Introduction

Lutetium is a difficult element to analyze in k0-neutron activation analysis due to its highly “non-1/v” thermal capture cross section. For this reason, it has in the past been necessary to include a separate standard for lutetium, in the Westcott formalism [1]. Recent innovative work has been done to eliminate this complication by using the modified extended Høgdahl convention [2, 3]. While this provides a promising new opportunity for the unification of methods in multi-elemental k0 analysis, there is still a great deal of value in the comparative method due to its simplicity and elimination of sources of uncertainty.

The preferred method for determining lutetium concentrations in instrumental neutron activation analysis (INAA) makes use of the 176Lu(n,γ)177Lu radiative capture reaction, with gamma spectrometry targeting the 208.4 keV emission. In the comparator method of INAA, a standard is irradiated alongside the analyzed sample. The net area of the relevant peak from the standard is used as a reference to calculate the concentration in the unknown. This precludes the need to characterize detector efficiency or neutron temperature. Assuming the standard and analyzed are irradiated and counted for the same duration (ti and tc respectively), and that flux and counting geometry are consistent, then the standard expression for the concentration of a target nucleus is

$$C_{u} = C_{k} \frac{{M_{k} }}{{M_{u} }}\frac{{N_{u}^{*} }}{{N_{k}^{*} }}\frac{{e^{{ - \lambda t_{dk} }} }}{{e^{{ - \lambda t_{du} }} }}$$
(1)

where Ck is the known concentration in the comparator, Mk and Mu are the masses of the known and unknown, N *k and N *u are the net counts of the are the net counts of the peak area for each sample, and the exponential terms are the normal decay correction after a decay period of td. The time dependency of the 176Lu activation products does not follow this trend due to self-interference effects (Fig. 1). It is possible to derive a new formula which incorporates the various sources of the 208.4 keV gamma emission resulting from activation of 176Lu as

$$C_{u} = C_{k} \frac{{M_{k} }}{{M_{u} }}\frac{{N_{u}^{*} }}{{N_{k}^{*} }}\frac{{A_{Lu} (t_{dk} )}}{{A_{Lu} (t_{du} )}}$$
(2)

where \(A_{Lu}\) is an apparent activity function of the 208.4 keV gamma emission, given by a summation of the various contributions to that peak. The difference in cross sections for each activation product suggest that this effect is minimal, and this likely explains why the phenomena is absent in the literature.

Fig. 1
figure 1

Activation and decay diagram for neutron activation of 176Lu. There are several contributions to the desired 208.4 keV signal resulting from this activation

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However, there exists a potentially problematic interference between the desired 208.4 keV from 177Lu and the 209.8 keV emission from 239Np, produced from the neutron absorption of 238U(n,γ)239U(t1/2 = 25 min) → 239Np(t1/2 = 2.35 d) reaction. Figure 2 contains an overlay of a typical 208.4 keV spectral peak showing the interference from this 239Np emission. Because this 239Np has a shorter half-life than 177Lu, it is possible to reduce the impact of this interference by allowing the sample to decay for a longer duration at the cost of poorer statistics. However, it is hypothesized that there may be more significant errors from these secondary signal contributions at longer decay times. It is anticipated that this effect is not significant for samples with low uranium mass fraction. However, in samples with significant uranium content, such as phopshogypsum (a waste product from manufacturing of phosphate fertilizers), this interference may be especially problematic. With the recent interest in recovering rare earth elements like lutetium from phosphogypsum, it is valuable to investigate all potential sources of error in lutetium analysis when waiting for high levels of 239Np to decay [4, 5].

Fig. 2
figure 2

The spectral features present in the region of 206–212 keV. The dark gray peak is the 209.8 keV emission from 239Np. The light gray peak is the sum of the 177Lu and 177mLu emissions

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Theory

Many recent publications documenting results of lutetium determination report the use of the neutron capture reaction 176Lu(n,γ)177Lu, and we see no reason to break with this convention [6,7,8,9]. The other candidate reaction is the neutron activation of 175Lu to 176mLu, which at first consideration may seem advantageous due to the greater isotopic abundance of 175Lu (97.41%) compared to 176Lu (2.59%). However, both the thermal capture cross section and resonance integral for 176Lu to 177Lu are several orders of magnitude greater than that for 175Lu [10]. Additionally, the activation product 176mLu has a half-life which is less favorable for routine analysis as compared to 177Lu (3.64 h as compared to 6.73 days). The shorter half-life requires inconvenient analysis times—to optimize the analysis, it may be necessary to allow for a total decay and counting period of several hours. Use of the longer-lived 176mLu allows the analyst to reduce the overall Compton background due to problematic medium-lived contributors such as 24Na (t1/2 = 15.02 h and 82Br (t1/2 = 35.3 h) by simply allowing them to decay away. This results in lower detection limits and a greater ratio of signal-to-noise.

The complications to the 208.4 keV signal which we seek to deconvolute arise from various parent-daughter relationships in the decay scheme of the possible 176Lu activation products. Because they each only rely on the number density of 176Lu targets, they can be considered individually and then summed over to find an overall “apparent” activity for the 208.4 keV peak. Parent-daughter activity relationships are canonically described by the Bateman equation—for an i-th isotope in a decay chain which is built in from the decay of the (i − 1)-th isotope, the time derivative of the population must obey

$$A_{i} \left( {t_{d} } \right) = -\,\lambda_{i} N_{i} \left( {t_{d} } \right) + A_{i - 1} \left( {t_{d} } \right)$$
(3)

For long decay chains, solutions to this sequence can be complicated. General solutions can be found analytically by using the Laplace transform. For single parent decays, the solutions are well known and can be classified by the ratio of the half-lives of parent and daughter. For the decays considered in this work, solutions can be written as follows:

$$A_{177m \to 177} \left( {t_{d} } \right) = \phi nt_{i} \lambda_{Lm} \sigma_{Lm} \left( {\chi_{IT} \frac{{\lambda_{Lm} }}{{\lambda_{L} - \lambda_{Lm} }}\left( {e^{{ - \lambda_{Lm} t_{d} }} - e^{{ - \lambda_{L} t_{d} }} } \right)} \right)$$
(4)
$$A_{177m \to Hf} \left( {t_{d} } \right) = \phi nt_{i} \lambda_{Lm} \sigma_{Lm} \chi_{\beta } e^{{ - \lambda_{Lm} t_{d} }}$$
(5)
$$A_{177} \left( {t_{d} } \right) = \phi nt_{i} \lambda_{L} \sigma_{L} e^{{ - \lambda_{L} t_{d} }}$$
(6)

where ϕ is the neutron flux, n is the number density of 176Lu targets, ti is the irradiation time, λLm and λL are the decay constants for 177mLu and 177Lu respectively, σLm and σL are the radiative capture cross sections at thermal neutron energy, χIT and χβ are the branching ratios for the isomeric transition from 177mLu to 177Lu and beta decay from 177mLu directly to 177mHf respectively, and td is the decay time. These values are readily available in most nuclear databases, allowing for straightforward hypothesis testing. Equation (3) describes the contribution due to the production of 177mLu which then decays into 177Lu. It is written using the normal form of the solution to the Bateman equation for isotopes in transient equilibrium, the derivation of which is covered in most introductory texts [11]. Equation (4) is similarly a solution to the Bateman equation, in this case for the secular equilibrium that arises due to the direct beta decay of 177mLu into 177mHf. Equation (5) describes simply the signal contribution due to the direct production of 177Lu which then decays into 177Hf. By summing over these terms, we find

$$A_{Lu} \left( {t_{d} } \right) = \lambda_{Lm} \sigma_{Lm} \left( {\chi_{IT} \frac{{\lambda_{Lm} }}{{\lambda_{L} - \lambda_{Lm} }}\left( {e^{{ - \lambda_{Lm} t_{d} }} - e^{{ - \lambda_{L} t_{d} }} } \right) + \chi_{\beta } e^{{ - \lambda_{Lm} t_{d} }} } \right) + \lambda_{L} \sigma_{L} e^{{ - \lambda_{L} t_{d} }}$$
(7)

These contributions have typically not been treated by previous studies, most likely because the cross section for production of 177mLu is orders of magnitude lower than that for 177Lu. We will investigate whether this approximation is valid for decay times relevant for suppression of the 239Np interference.

Experimental

A custom mixed rare earth standard purchased from Inorganic Ventures, of 10 µg/g concentration in water and 5% volume fraction HNO3, as well as a sample of NIST standard reference material 1633c (Trace Elements in Coal Fly Ash) were chosen for use in this work. Approximately 1.0 mL of the liquid and 0.2 g of the NIST SRM were prepared in 1.02 cm polyethylene dram vials and encapsulated in an outer 5.08 cm vial. Both the inner and outer vials for the liquid were heat sealed to prevent leakage.

Irradiations were performed at the University of Texas, Austin’s TRIGA Mark II reactor in the rotary specimen rack (RSR) facility. In this facility, samples are placed in a circular rack and rotated around the perimeter of the reactor, ensuring that the flux profile applied for each sample is consistent. The irradiation was performed for 1 h at a reactor power of 950 kW corresponding to a thermal neutron flux of ~ 4.5 × 1012 n cm−2s−1. Samples decayed for 7 days in a shielded storage area before being counted. Samples were counted using an ORTEC HPGe with a FWHM of 2.0 keV for the 1332 keV emission of 60Co. The spectrum of detected energies is sorted by an ORTEC multi-channel analyzer. Decisions regarding analysis optimization were made in consideration of the possible detection limits. Detection limits (LD) are calculated using the definition derived in [12] as \(L_{D} = 2.71 + 4.65\sqrt {\mu_{B} }\), where µB is the background under the peak. Samples were counted daily for 4 h each after a 1-week decay period. This was done to observe the time dependence of the activity of the 208.4 keV peak. A comparison our time series with the predictions made by both the simple prediction of Eq. (1) as well as the predictions made by our model in Eqs. (2) and (6) was done.

Due to the interference from the 209.8 keV emission from 239Np, it was necessary to use peak fitting software in order to resolve the 177Lu peak. For this purpose, the Interactive Peak Fit functionality in Canberra Industries’ Genie 2000 Gamma Acquisition and Analysis software was used. This function produces non-linear least squares fits within a range of energies and background regions specified by the user.

Results and discussion

Figure 3 depicts the results of lutetium determinations using Eqs. (2) and (6) for the independent measurements taken over 3 weeks. Uncertainty is estimated using the propagated Poisson error. The mean of these individual determinations was 1.32 µg/g, with a standard deviation of 0.07 µg/g across the data set. These results agree with the references values within these error bars for each individual determination for all but the earliest two measurements. In the early measurements, the lutetium was determined to be more than one standard deviation above the mean, indicating a significant error. This was most likely due to the interference with the 239Np emission, which should have the effect of increasing the apparent mass fraction. After 8 days (3.4 half-lives of 239Np), it was found that, for this sample with a U mass fraction of 9.25 ± 0.45 µg/g, the error produced by the Np interference caused insignificant error relative to the counting uncertainty, as evidenced by the clustering around the certified mass fraction.

Fig. 3
figure 3

Determinations of lutetium mass fraction based on separate measurements over the 3 week decay period for the sample of NIST 1633c. The solid and dashed lines indicate the reference value of 1.32 ± 0.03 µg/g

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Figure 4 depicts a log plot of the measured activity for the 208.4 keV peak over time, where the error is estimated as the standard Poisson uncertainty of \(\sqrt A\). These values are scaled by the initial detected activity, in order to establish a convenient comparison with the proposed model in Eq. (2). The dashed line shows the prediction of this model over the counting period. The root-mean-square error (RMSE) value for this model is calculated to be 0.011 Ao, which indicates a strong agreement between this and the observed activity.

Fig. 4
figure 4

The activity (as defined by the net counting area) of the 208.4 keV peak over the observation period, scaled by the initial activity Ao. We assume the standard Poisson-statistical error for the uncertainty. The dashed line shows the expected trend of Eq. (2)

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Although we have found there to be good agreement with the proposed model and our data, this is also true for the simpler assumption of Eq. (1). Evaluating RMSE for the exponential decay model gives a value of 0.011 Ao. These results indicate the information gained by including the additional terms from the decay of 177mLu is minimal for decay times of 6–21 days. This is not a great surprise. The relative contributions of these terms are dominated by the values of their coefficients, \(\lambda_{Lm} \sigma_{Lm} \left( {\chi_{IT} \frac{{\lambda_{Lm} }}{{\lambda_{L} - \lambda_{Lm} }}} \right)\) (4) and \(\lambda_{Lm} \sigma_{Lm} \chi_{\beta }\) (5) for the terms produced by 177mLu decay, and \(\lambda_{L} \sigma_{L}\) (6) from the direct production of 177Lu. Evaluating these coefficients using the relevant nuclear data yields the following ratios:

$$\frac{{\lambda_{Lm} \sigma_{Lm} \left( {\chi_{IT} \frac{{\lambda_{Lm} }}{{\lambda_{L} - \lambda_{Lm} }}} \right)}}{{\lambda_{L} \sigma_{L} }} = 5.34*10^{ - 7}$$
(8)
$$\frac{{\lambda_{Lm} \sigma_{Lm} \chi_{\beta } }}{{\lambda_{L} \sigma_{L} }} = 4.40* 10^{ - 5}$$
(9)

Therefore, these interferences produce approximately one count out of every 109 immediately following irradiation. In truth, we should expect these relative contributions to increase significantly as the original 177Lu population decays. It is possible to find the approximate time at which these contributions take over by finding the decay time td at which each term (3) and (4) are equal to the term for the direct production of 177Lu (5). By setting them equal and solving for td, it can be found that

$$t_{177m > 177} = \frac{{\ln \left( {1 + \frac{{\lambda_{L} \sigma_{L} (\lambda_{L} - \lambda_{Lm} )}}{{\lambda_{Lm}^{2} \sigma_{Lm} \chi_{IT} }}} \right)}}{{\lambda_{L} - \lambda_{Lm} }} = 146.4\;{\text{days}}$$
(10)
$$t_{Hf > 177} = \frac{{\ln \left( {\frac{{\lambda_{L} \sigma_{L} }}{{\lambda_{Lm} \sigma_{Lm} \chi_{\beta } }}} \right)}}{{\lambda_{L} - \lambda_{Lm} }} = 101.7\;{\text{days}}$$
(11)

These decay times far exceed the expected lifetime of the 239Np, indicating that these interferences are negligible source of error even in samples with high uranium content. By similar arguments, it can be found that the sum of the two interference terms only reaches 1% of the signal at a decay time of approximately 55 days, which also far exceeds the lifetime of 239Np. It is anticipated that for such a long period after irradiation, the 208.4 keV peak will not be distinguishable from the background regardless.

Conclusions

We have investigated various potential sources of error in lutetium determinations using comparative neutron activation analysis. We have found that these sources, originating from the production of 177mLu, do not cause a significant contribution to the targeted signal of 208.4 keV. This is likely due to the orders of magnitude separation between the relevant nuclear cross sections and decay parameters. This holds true even for any realistically long decay time, which may be desirable when attempting to resolve the 208.4 keV gamma ray from the problematic 209.8 keV gamma ray from 239Np. Such a technique may be desired for investigating the valuable rare earth element content of phosphogypsum byproducts. To that end, further work should be done in possible optimizations in decay duration to minimize this interference.