General Activation and Decay Formulas and Their Application in Neutron Activation Analysis with k 0 Standardization Stefaan G. Pomme ´,* ,² Frank E. M. C. Hardeman, ² Piotr B. Robouch, ‡ Nestor Etxebarria, ‡ Frans A. De Corte, § Antoine H. M. J. De Wispelaere, § Robbert van Sluijs, | and Andras P. Simonits ⊥ Studiecentrum voor Kernenergie, SCK‚CEN, Boeretang 200, B-2400 Mol, Belgium, Joint Research Centre, European Commission, Institute for Reference Materials and Measurements, Retieseweg, B-2440 Geel, Belgium, Laboratory of Analytical Chemistry, Institute for Nuclear Sciences, University of Gent, Proeftuinstraat 86, B-9000 Gent, Belgium, Radioisotope Applications & Support, DSM Research B.V., P.O. Box 18, 6160 MD Geleen, The Netherlands, and KFKI-Atomic Energy Research Institute of the Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary A general formula for nuclear activation and decay is presented. Besides decay processes, it deals with burnup and successive activation. The representation is compact and has a structure comparable to the classical NAA formulas, used in absence of burnup and activation of daughter products. A solution is presented to the math- ematical singularities appearing when different products have equal time constants. Explicit expressions are given for typical situations encountered in neutron activation analysis. Its incorporation into the k 0 standardization is demonstrated for several decay types. The processes occurring in neutron activation analysis (NAA) are governed by sets of linear first-order differential equations. They belong to a class of mathematical problems with important applications in a variety of fields, such as physics, chemistry, biology, and health physics. As a consequence, solutions for these systems have already been developed. In the course of time, refinements and alternative representations have become available. The pioneering work of Bateman 1 and Rubinson 2 is fully recognized in the field. General equations have been reported also by others. 3-9 The general solution was translated into recursive formulas by Hamawi, 3 later extended by Scherpelz 7 to deal with the problem of identical time constants in a single chain and by Miles 8 for use in complex chains with branching. Blaauw 9 tackled also the problem of backward branching. The general formula presented here is comparable with the Rubinson solution 2 but is more compact and eases the link with the more classical expressions used in NAA. Also, alternative expressions are presented for cases in which different products have equal disappearance and/ or decay rates. Besides demon- strating the general use of the formula, we also focus on its place in the definitions concerning the k 0 standardization in NAA. 10-13 The concept of k 0 standardization of (n, γ) reactor NAA was launched in 1975 10 to make better use of the potential power of this analytical tool (see, e.g., ref 13 and references therein). The technique enjoys growing popularity, for it adequately exploits the accuracy, traceability, and multielement capability of NAA. (1) BASIC ACTIVATION AND DECAY FORMULAS Activation. Aware of the general applicability of the equations to be discussed, we will nevertheless interpret the symbolic information merely in the framework of the nuclear activation and decay processes involved in NAA. First we consider the simple case of a linear series of “compartments”, in which matter flows unidirectionally and sequentially from the “first” to the “last” compartment. Afterward, the solution will be extended to complex networks of directly or indirectly interrelated compartments. The simplified case is applicable in most of the common neutron activation reactions. The NAA technique basically gives information about the abundance of an element (1) through the † SCK‚CEN. ‡ European Commission. § University of Gent. | DSM Research B.V. ⊥ KFKI. (1) Bateman, H. Proc. Cambridge Philos. Soc. 1910 , 15, 423. (2) Rubinson, W. J . Chem. Phys. 1949 , 17, 542. (3) Hamawi, J. N. Nucl. Technol. 1971 , 11, 84. (4) Clarcke, R. H. Health Phys. 1972 , 23, 565. (5) Skrable, K.; French, C.; Chabot, G.; Major, A. Health Phys. 1974 , 27, 155. (6) Skrable, K.; French, C.; Chabot, G.; Major, A.; Ward, K. Nucl. Saf . 1975 , 16, 337. (7) Scherpelz, R. I.; Desrosiers, A. E. Health Phys. 1981 , 40, 905. (8) Miles, R. E. Nucl. Sci. Eng. 1981 , 79, 239. (9) Blaauw, M. The Holistic Analysis of Gamma-ray spectra in Instrumental Neutron Activation Analysis. Ph.D. Thesis, Delft University of Technology, Delft, The Netherlands. (10) Simonits, A.; De Corte, F.; Hoste, J. J . Radioanal. Chem. 1975 , 24, 31. (11) De Corte, F.; Simonits, A. J . Radioanal. Nucl. Chem., Articles 1989 , 133, 43-130. (12) De Corte, F.; Simonits, A.; Bellemans, F.; Freitas, M. C.; Jovanovic, S.; Smodis ˇ, B.; Erdtmann, G.; Petri, H.; De Wispelaere, A. J . Radioanal. Nucl. Chem., Articles 1993 , 169, 125-158. (13) De Corte, F.; Simonits, A.; Lin Xilei; Freitas, M. In Nuclear Analytical Methods in the Life Sciences 1994; Kuc ˇ era, J., Obrusnı ´k, I., Sabbioni, E., Eds.; Humana Press: Totowa, NJ, 1994; pp 19-31. Figure 1. Sequential activation and decay series. Anal. Chem. 1996, 68, 4326-4334 4326 Analytical Chemistry, Vol. 68, No. 24, December 15, 1996 S0003-2700(96)00440-4 CCC: $12.00 © 1996 American Chemical Society