Influence of the Neutron Flux Characteristic Parameters in the Irradiation Channels of Reactor on NAA Results Using k 0-Standardization Method

An approximation method using to estimate the influence of the uncertainties of the neutron flux characteristic parameters in the irradiation positions on the NAA results using k0-standardization technique was presented. Those are the epithermal reactor neutron spectrum shape-factor α, the effective resonance energy r E for a given nuclide and the thermal to epithermal neutron flux ratio f. The method is applied to estimate the effect of the uncertainties in the determination of α, r E and f on final NAA results for some irradiation channels of the Dalat reactor. It also shows that presented method is suitable in practical use for the estimation of the errors due to the uncertainty of the neutron flux characteristic parameters at the irradiation position.


Introduction
Since the k 0 -standardization method was introduced in NAA [1], it has been broadly applied in the reactor in the world.The fundamental concept of k 0 -method was being elaborated previously in great detail [1-3].The concentration of an element in the k 0 -method is calculated by: ( ) ( ) ( ) with k 0 in Equation (1) defined as: In Equations (1) and ( 2 f-thermal to epithermal neutron flux ratio; Q 0 (α) = I 0 (α)/σ 0 ; I 0 (α)-resonance integral corrected for a non-ideal epithermal neutron flux distribution (assumed 1/E 1+α ); ε p -detector's efficiency; When the epithermal neutron flux distribution deviates from ideality, i.e. it does not follow the 1/E-law, Q 0 (α) of nuclide i can be written by: ( ) ( )( ) 0 0 0.429 0.429 2 1 0.55 with α-neutron spectrum shape factor deviating from the 1/E-law, independent of neutron energy and . 1 α  ri The asterisks in Equations (1) and (2) refers to the comparator, which is suitable for coirradiation with the sample; in most case, Au is used as a comparator.The k 0 -factors to Au for interested isotopes in NAA were experimentally determined and tabulated in report [4] with an accuracy which better than 2% (average ~1%).The relevant nuclear data as Q 0i and E -effective resonance energy of nuclide i. ri can be found in a tabulated form or in a computer library.α, f and ε p must be experimemently determined and they depend on spe-E T. VAN HUNG   251   cific irradiation channel and detector, which are used in practice.The detector's efficiency (ε p ) can be determined with an uncertainty about 2%; but the uncertainty of α can be more than 10%, even bigger, depend on the irradiation channels in reactor.Since the term [f + 0 (α)]/ [f + Q 0 (α)] in Equation (1), it is clear that an additional parameter, Q * ri , should be considered, because the uncetainties of E ri E of some nuclides are about 20% [4,5].
The accuracy and the applicability of the k 0 -standardization method were detailly presented in paper [5]  by F. De CORTE et al.In paper [6], J. OP De BEEK evaluated the effect of errors of α and ri on the results in terms of concentration, based on the 197 Au comparator; in that Q 0i (α) was approximated by : However, with this approximation, it led that some results in paper [6] have to be put to discussion (see below).
In this work, we carry out an approximation method to evaluate the effect of errors of α and ri on the NAA results in the k 0 -standardization method.The obtained results showed that the approximate method in this work is acceptable with confident accuracy.

Base of Approximation
As we know, α value is smaller than unity in absolutte value.In practice, in irradiation channels of reactor, absolute value of α is less than 0.2 (in most cases, α < 0.1 and this condition is satisfactory in reactor core).In the approximation of J. OP De BEEK, it is good for the nuclides having Q 0i >1, but is not for the nuclides with , in paper [7,8], we suggest substituting 0,i from Equation (3) by the following approximated formula: where a i is constant for each nuclide and determined by fitting the values of Q 0i (α), which are calculated from Equation (3) in range 0.2 , then fitting according to function (5) (see reference [7,8]).Note that, a i of each nuclide depends on the sign of α.The values of a i for the interested nuclides in NAA are given in Table 1.Seeing the Equation (5), it differs to Equation (4) of J. OP De BEEK by a correctional coefficient a i .However, it can be used good for all nuclides with uncertainties of the calculated 0,i less than about 5% for the nuclides having Q 0i < 1 and less than about 2% for Q 0i > 1 with 0.2 α ≤ .Indeed, we carried out a survey of the ratios of Q 0i (α) calculated from Equation (5) (in this work) and Equation (4) (of J. OP De BEEK) to Equation (3) (accurate expression) for Q 0i from 0.44 ( 46 Sc) to 248 ( 97 Zr) with α = -0.1.The results are presented in Figure 1 and some results are presented in Table 2. Clearly, the approximated expression in this work is better than one of J. OP De BEEK.Moreover, the calculated Q 0i (α) from three expression Equation (3), Equation (4) and Equation (5) for 45 Sc(n, γ) 46 Sc presented in Table 3.The another nuclides presented in papers [7,8] also confirm the above conclusion.
Value of α From Table 1, it shows that coefficients a i of nuclides having Q 0i > 1 are close to unity, but a i of the nuclides having Q 0i < 1 differs more than unity.Therefore, the approximation of Equation ( 4) in paper [6] is only acceptable for the nuclides having Q 0i > 1, but for the nuclides having Q 0i < 1, it is not reliable.
In this work, we use the approximation expression; Equation (5), to evaluate influence of the uncertainties of α, f and ri E on the final element concentration in k 0 -method in the channels; 7 -1, neutron trap of Dalat reactor (Vietnam) and channel 17 of THETIS reactor (Belgium) for the nuclides; 45 Sc, 59 Co, 94 Zr, 186 W, 197 Au, 98 Mo, 96 Zr.We choose these nuclides, because they differ considerably in Q 0i and ri E values.The numerical data of concerning isotopes and irradiation channels used in this work are summarized in Tables 4 and 5.

Results and Discussion
The absolute uncertainty in ρ can be calculated from the uncertainties of the variables (denoted x j ) which determine ρ in Equation (1): where ∂α/∂x j are the corresponding partial derivatives.According to the customary error propagation theory, ( ) and relative error is: ( )

Influence of Uncertainty of ri E on NAA Results
From Equation ( 8), the uncertainty of the concentration (ρ) in k 0 -method due to the uncertainties of the effective resonace energies can be written by: Using Equation ( 7) for the effective resonance energy of the nuclide i, we obtain: ( ) ( ) The values of calculated ( ) ri Z E ρ for chosen nuclides are presented in Table 6.The effect of the effective resonance energy on NAA result include the uncertainties of the effective resonance energies of analytical and comparator nuclides.In this case, Au used as comparator with Au E of 5.65 eV and uncertainty of 7.1% from paper [5], the contribution of the uncertainty of Au to the error of NAA result in channels 7 -1, neutron trap of Dalat reactor and channel 17 of THETIS reactor is 0.17%, 0.077% and 0.13%, respectively.Clearly, the effect of the uncertainty of the effective resonance energy of Au is negligible and can be overlooked in the evaluation.

E
The analysis for 94 nuclides used in NAA showed that the uncertainties of their effective resonace energy are from 0 to 20%, except 75 As (34%) [4].In this measure, we are able to realize that the effect of them on NAA result is also negligible.For example, 45 Sc ( r = 5130 eV, E r = 17%) and 95 Zr ( E Δ r = 338 eV, E r Δ = 2.1%), the contribution of the uncertainty of the effective resonance energy to the error of NAA result in three above channels is less than 0.01% for 45 Sc and 0.1% for 95 Zr.

E
In epicadmium neutron activation analysis (ENAA), the f-term in Equation ( 10) should be omitted.The error propagation function of ri E can be written: ( ) The calculated results of for the nuclides; 45 Sc, 59 Co, 94 Zr, 186 W, 197 Au, 98 Mo, 96 Zr in ENAA are carried in Table 7.In this case, the error propagation function is higher than in the one of irradiation without cadmium.Generaly speaking, a i < 1 and if α 1, the contribution of ΔE ri to the error of NAA result for al- most analytical nuclides is less than 1% and can be omitted in the calculation.

Results
Also from Equation ( 8), the uncertainty of ρ due to the uncertainty of α can be written: and error propagation function of α: The values of the error propagation function of α in the channels; 7 -1 and neutron trap of Dalat reactor (Vietnam) and channel 17 of THETIS reactor (Belgium) for the nuclides; 45 Sc, 59 Co, 94 Zr, 186 W, 197 Au, 98 Mo, 96 Zr were shown in Table 8.From Table 8, for the nuclides having Q 0 < Q 0Au in three these channels, the contribution of the uncertainty of α to the error of NAA result is not significant, about less than 1%.But for nuclides having Q 0 Q 0Au , this effect is noticeable.For instance, in channel 7 -1 of Dalat reactor (α = -0.044,Δα = 12%  [7,8]), the contribution of the uncertainty of α on the error of result of 45 Sc (Q 0 = 0.44) is 0.42%, but for 99 Mo and 96 Zr is 1.36% and 2.4%, respectively.As a comment, for RNAA using 197 Au comparator, the systematic effect for α value up to 0.1 is practically negligible for all nuclides with a low enough Q 0 value (e.g. 45 Sc, 59 Co, 58 Fe, ect.).On the other hand, for nuclides with a relatively large Q 0 value, a correction for the α effect becomes really necessary.To reduce the α effect, it is either to develop more accurate and precise techniques for α determination or to choose the irradiation channels with the α value low enough.
In the case of the epicadmium neutron activation, Equation (13) can be changed into: The values of the error propagation of α in this case were carried in Table 9.In this case, it clearly shows the inaccuracy of the approximation expression in [6] (Equation (4) in this report).Really, according to Equation (4), the error propagation function of α in the irradiation with cadmium can be written: Equation ( 15) is different to Equation ( 14) by the correctional coefficients a i .However, the value of the error propagation function in channel 7 -1 of Dalat reactor, for 45 Sc is 0.0083 from Equation ( 14) and 0.2997 from Equation (15).If the uncertainty of α in experiment is 100%, the contribution of uncertainty of α on NAA result is 0.83% and 29.97%, respectively.It differs by a factor of 30 (!).Similarly, in channel 17 of Thetis reactor, the error propagation function for 45 Sc is 0.0053 and 0.1907.The difference is huge.This comment is also correct for nuclides having Q 0 < 1.It once more confirms that the approximation expression in paper [6] is not good for nuclides having Q 0 < 1.
From Equation (13) or Equation ( 14), we easily estimate the influence of α on NAA results, if we know uncertainty of α in the irradiation channel.However, for ENAA (epicadmium neutron activation analysis) the situation is much more dramatic, especially for nuclides with low Q 0 value.

Influence of Uncertainty of f on NAA Results
The error propagation function Z ρ (f) can be written: The values of the error propagation function of f in the channels; 7 -1 and neutron trap of Dalat reactor and channel 17 of THETIS reactor for the nuclides; 45 Sc, 59 Co, 94 Zr, 186 W, 197 Au, 98 Mo, 96 Zr were carried in Table 9.The uncertainty of f contributes on the error of NAA results is: bution of the uncertainty of f on the error of NAA result (17) Generally seeing, the uncertainty of f in experiment is about less than 4%, therefore, from However, as discu e, the ssion abov r E effect is ne gi

REFERENCES
[1] A. Simonits, ste, "Single-Comglible and can be omitted in Equation (18).Thus, the contribution on error of NAA results in this case is primarily due to the uncertainties of α and f.Finally, as well as estimation above, this overall contribution of α and f is about 2% on the error of NAA results.It was also confirmed by actual analysis.
For α in the irradia sition relatively small ( 1  ), Equation ( 5) is a good approximation to estimat uence of the neutron flux characteristics on NAA result using the k 0 standardization method.From this approxi-mative expression, the error propagation functions of the parameters were presented.They can be used for the estimation of the errors on NAA due to the uncertainty of the neutron flux characteristic parameters at the irradiation position.From the results of this report, it was also confirmed that the approximation in paper [6] is only acceptable for the nuclides having Q 0i > 1, but not for the nuclides having Q 0i < 1.

Table 5 . Characteristics of irradiation channels considered: channles 17 of Thetis reactor, Belgium [9], 7 -1 channel and neutron trap of Dalat reactor, Vietnam.
the error propagation functions can be written as: