CMS NOTE 1998/086 The Compact Muon Solenoid Experiment CMS Note Mailing address: CMS CERN, CH-1211 GENEVA 23, Switzerland December 30, 1998 Decay and Transmutation of Nuclides P. Aarnio Laboratory of Advanced Energy Systems Helsinki University of Technology FIN-02015 HUT, Finland Abstract We present a computer code DETRA which solves analytically the Bateman equations gov- erning the decay, build-up and transmutation of radionuclides. The complexity of the chains and the number of nuclides are not limited. The nuclide production terms considered in- clude transmutation of the nuclides inside the chain, external production, and fission. Time dependent calculations are possible since all the production terms can be re-defined for each irradiation step. The number of irradiation steps and output times is unlimited. DETRA is thus able to solve any decay and transmutation problem as long as the nuclear data i.e. decay data and production rates, or cross sections, are known. 1 Introduction The CMS detector will be a subject of significant activation during its operation. This is mainly due to the secondaries emerging from the vertex but also due to the beam halo. For the purposes of detector design and safety, as well as due to the requirements set by the disposal of radioactive waste, the spatial and temporal distribution of the radioactive isotopes must be known. This requires knowledge on the isotope production and decay. Radionuclide production in CMS can be divided in two quite distinct categories: low energy neutron activation and radionuclide production by high energy hadrons. The borderline between these two cannot be exactly defined. In practice 20 MeV is commonly used as a transition energy between these two models. Since the cross sections of low energy neutron activation are experimentally well known, the best method to calculate the isotope production is to produce Monte Carlo estimates of the neutron flux and fold the results with the known cross sections. High energy hadron activation is a different story. First, the number of allowed reaction channels is very high; essentially all particle stable nuclides with A +1 masses are produced. Second, exclusive cross sections are poorly known except for intermediate energies and relatively light elements [1]. This means that the best method to estimate the production is a direct microscopic simulation of particle nucleus collisions. We have used the FLUKA code [2] to obtain both the low energy neutron fluxes and to simulate the high energy hadron production of residual nuclides in CMS. Once the production rates of radionuclides and their spatial distribution are known their build-up and decay can be calculated. For this relatively complete data exists [3], although it is not always fully consistent. However, since high-energy hadronic reactions can form nuclides very far from the stability line, the decay chains are often complicated so that the solution of their time development is far from trivial. Having nuclides far from the stability line requires a very comprehensive radionuclide library. Additionally, the missing data between the last nuclide with known decay properties and between the first particle unstable nuclide must be reasonably treated. Most of these exotic nuclei are very short lived and thus of little importance for safety. However, they must be considered when estimating the induced activity background to detectors during CMS operation. At the other extreme, reliable information about build-up and decay of long-lived radionuclides is needed in order to estimate the disposal requirements for LHC equipment and detectors. Although good data are usually easy to find for the long-lived nuclei these tend to be embedded in decay chains which can be quite complicated at the LHC. Therefore proper solutions of the full decay chains of even the long-lived isotopes are mandatory in order to obtain reliable results. The DEcay and TRAnsmutation code DETRA has been developed to provide a fully automatic interface to the most complete radionuclide libraries currently available and to solve the equations governing the time dependence of nuclear abundances during any arbitrary irradiation and cooling cycle. Although DETRA has been developed in view of LHC needs, it is much more general, enabling to extract complete time dependent radiation spectra as long as the radionuclide yields and irradiation conditions are known. The major features of DETRA are: accelerator production, or any external production, is properly accounted for can treat fission production from any number of fissile isotopes or fission types accepts arbitrary transmutation or activation of nuclides inside the decay chains accelerator production, transmutating flux, transmutating spectrum, fission power, fission energy 1 and fission fraction can be changed between irradiation steps making time dependent calculations possible accepts any number of irradiation steps accepts any number of output times calculates decay heat production The complexity of decay chains is not limited in any way i.e. the lengths of the decay chains are unlimited and nuclides can have any number of direct parents. The solution of the decay and transmutation equations is given in section 2, where the method of the solution has been adopted from Kanji Tasaka [4]. In section 3 we give instructions for the program input and in section 4 the program output is described. The nuclear data libraries and their editing is discussed in section 5 and an example of a calculation of gamma emission rates due to induced radioactivity in the LHC beam pipe is given in section 6. Some useful tools are introduced in appendix A. Finally the compiling and porting are discussed in appendix B. 2 Solution of the Decay and Transmutation Equation Decay and transmutation of nuclides can be expressed as X X dN i = S + b N (t)+ g N (t) N (t) N (t) ji j j ji j j i i i i i (1) dt j j where N i i is the abundance of nuclide , S i i is the external production rate of nuclide , b j i ji is the branching ratio of the decay of nuclide to nuclide , i i is the decay constant of nuclide , g j i ji is the fraction of absorptions in nuclide leading to nuclide , i i is the spectrum averaged absorption cross section of nuclide ,and is the transmutating flux. The first term on the right is the source term. It is any external production rate of nuclides i.e. it does not depend on any of the abundances N . Below we divide the source term in two components – one is due to fission and the other due to accelerator production. i N The second term describes the build-up rate of the nuclide due to decay of other nuclides, j . i N The third term describes the transmutation production rate of nuclide from other nuclides, j .This term includes both the transmutation of unstable isotopes and activation of the stable ones. Note that, even though the accelerator production of the source term above could also be called transmutation or activation, we have reserved the word transmutation for this reaction only. i The fourth term gives the decay rate of nuclide i, and the last term defines the depletion of the nuclide due to its transmutation. In the solution of Eq. 1 we assume that all the quantities, except N , above are constant i.e. do not depend on time. The assumption is usually well satisfied. In DETRA Eq. 1 is solved for each time step given in 2 input and all the constants are recalculated for each step. That is, DETRA can be used to solve also the time dependent problems provided that short enough time steps are used. Below we discuss the solution of Eq. 1 and each of its terms in more detail. The solution is based on Refs. [4, 5]. We have added the treatment of accelerator production and modified the way the complex decay chains are decomposed. For consistency reasons we have also made some changes in the notation. 2.1 Solution of linear chains – the Bateman equations The decay of radionuclides is governed by the differential equation dN N = (2) dt where N is the amount of a given nuclide and its decay constant. In the case of several radionu- clides forming a linear decay and build-up chain this leads to a recursive dependence, also known as the Bateman equations dN i = N + N i i1 i1 i (3) dt N =0 =0 0 where we define 0 and . These equations were first formulated by Rutherford and he also =1;2;3 i worked out the solutions for i [6]. The general solution for all is due to Bateman [5]. The Bateman solution, with somewhat different notation, is derived below. We introduce the following auxiliary quantities Z 1 xt n (x)= e N (t)dt i i (4) 0 It is easily seen that Z Z 1 1 dN i xt xt 0 dt = N (0) + x e e N (t)dt = N + xn i i i (5) i dt 0 0 0 n n (x) N N (0) i i where i is written for and for . i xt Now, multiplying Eq. 3 by e and integrating from zero to infinity we obtain Z Z Z 1 1 1 dN i xt xt xt e dt = e N (t)dt + e N (t)dt i i i1 1 i (6) dt 0 0 0 which gives 0 N + xn = n + n i i i1 i1 i (7) i 0 =0 i> 1 Assuming, for the time being, that N for all we obtain the recursive dependence for the i solution of Eq. 7 i1 n = n i 1 i (8) x + i 1 which can be written explicitly for the nuclide i> as Q i1 j j =1 n = n Q 1 i (9) i (x + ) j j =2 =1 For i we obtain 0 N 1 n = 1 (10) x + 1 3 0 Thus we have a general solution for a situation where only N differs from zero as 1 Q i1 j j =1 0 n = N Q i (11) 1 i (x + ) j j =1 Expressing this as partial fractions we have i X d ij 0 n = N i (12) 1 x + j j =1 where Q i1 k k =1 d = Q ij (13) i ( ) k j k =1 k 6=j 1 =1 j =1 d =1 Note that if i then also and .