported

Results and discussion

The Online Resource 1 illustrates the linear fits on the Y _α versus α plots obtained for 25 investigated isotopes. The best α-vector was obtained after several iterations, aiming at the lowest sum of residuals on all the linear regressions from the calibration isotopes. This can be done, for instance; with the Solver tool provided in the software Excel [22]. In general, good agreement is observed between our linear fits (dashed black lines) and the expected values (dotted blue lines) when comparing biases in intercept (b) and slope (m) divergences. The extrapolated $\bar{E}_{r}$ and Q ₀ values and their uncertainty propagation are summarized in the Online Resource 2. A comparison is made with the official literature values from [6, 23, 24] and other sources [25, 26].

Only 11 isotopes reported a >10 % relative difference to the official Q ₀ values. However 9 of them have adopted Q ₀ values (20 % relative uncertainty; 2s) and are of very low magnitude. Under bare irradiation conditions, the accurate knowledge of a low-Q ₀ isotope is not necessary (hence the reason for the adoption). Nowadays, irradiations under Cd are useful for avoiding spectral interferences from activation of unwanted nuclides such as ²⁷Al, ⁵¹V, ²³Na. This is sometimes necessary for instance when performing trace analysis on pure Al and/or Ti matrices, vitamins, etc. Although the idea is to substantially reduce the activation of these unwanted nuclides, one can be still interested in accurately quantifying these elements for practical reasons (internal comparator/control), for which the accurate knowledge of the Q ₀ factor is fundamental. Hence, we recommend our values over the adopted ones, especially for the As and Ba isotopes (high Q ₀ ) since our findings are also in agreement with other authors [27]. The remaining discrepancies were for ⁸⁷Rb and ⁸⁹Y (11 and 41 %, respectively), but unfortunately these isotopes could only be tested on our sole irradiation channel suited for short-lived investigation (S84). Our Q ₀ value for ⁸⁷Rb is not statistically significant at 3s level. For ⁸⁹Y, we tested first the same compound (pure yttrium oxide) employed in [5]. In virtue of the discrepancy, another matrix (pure yttrium foil) was tested, which confirmed our first results. The observed difference could not be explained in terms of self-shielding effects (negligible; see Table 1), F _Cd (equal to unity), decay scheme (I; normal) and half-life (not changed).

Our Q ₀ results can be weighted along the official ones if the same $\bar{E}_{r}$ has been adopted (the classical method). Under an experimental approach, then (Q ₀ , $\bar{E}_{r}$) values obtained from the α-vector method should be chosen instead. ⁹⁴Zr is quite a difficult case because of its high $\bar{E}_{r}$ value. It has been re-determined several times, with results differing from author to author. Our Q ₀ results are in better agreement with those reported in [4, 5, 28, 29].

Our relative differences to the official $\bar{E}_{r}$ values were ≥25 % for 7 isotopes. This had a <4 % influence on the resulting Q ₀ when alternating both inputs. In this respect, the choice of $\bar{E}_{r}$ value does not play a significant role. However, 6 isotopes have adopted Q ₀ factors so the accuracy of their $\bar{E}_{r}$ values (adopted) has not been studied before. Several of our experimental $\bar{E}_{r}$ values are in better agreement with reported values from others authors (i. e. for Mn, Zn, As, Sc and Mo isotopes).

The F _Cd factors are not quoted in the latest official references [6, 24] but can be found in [5, 23], from where we took F _Cd  = 0.98 for ¹⁸⁵Re, 1.034 for ⁶⁵Cu, 0.99 for ¹⁶⁵Ho, 0.908 for ¹⁸⁶W and 0.991 for the comparator (¹⁹⁷Au). No discrepancies were observed in Q ₀ factors for these isotopes except for ¹⁸⁵Re (5 %). The 1.5 % lower half-life employed in the past [5] does not seem to be the reason for the disagreement.

The k ₀ factors and thermal neutron cross-sections, derived with aid of data from [30–34] and the k ₀ definition, are given in the Online Resource 3. In general, the percent relative differences between our k ₀ factors and the official ones are usually ≤4 %, which is reasonable, since this is our worst expected precision (see Table 3) at 2s level. The results are satisfactory because less than 20 isotopes reported discrepancies >2 % (yet ≤4 %). For 13 isotopes the discrepancies are ≤1 %, which confirms the accuracy of our efficiency curves for most of the energy range dealt with. The k ₀ -data for ¹³⁴Ba(n,γ) ^135mBa and ¹⁸⁸Re (under decay scheme IV/b) are proposed here for the first time.

The only k ₀ -factors with observed >5 % relative difference correspond to the ¹⁰⁸Ag, ⁸⁸Rb, ¹⁵²Sm and ¹³¹Ba isotopes. The 434 keV line of ¹⁰⁸Ag seems biased in 6 % but the 633 keV line is in agreement with the literature instead. The k ₀ standardization of ¹³¹Ba has been a historical problem, worsened because of accidental typos or inaccurate Q ₀ adoptions, as has been explained in [27]. Our Q ₀ and k ₀ factors for this reaction are in better agreement with [35]. The 511 keV k ₀ factors for ⁶⁴Cu and ⁶⁵Zn are 10 % different and had always represented a spectrometric challenge. We aimed at ≤0.7 % uncertainty in counting statistics in order to facilitate the deconvolution while increasing the number of measurements to compensate. For ¹⁵²Sm, the 97.4 keV line should not be recommended as it is severely affected by the Compton from the more prominent 103.2 keV line.

Although our results are from 1 independent laboratory, we had a clear advantage on several technological improvements and knowledge of nuclear data over researchers from 30 years ago. For instance, a typical ⁷⁵As spectrum is almost entirely made up of duplets, a limiting factor depending on the resolution of the detectors employed. We recommend the use of the 657.1 keV line over the others. For ⁹⁵Zr, the values are in agreement with [3–5, 28, 36] and the 765.8 keV factor for ⁹⁵Nb is only 1 % different when employing the official F _i values. Nonetheless, coherence with its mother nuclide (⁹⁵Zr) is not observed in terms of cross-sections. Instead, when the more recent and accurate F _i values from [32] are employed (where F ₂₄ /F ₂ F ₃ is 5 % lower), the agreement is reached. We computed this k ₀ factor for both cases but we recommend the use of the underlined one along the updated F _i values.

Conclusions

The validity of the Breit-Wigner expression and the $\bar{E}_{r}$ concept was demonstrated here through the α-vector method. This method also offers a mechanism for studying the linearity of α and the generally assumed α-independence on the $\bar{E}_{r}$ value. Apart from the half-lives and decay branching fractions, the $\bar{E}_{r}$ values are the only values left that are still borrowed directly from the literature, so the accuracy of a theoretical $\bar{E}_{r}$ value can be tested through this method. On overall, the experimental $\bar{E}_{r}$ values from this work were ≥25 % different than the literature ones, but this had little influence on the Q ₀ determination. Yet, since these are hypothetical (or effective) values calculated from approximate models and sometimes incomplete resonance data, we would like to encourage the adoption of our experimental values instead. This is because the calibration (f and α-determination) of a reactor irradiation channel does rely on the accurate knowledge of $\bar{E}_{r}$ factors over the chosen energy range. The validation of a given $\bar{E}_{r}$ though a high rank (and spread) α-vector aims at expanding the diversity of isotopes available for calibration (or comparator) purposes. We believe that our rank-3 α-vector satisfied these requirements but it would be better to construct (include) larger datasets from coordinated worldwide exercises.

When considering how our factors would influence a previous analytical result (elemental concentration), it is clear that differences in k ₀ values will propagate entirely to it. The differences in Q ₀ factors will propagate fully if α = 0 and/or cadmium irradiations were performed (f = 0), but become negligible if f is sufficiently high (f >> Q _α ) under bare irradiations. Our Q ₀ factors are ≤10 % different and the k ₀ factors ≤4 %, on average. Our uncertainties are usually lower. New k ₀ and Q ₀ factors were proposed for ¹⁸⁸Re under the decay scheme IV/b and for ¹³⁴Ba(n,γ)^135mBa. Several recommended k ₀ -factors are introduced for multi-gamma radioisotopes (i.e. ⁷²Ga, ¹⁴⁰La and ⁷⁶As).

After more than 30 years of development and applications of this technique, several improvements in instrumentation, software concepts, etc., we believe that more laboratories should embrace a general re-determination of all these factors or new ones (i.e. ¹⁹²Ir) when possible, as it is the only way to enrich the statistical pool if an accuracy better than 4–5 % at 2s is desired.