Decay and Transmutation of Nuclides

CMS NOTE 1998/086

The Compact Muon Solenoid Experiment CMS Note Mailing address: CMS CERN, CH-1211 GENEVA 23, Switzerland

December 30, 1998

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 governing 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 include 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

= S + b N (t)+ g N (t) N (t) N (t)

ji j j ji j j i i i i

i (1)

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 is the decay constant of nuclide ,

g j i

ji is the fraction of absorptions in nuclide leading to nuclide ,

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

= (2)

dt where N is the amount of a given nuclide and its decay constant. In the case of several radionuclides forming a linear decay and build-up chain this leads to a recursive dependence, also known as the

Bateman equations

= N + N

i i1 i1

i (3)

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

n (x)= e N (t)dt i

i (4) 0

It is easily seen that

Z Z

1 1

xt xt 0

dt = N (0) + x e e N (t)dt = N + xn

i i

i (5)

0 0

n n (x) N N (0)

i i

where i is written for and for .

Now, multiplying Eq. 3 by e and integrating from zero to inﬁnity we obtain

Z Z Z

1 1 1

xt xt xt

e dt = e N (t)dt + e N (t)dt

i i i1 1

i (6)

0 0 0

which gives

N + xn = n + n

i i i1 i1

i (7)

=0 i> 1

Assuming, for the time being, that N for all we obtain the recursive dependence for the i

solution of Eq. 7

n = n

i 1

i (8)

x +

i 1

which can be written explicitly for the nuclide i> as

j =1

n = n

Q 1

i (9)

(x + )

j =2 =1

For i we obtain

n =

1 (10)

x + 1

3 0

Thus we have a general solution for a situation where only N differs from zero as

j =1

n = N Q

i (11)

(x + )

j =1

Expressing this as partial fractions we have

n = N

i (12)

x +

j =1

where

k =1

d = Q

ij (13)

( )

k j

k =1

k 6=j

=1 j =1 d =1

Note that if i then also and . 1

Now Z

xt t

e e dt

= (14)

x +

0 n

Returning to the deﬁnition of i in Eq. 4 and taking that together with Eqs. 12 and 14 it is easily seen that 0

for i>

0 t

N (t)= N d e ij

i (15)

j=1 where the constants have the values as given by Eq. 13.

Eq. 15 is a solution for a single linear chain of length i where only the top nuclide has a non-zero initial

0 i

value, namely N . The general solution can be obtained by summing linear chains with lengths,

0 0

m +1 i N N

i , varying from to 1 and initial values from to

1 i

i i

X X

0 t

N (t)= N d e ij m

i (16)

m=1

j =m

wherewehavedeﬁned

k =m

; 1 m j i: d = Q

ij m (17)

( )

k j

k =m

k 6=j Eqs. 16,and 17 are known as the Bateman solution.

2.2 Source term

S d m d

Let m be the amount of nuclide produced during a small time interval . The contribution of the

t S d

source term during the decay time can be obtained by substituting m for the initial abundance in

Eq. 16 and integrating from 0 to t

i i

X X

N (t)= S d d e

m ij m

i (18)

m=1

j =m

and we obtain

i i

X X

1 e

d N (t)= S

ij m m

i (19)

m=1

j =m

In DETRA the source term is deﬁned as

S = R + P y f Q

m a km k k

m (20)

k =1

where

R m

m is the accelerator production rate per unit ﬂux of nuclide ,

a is the accelerator production ﬂux,

P is the ﬁssion power,

K is the number of ﬁssion types deﬁned in the nuclide library,

y m k

km is the direct ﬁssion yield of nuclide due to ﬁssion type ,

f k

k is the fraction of ﬁssions of type ,and

Q k

k is the amount of energy released in the ﬁssion type .

R P f Q

a k k

m , , , ,and are given in input and can be different for each time step. The direct ﬁssion yield, y

km is calculated from the cumulative ﬁssion yields obtained from the nuclear library. The available

ﬁssion types, k , depend on the nuclear data library. See section 5.2.

2.3 Complex chains

In contrast to linear chains, we call a decay chain complex if any member of the chain has more than

one direct parent. Any complex decay chain can be decomposed into linear chains when the branching

b =1 b i j ij

ratios ij from nuclide to are known. Note that . As an example, consider the decay chain i depicted in Fig. 1a. To solve the decay and transmutation of nuclide 4, for example, we decompose the

complex chain into to the two linear chains in Fig. 1b. As far as the time behaviour of the nuclide 4 is

b =1

concerned 4? i.e. it represents the sum of all the available decay channels. We change the indexing

b =1=2

to be internal to the linear chains according to Fig. 1c. We set 3 for both the linear chains i.e. the branching ratio must be divided by the number of direct parents. It is obvious that, in this schema, each nuclide has to be weighted with its branching ratio, as in Eq. 1, and the complex chain solution for a nuclide will be the sum over the solutions of the corresponding linear chains.

Now we can rewrite Eq. 16 as

i i i

X X Y X

0 t

N (t)= ( b )N d e

l ij m

i (21)

m=1 j =m

l=m

linear

chains

and Eq. 19 as

i i i

X X Y X

1 e

N (t)= ( b )S d

l m ij m

i (22)

m=1 j =m

l=m

linear

chains

where the summing over m is over the three subchains as depicted in Fig. 1d. In the ﬁgure we have

m N S

encircled the nuclides having an initial amount or a source term m . Note that instead of having

m b

the branching ratio 3 deﬁned as above we could have used only one instance of the corresponding

b =1

subchain i.e. we could have kept 3 and not have summed over the subchain in parentheses. The approach we have chosen allows, however, to treat all the linear chains under eQual footing.

5 1 b 2 b 12 2? λ λ 1 2 b 24 b 13 34b 4? a complex chain λ b λ 3 34 4

124b b 12 b 24 4?

134b b b 13 34 4? blinear chains

123 b 1 b 2b3

1bb2 3 b 123Cre-indexed linear chains

d subchains ()

Figure 1: Decomposing a complex decay chain for nuclide 4. See text.

2.4 Transmutation of nuclides in decay chains

The transmutation induced build-up and depletion of nuclide i in Eq. 1 is

g N N

j j i i

ji (23) j

and the build-up and depletion due to decay

b N N

j j i i

ji (24) j

Now deﬁning analogously to the form of Eq. 24

X X

b N b ( + g )N ( + )N N =

j ji j ji j j i i i

i (25)

ji j i

j j

and solving for and we obtain

j ji

= + j

j (26)

b + g

ji j ji j

= (27)

j j

b b ji

Now if we replace j and in Eqs. 21 and 22 with and , respectively, we obtain the solution

j ji

which includes both the decay and transmutation induced build-up and depletion. g

In DETRA the fractions ji are read in from the library under equal footing with the branching ratios.

In DETRA the transmutation cross sections are deﬁned as

p j

j (28)

where is the multigroup cross section and the multigroup ﬂux. The multigroup cross sections are j

read in from the library and the multigroup ﬂux, i.e. transmutating spectrum, is given in input. It is

P p

also possible to keep the transmutating spectrum unnormalised. Then the division by will not be p performed.

2.5 The full solution

Combining the solutions in Eqs. 21 and 22 and applying the deﬁnitions in Eqs. 26 and 27 we ﬁnally

obtain the full solution of Eq. 1

! " # !

i i i K

X X Y X X

1 e

N (t)= b d N e + R + P y f Q

i ij m a km k k

m (29)

l m

m=1 j =m

l=m k =1

linear

chains

2.6 Nuclides with equal decay constants

The Bateman solution, Eqs. 16 and 17, is singular if there are two nuclides with equal decay constants in the chain. This is a very rare incident and if it happens we only adjust the decay constants to circumvent the problem. A better and more general solution will hopefully be implemented in the future.

2.7 Cyclic chains

A cyclic chain of nuclides formed by radioactive decay only is, of course, impossible. However, since DETRA can calculate cases where there are both transmutation and decay, also cyclic chains occur. Bateman Eqs. 3, which are formulated for linear chains only, cannot directly treat these cases. An approximation to a cyclic chain can be made where the linear chain is stopped if a nuclide occurs twice in the chain. That is, for every nuclide in a cyclic chain there is a linear chain where the nuclide of interest appears twice: once at the bottom and once at the top. It has been shown by Kanji Tasaka that ”when the time step was smaller than one-third of the longest equivalent-half-life in the chain, the computational

7 error of the accumulation of each nuclide was less than 0.3% for the calculation of three time steps”, and that ”the accuracy of the calculation is poor without the consideration of the same nuclide at the top of the linear chain [4]”. Since the top and the bottom nuclides are always the same in a cyclic chain its calculation involves equal decay constants i.e. the use of methods described above in section 2.6.

3 Program Input

DETRA should be run under Unix by typing > detra < detra.in detra.out & where detra.in is the input file and detra.out the printout file. In addition DETRA requires a special format nuclear data library and produces a results file discussed in more detail in sections 5 and 4, respectively. Program input file consists of a set of commands described below. Each line of the input must start with a command, or with an asterisk (*) denoting a comment line, and must not be longer than 150 characters. A command may be followed by one or more parameters which are in the same line with the command. The parameters are separated by blanks. All the parameters need not to be given. The default values mentioned together with the parameters below are the values which are applied if the corresponding parameter is not given. Because of the syntax chosen it is, of course, impossible to give parameter WHAT(n+1) if WHAT(n) is not given. Note that numerical parameters must always be given as real, i.e. containing the decimal point, even if they are inherently integers. Most of the commands are optional and their number is limited only by their memory requirements. The order in which the commands are given is significant, i.e., DETRA executes them in the order of their occurrence. There are, however, exceptions to these rules. They are mentioned in connection with the description of the commands.

3.1 CALCULATE

CALCULATE calculates the time evolution of the nuclear abundances based on the current data. Several calculate commands can be given in the same run. Only the following commands can be given between CALCULATE commands: INIT, OUTPUT and TITLE. Default (CALCULATE not given): No calculations will be done, i.e., only the syntax of the input is checked.

3.2 ENERGY

ENERGY defines the fission energies for each fission type during the next irradiation step.

WHAT(N) = fission energy [MeV/fission] of fission type N. One must give as many WHAT-parameter values as there are fission types in the library. This may require several ENERGY commands. Default = 0.0

Default: the values of the previous ENERGY command are used. If no ENERGY command has been

8 given all the fission types have the fission energy corresponding to thermal neutron fission of 235 U, i.e, 201.4 MeV/fission. The fission types are determined by the nuclear data library. See section 5.2.

3.3 FRACTION

FRACTION deﬁnes the fraction of different ﬁssion types during the next irradiation step.

WHAT(N) = absolute fission fraction of fission type N i.e. number of fissions of type N divided by total number of fissions. One must give as many WHAT-parameter values as there are fission types in the library. This may require several FRACTION commands. Default = 0.0

Default: the values of the previous FRACTION command are used. If no FRACTION command has been given the first fission type happens with a fraction of 1.0. All the other fission types have a fraction 0.0. The fission types are determined by the nuclear data library. See section 5.2.

3.4 INIT

Initialises the nuclide abundances and accelerator production rates. This command may be followed by the list of nuclides to be initialised.

WHAT(1) = index of the ﬁrst nuclide in the library to be initialised = 0.0 the list of nuclides to be initialised follows WHAT(2) = index of the last nuclide in the library to be initialised Default = the last nuclide in the library

WHAT(3) >0 initial number of atoms of nuclide having an index between WHAT(1) and WHAT(2)

<0 initial activity [Bq] of nuclide having an index between WHAT(1) and WHAT(2) Default = 0.0

WHAT(4) = accelerator production rate per unit flux in [cm2 ] of the nuclide index i.e. WHAT(4) times the accelerator production flux is the production rate per second. The accelerator production flux is given with the POWER-command. Note that this rate does not depend on the abundance of ANY nuclide. Default = old values are retained WHAT(5) = control parameter for the decay of nuclides not found in the library. See section 5.3.

9 = 0.0 the nuclides are forced to beta decay towards the valley of stability until the resulting nuclide is found in the library. The original nuclide will be replaced with the one found and for each decay a virtual nuclide with zero half life is created. See section 5.3. = 1.0 like 0.0 above but the virtual nuclides are not created. = 2.0 the nuclide is ignored. Default = 0.0

Default (INIT not given): the initial amounts and accelerator production rates are all zero. The format for each line of the list of nuclides is:

Nuclide WHAT(3) WHAT(4)

The maximum langth of the nuclide name Nuclide is eight characters. The list must end with nuclide named END and the nuclide names must be in exactly the same format as in the library.

3.5 IRRADIATE

IRRADIATE gives the length of the next irradiation time step.

WHAT(1) = irradiation time step length in days Default = 0.0 WHAT(2) = irradiation time step length in hours Default = 0.0 WHAT(3) = irradiation time step length in minutes Default = 0.0 WHAT(4) = irradiation time step length in seconds Default = 0.0

The total length of the irradiation time step is thus WHAT(1)d+WHAT(2)h+WHAT(3)m+WHAT(4)s. Any number of IRRADIATE commands can be given. The irradiations take place in the same order as the commands are given. Default (IRRADIATE not given): no irradiation takes place.

3.6 LIBRARY

LIBRARY gives the name of the nuclear library.

WHAT(1) = name of the nuclear library

This command is compulsory and must contain a name of a valid nuclear library.

10 3.7 NUCLIDES

NUCLIDES deﬁnes which nuclides in the library will be calculated.

WHAT(1) = number of the ﬁrst nuclide in the library to be calculated Default = list of nuclides to be calculated follows WHAT(1) = number of the last nuclide in the library to be calculated Default = last nuclide in the library

Default (NUCLIDES not given): all the nuclides in the library are calculated. Note that the program will have no knowledge of the nuclides outside of this list. For example, if the last member of a decay chain is omitted, it will never be produced and the number of nuclides is not conserved. The format of the optional list of nuclides immediately following the NUCLIDES command is (8A8). Nuclides with blank names are ignored. The list must be ended with a nuclide named END.

3.8 OUTPUT

OUTPUT defines the amount of output and the name of results file. LSD below means ”Least Significant Digit”.

WHAT(1) = 1st LSD = 1: print the time step information 2nd LSD = 1: print the nuclear abundances and activities 3rd LSD = 1: print the table of nuclear properties 4th LSD = 1: print the nuclear library 5th LSD = 1: print the total heat production 5th LSD = 2: print the heat contribution by nuclide Default = old values are retained WHAT(2) = name of the abundances ﬁle. If given the nuclear abundances are output to this ﬁle. Default = old name is retained

Default (OUTPUT not given): WHAT(1) = 00000.0 and abundances are not output to a separate ﬁle. The program output is described in more detail in section 4.

3.9 POWER

Defines the fission power, transmutating flux and accelerator production flux to be applied during the next irradiation step.

WHAT(1) >0 ﬁssion power [W]

Default: 0.0

2 1

WHAT(2) >0 transmutating ﬂux in [cm s ]

2 1 1

j j

<0 WHAT(2) is the transmutating flux in [cm s W ] i.e. the flux is obtained j as a product of jWHAT(2) and the fission power WHAT(1). For the energy spectrum of transmutating particles see the command SPECTRUM.

Default = 0.0

2 1 WHAT(3) = accelerator production ﬂux in [cm s ] Default = WHAT(2)

Default: the values of the previous POWER command are used. If no POWER command has been given fission power and fluxes will all be 0.0. The transmutating and accelerator production fluxes have a different meaning. For transmutating flux see section 2.4 and for the accelerator production section 2.2.

3.10 SELECTOR

SELECTOR selects which processes are on during the next irradiation step.

WHAT(1) = 1.0 transmutation production is enabled WHAT(2) = 1.0 accelerator production is enabled WHAT(3) = 1.0 ﬁssion production is enabled WHAT(4) = 1.0 all the production is instantaneous at the beginning of the irradiation step. WHAT(5) = 1.0 decay heat production is calculated.

Default: the values of the previous SELECTOR command are used. If no SELECTOR command has been given: 1.0, 1.0, 1.0, 0.0, and 0.0.

3.11 SIZE

SIZE deﬁnes the maximum size of the problem.

WHAT(1) = maximum number of irradiation time steps Default = 20 WHAT(2) = maximum number of of output time points Default = 100 WHAT(3) = maximum length of any linear decay chain Default = 20 WHAT(4) = maximum number of linear chains for any nuclide Default = 20

Default (SIZE not given): all the above defaults apply. The size command is only needed for the storage allocation. If the actual size of the problem is larger than deﬁned by the SIZE command the user will be requested to increase the storage allocation. The

12 actual size of the problem may be smaller. Only the ﬁrst SIZE command is acknowledged.

3.12 SPECTRUM

Deﬁnes the transmutating spectrum in the group structure for the next irradiation step.

>=0 p

WHAT(p) relative multigroup ﬂux in group . The input will be normalised by DETRA.

0 p < absolute multigroup ﬂux in group . The input will not be normalised by DETRA.

Default: the values of the previous SPECTRUM command are used. If no SPECTRUM command has been given the ﬂux is 1.0 in the ﬁrst group. In all other groups it is zero. One must give as many WHAT-parameter values as there are group cross sections in the library. This

may require several SPECTRUM commands.

Transmutation is determined by the multigroup ﬂux , given here, by the transmutating ﬂux, ,givenby p

the POWER command, and by the multigroup cross sections, , given in the nuclear data library. The

total transmutation rate is p .IfanyoftheWHAT( ) parameters is negative the normalisation by

P p

is not performed. See also section 2.4. p

3.13 TIME

TIME deﬁnes an output time.

WHAT(1) = output time in days Default = 0.0 WHAT(2) = output time in hours Default = 0.0 WHAT(3) = output time in minutes Default = 0.0 WHAT(4) = output time in seconds Default = 0.0

The output time is thus WHAT(1)d+WHAT(2)h+WHAT(3)m+WHAT(4)s. Note that this a time relative to t=0, not a time step. Time t=0 corresponds to the start of the ﬁrst irradiation time step. Any number of TIME commands can be given. They can be freely mixed with TIMES-LIN and TIMES- LOG commands. If no TIME, TIMES-LIN or TIMES-LOG command is given the nuclide abundances will not be calculated.

3.14 TIMES-LIN

TIMES-LIN deﬁnes a set of of output times with constant time difference.

13 WHAT(1) = first output time in days Default = 0.0 WHAT(2) = first output time in hours Default = 0.0 WHAT(3) = first output time in minutes Default = 0.0 WHAT(4) = first output time in seconds Default = 0.0 WHAT(5) = length of the time increment in days Default = 0.0 WHAT(6) = length of the time increment in hours Default = 0.0 WHAT(7) = length of the time increment in minutes Default = 0.0 WHAT(8) = length of the time increment in seconds Default = 0.0 WHAT(9) = number of output times

Thus, the ﬁrst output time is WHAT(1)d+WHAT(2)h+WHAT(3)m+WHAT(4)s, and the length of the

time increment is WHAT(5)d+WHAT(6)h+WHAT(7)m+WHAT(8)s. Note that the set of times thus

=0 t =0 formed are relative to t , not time steps. Time corresponds to the start of the ﬁrst irradiation time step. Any number of TIMES-LIN commands can be given. They can be freely mixed with TIME and TIMES- LOG commands. If no TIME, TIMES-LIN or TIMES-LOG command is given the nuclide abundances will not be calculated.

3.15 TIMES-LOG

Deﬁnes a set of of output times evenly distributed on a logarithmic scale.

WHAT(1) = first output time in days Default = 0.0 WHAT(2) = first output time in hours Default = 0.0 WHAT(3) = first output time in minutes Default = 0.0 WHAT(4) = first output time in seconds Default = 0.0 WHAT(5) = number of output times per order of magnitude

14 WHAT(6) = total number of output times WHAT(7) = time to second output time in seconds. The time differences between succes- sive output times will be exponentially increasing so that a tenfold increase is

reached after WHAT(5) output times.

t = i

The ﬁrst output time is thus 0 WHAT(1)d+WHAT(2)h+WHAT(3)m+WHAT(4)s and the :th output

i j=n

t = t +t 10 n = t = 0

time i ,where WHAT(5) and WHAT(7)s. Note that the set of times thus

j=1

=0 t =0 formed are relative to t , not time steps. Time corresponds to the start of the ﬁrst irradiation time step. Any number of TIMES-LOG commands can be given. They can be freely mixed with TIME and TIMES- LIN commands. If no TIME, TIMES-LIN or TIMES-LOG command is given the nuclide abundances will not be calculated.

3.16 TITLE

TITLE gives the title of the job.

WHAT(1) = title of the job, max. 72 characters. Default =

Default (TITLE not given): DETRA version 1.0.

4 Program Output

DETRA provides a printout to the standard output file and to a binary output file. The name of the binary output file can be given as the second parameter of the OUTPUT-command. The optional printout is controlled by the first parameter of the OUTPUT-command. See section 3.

4.1 Printout

The printout should be self-explanatory and is thus only brieﬂy discussed here. It depends on the nuclear library and on the calculations performed. The printout consists of the following items:

1. Header and storage information.

2. The input commands as interpreted by DETRA. This helps in ﬁnding input errors.

3. Library printout (optional). The printout contains a list of all the nuclides in the nuclear data library with their names, decay constants, Q-values, beta decay energies, gamma decay energies, cumulative ﬁssion yields, and their group transmutation cross sections. The names of the direct mothers of the nuclides are also given together with their decay modes and branching ratios. This is a printout of the whole nuclide library and can thus be very long.

15 4. Table of nuclear properties (optional). The table provides the list of nuclides with their names, decay constants, direct fission yields, transmutation cross sections, accelerator production rates per unit flux and the number of direct mothers. This output is given at the first output time only. Note that since the fission yields and transmutation cross sections can change between the irradiation time steps, this output is representative for the first time step only. This is essentially a printout of the whole nuclide library and can thus be very long. 5. List of time steps (optional). Lists all the irradiation time steps with their power and fluxes. Lists all the output times. 6. Information of the current time step (optional). This printout is given for every time step whether followed by the output or by the end of an irradiation time step. The following information is provided: length of the time step, type of the time step, power, transmutating flux, accelerator production flux, fission rate and whether instantaneous production, transmutation, accelerator production and heat calculations are performed or not. This is a relatively long printout to be used for debugging the irradiation and output times. 7. Nuclide abundances (optional). This printout is given at each output time defined. It prints the current time, and the amount and activity of each nuclide. 8. Totals. This printout is given at each output time defined. It prints the total amount and activity of nuclides, and the current time. 9. Heat production (optional). This printout is given at each output time defined. It prints the current time, the absolute and relative heating power decomposed to components due to beta emission, gamma emission and due to each nuclide.

4.2 Binary output

The binary output ﬁle is written as

DOUBLE PRECISION RNUM,TNOW,ACTIV(*) CHARACTER*72 TITLE CHARACTER*8 CHNUC(*) WRITE(IUUCH)’BIN ’,RNUM WRITE(IUUCH)TNOW WRITE(IUUCH)TITLE DO N=1,NUMNUC IF(ACTIV(N).GT.1.E-20)THEN WRITE(IUUCH)CHNUC(N),ACTIV(N) ENDIF ENDDO where the TITLE is the title of the calculation deﬁned by the TITLE-command, ’BIN ’ is a separator in the output used between the CALCULATE-commands, RNUM counts the CALCULATE-commands, TNOW is the current time, CHNUC(N) is the identiﬁer of the nuclide N and ACTIV(N) is its activity.

5 Nuclear Data Library

5.1 Libraries

Currently there are two nuclear data libraries available nudat.bin and kanji.bin. The library nudat.bin has been processed from a PCNuDat retrieval of the NUDAT data base [7] which is based

16 mainly on the Evaluated Nuclear Structure Data file, ENSDF [3]. nudat.bin contains 4681 nuclide ground and metastable states with their decay constants and decay chains. It should be considered comprehensive and currently up to date. It lacks, however, decay beta and gamma energies, fission yields and transmutation cross sections. How nudat.bin has been processed is described in more detail in Ref. [8]. The second library, kanji.bin, is based on the original library described in Ref. [4]. It contains 1170 nuclides with atomic masses 66 and higher. The data is in many ways obsolete and should not be used if only the decay properties are needed. The library, however, contains fission yields as described below in section 5.2. Neutron transmutation cross sections are also defined for 29 important nuclides. There are 25 energy groups from 10.5 MeV to 0.465 eV. The first 3, next 2, next 3, and last 17 groups have lethargy widths of 0.48, 0.57, 0.69, and 0.77, respectively. kanji.bin contains also the decay Q-values and decay beta and gamma energies, which are needed for decay heat calculations.

The printout of the contents of a library can be obtained from DETRA by using OUTPUT 1000.0 command.

5.2 Library format and data

New libraries can be created by merging the old ones, or by editing them with any text editor, or by converting from other data bases. The two latter methods require the knowledge of the format of the DETRA ASCII nuclear library. The library data is read in as

READ(IN,’(4I6)’) NUMNUC,NPAR,NFIS,NGS DO N=1,NUMNUC READ(IN,’(A8,1P,4E12.5,I4)’ )CHNUC(N),RAMDA(N),QVALUE(N),EBB(N),EGG(N),NCH(N) READ(IN,’(6(I6,A8,1PE12.5))’)(NBIC(N,M),CHMOTH(N,M),PBIC(N,M),M=1,NPAR) READ(IN,’(10(1PE12.5))’ )(GAMMAX(N,M),M=1,NFIS) READ(IN,’(12(1PE12.5))’ )(XSEC(I,N),I=1,NGS) ENDDO

The meaning of the input parameters in the ﬁrst line is

NUMNUC = Number of nuclides in the library NPAR = Maximum number of direct parents that any nuclide may have NFIS = Number of ﬁssion types deﬁned NGS = Number of groups in transmutation cross sections

Then follows the information for each nuclide with the input parameters

CHNUC = Nuclide identiﬁer RAMDA = Decay constant [1/s] QVALUE = Decay Q-value [MeV] EBB = Total beta decay energy [MeV] EGG = Total gamma decay energy [MeV] NCH = Number of direct mother nuclides NBIC = Decay type of each direct mother nuclide

17 1 = beta minus decay 2 = isomeric transition 3 = transmutation reaction 4 = EC or positron emission 5 = alpha decay 6 = delayed neutron emission 7 = other decay CHMOTH = Nuclide identifier of each mother nuclide PBIC = Branching ratios from each mother nuclide [fractions, NOT %] GAMMAX = Cumulative fission yields from each fission type [%] XSEC = Group transmutation cross section for each group [barn]

The meaning of most of the parameters above is self evident. The less obvious ones are further described here. The decay Q-value, total beta decay energy, and total gamma decay energy are used only for decay heat calculations. The solution of the decay and build-up is based on the decay chains. They are formed using the parameters CHMOTH and PBIC above. That is, the mother-daughter relationships are NOT based on the decay type NBIC. The decay type flag values 1, 2 and 4 are needed only for the construction of the direct fission yields from the cumulative ones. The decay type 3 is only used to flag the nuclides for which the group transmutation cross sections are defined in the library. For the reason to treat transmutation as decay see chapter 2.4. The flag values 5, 6 and 7 are ignored. If the above rules are followed any decay can be accommodated under type 7, other decay. That is, as long as the mothers in the decay chain library are correctly given the actual decay mode itself is insignificant. The decay type 3 can be any reaction as long this reaction leads from the mother to the daughter and as long as the group transmutation cross sections in the library are defined for this reaction. Currently only the library kanji.bin includes these cross sections. Since they are for neutron transmutation the DETRA printout titles must be changed if libraries with other cross sections are to be used.

Currently only the library kanji.bin includes ﬁssion yields. The ten ﬁssion types in this library are:

235 235

thermal neutron fission of 235 U, fast neutron fission of U, 14 MeV neutron fission of U, fast neutron

238 239 239

fission of 238 U, 14 MeV fission of U, thermal neutron fission of Pu, fast neutron fission of Pu,

233 232 thermal neutron fission of 241 Pu, thermal neutron fission of U and fast neutron fission of Th. Some titles of the DETRA output contain references to these fission types. If a library with different fission types were to be used the printout should be changed accordingly.

5.3 Incomplete and missing data

DETRA can handle most of the incomplete and missing data and produce reasonable results as long as the format of the library is correct. If, for example, production terms (cross sections, or ﬁssion yields) are missing the production is not activated. However, if the decay chains are inconsistent the results are unpredictable.

18 The library nudat.bin has been checked for consistency. It is possible, however, that the user deﬁnes in the INIT command nuclides that are not known in the library. These nuclides are far from the valley of stability, at least when the very comprehensive library nudat.bin is used, and their properties are not well known. They are usually very short lived and may decay in various modes. They may even be particle unstable. In DETRA we assume that their half life is zero and that they decay via beta emission towards the valley of stability. This is accomplished by replacing the input nuclide in question with the nearest nuclide along the isobar in the library. The replacement is done both for the initial amount and

for the accelerator production of the nuclide. Additionally a new virtual nuclide is created. This nuclide N has an initial amount and accelerator production of N times the value of the original nuclide. Here is the number of beta decays needed to reach the nearest nuclide in the library. If there are more than one nuclide, which is not found in the library, they are all replaced and their initial amounts and production rates per unit ﬂux are summed together. The amount of this virtual nuclide is printed normally in the output where the nuclide is named ‘Virtual’. These virtual nuclides are not a part of any decay chain and do not have decay energies. In addition to the above default treatment of non-existing nuclides the user can also choose to ignore them completely, or only to replace them with the nearest nuclide without creating the virtual nuclides.

5.4 Library format converter

In order to save disk space, and also to gain some speed, nuclear data libraries used by DETRA are in a binary format. This format, however, does not lend itself for easy porting between computers and operating systems. That is why we have provided the program DETRALIB whichisabletoconvertthe portable ASCII format to binary, and back to ASCII. It also facilitates the merging of binary libraries.

DETRALIB is a simple interactive program which asks the user for the necessary input parameters and provides a converted or merged nuclear data library as an output. When combining two libraries, A and B, it is possible to select whether to take the data from A or B or from both. If the data is taken from both libraries it is possible to select whether the data in A or B will override. The selection can be separately made for the nuclides themselves, for their decay chains, for their fission yields, and for their transmutation cross sections. It is thus possible, for example, to take the nuclides from both libaries but use the transmutation cross sections of the library B, if they exist. It should be noted that the nuclides are identified by their symbolic names and thus it must be verified that the notation used in the libraries to be merged is identical.

6 Example

As an example we show below an input ﬁle calculating activation in the LHC during 10 years of operation. Each year there are three 60 days operation periods with two 14 days shut-down periods in between. The year is ended with 157 days of annual shut-down. The ﬁrst year is assumed to be run with 10%, the second with 33%, the third with 67%, and the rest with 100% of the nominal luminosity. Only the

irradiation steps during the ﬁrst year are shown. The output times are distributed logarithmically starting

=0;t =1 from t s with 30 output times per order of magnitude and with the total of 290 output times extending thus over 100 years. Two cases are calculated. They describe the production and decay of radionuclides in aluminum and in stainless steel near an interaction point. The radionuclide production cross sections have been obtained from FLUKA simulations. Note the number of radionuclides produced. For aluminum the number is only

19 4 and for steel already 36. For heavier elements the corresponding numbers can be very much higher.

TITLE Activation of Al vs. Steel in the LHC. SIZE 20 300 20 20 LIBRARY nudat.bin OUTPUT 0.0 activation.res * first year with 10% luminosity (flux = 0.1) POWER 0.0 0.0 0.1 * only accelerator production is ON SELECTOR 0.0 1.0 0.0 0.0 0.0 0.0 IRRADIATE 60.0 * cooling period, all production OFF SELECTOR 0.0 0.0 0.0 0.0 0.0 0.0 IRRADIATE 14.0 SELECTOR 0.0 1.0 0.0 0.0 0.0 0.0 IRRADIATE 60.0 SELECTOR 0.0 0.0 0.0 0.0 0.0 0.0 IRRADIATE 14.0 SELECTOR 0.0 1.0 0.0 0.0 0.0 0.0 IRRADIATE 60.0 SELECTOR 0.0 0.0 0.0 0.0 0.0 0.0 IRRADIATE 157.0 * irradiation of the 9 subsequent years not shown

TIMES-LOG 0.0 0.0 0.0 0.0 30.0 290.0 1.0 TITLE Aluminum INIT 0.0 Be 7 0.00000E+00 7.16100E+03 Na 22 0.00000E+00 1.56600E+04 Na 24 0.00000E+00 6.51000E+03 Na 24m 0.00000E+00 6.51000E+03 END CALCULATE TITLE Steel INIT 0.0 Be 7 0.00000E+00 1.06000E+04 Na 22 0.00000E+00 2.90700E+03 Na 24 0.00000E+00 1.97350E+03 Na 24m 0.00000E+00 1.97350E+03 Mg 28 0.00000E+00 4.32100E+02 Al 26 0.00000E+00 1.33200E+03 Al 26m 0.00000E+00 1.33200E+03 Cl 38 0.00000E+00 1.72400E+03 Cl 38m 0.00000E+00 1.72400E+03 Cl 39 0.00000E+00 1.49800E+03 Ar 41 0.00000E+00 5.93800E+02 K 42 0.00000E+00 4.57400E+03 K 43 0.00000E+00 1.34200E+03 Ca 47 0.00000E+00 5.84300E+01 Sc 44 0.00000E+00 6.62500E+03 Sc 44m 0.00000E+00 6.62500E+03 Sc 46 0.00000E+00 6.00000E+03 Sc 46m 0.00000E+00 6.00000E+03 Sc 47 0.00000E+00 2.84700E+03 Sc 48 0.00000E+00 6.86700E+02 Ti 45 0.00000E+00 7.47700E+03 V 47 0.00000E+00 1.00700E+04 V 48 0.00000E+00 1.78700E+04 Cr 48 0.00000E+00 7.96800E+02 Cr 51 0.00000E+00 4.70400E+04 Mn 52 0.00000E+00 5.10000E+03 Mn 52m 0.00000E+00 5.10000E+03 Mn 54 0.00000E+00 5.67700E+04 Mn 56 0.00000E+00 3.87400E+03 Fe 52 0.00000E+00 9.12000E+02 Fe 52m 0.00000E+00 9.12000E+02 Fe 59 0.00000E+00 5.45100E+02 Co 55 0.00000E+00 7.81500E+02

20 10 5 10 5 Total Mn-54 Total 4 4 10 Na-24 Na-22 10 Na-24m Sc-46 Co-56 Na-22 Gamma activity (MeV/s/ccm) 10 3 10 3

V-48

10 2 Be-7 10 2

3000 3050 3100 3150 3200 3000 3050 3100 3150 3200 5 5 t(s) x 10 t(s) x 10

Figure 2: Build-up and decay of the activity in aluminum (left) and in steel (right) after ten years of irradiation in the LHC.

Co 56 0.00000E+00 3.22600E+03 Co 57 0.00000E+00 3.09000E+02 Co 58 0.00000E+00 6.08000E+01 Co 58m 0.00000E+00 6.08000E+01 END CALCULATE

The FLUKA output of production rates has been converted to DETRA input using the RESNUC program. See appendix A. The results have been converted to gamma emission rates using a comprehensive decay radiation library [9] and the SELECT program. See appendix A. They are plotted in Fig. 2. It can be seen that immediately after the shut-down the gamma activity in steel (per cm3 ) is signiﬁcantly higher than in aluminum.

7 Conclusions

DETRA solves the decay and transmutation equations without numerical integration. That means that the results are accurate and CPU time requirements quite reasonable. There are also no limitations to the complexity of the problem; arbitrarily long and complex decay chains can be calculated.

The decay data library of DETRA is consistent, comprehensive and up-to-date. This has required quite a lot of work since the nuclear data libraries used [3, 7] were not always fully consistent with each other, or even with themselves. Currently the DETRA library contains close to 5000 nuclide ground and metastable states with their decay constants and decay chains. The very exotic nuclides not found in the library are reasonably treated by forced beta decay towards the valley of stability.

DETRA calculations may include any type of isotope production including, but not limited to, activation, transmutation and ﬁssion. There is a neutron transmutation and ﬁssion yield data library included [4].

21 More comprehensive libraries including other production modes can be produced from available data using DETRALIB .

In CMS we have so far succesfully applied DETRA to calculation of beam pipe activation. We have also calculated the increase of occupancy in the CMS silicon tracker due to decay photons and betas from induced radioactivity[10]. In addition, we have studied the background in the innermost barrel muon chambers caused by the gamma glow of the passing cooling liquid that has been activated in the tracker [11]. A fourth application has been the evaluation of the activity and isotope contents of

activated ECAL crystals. The results will be compared to gammaspectrometric measurements of PbWO 4 irradiated in the pion beam of the PSI [12].

References

[1] A. Iljinov et al., Production of Radionuclides at Intermediate Energies, Landolt-B¨ornstein, New Series Vols 13a-e, ed. H. Schopper, Springer (1991-1993).

[2] P. A. Aarnio et al, CERN TIS-RP/168 (1986) and CERN TIS-RP/190 (1987). A. Fassò et al, Proc IV Int. Conf. on Calorimetry in High Energy Physics, La Biodola, Sept 20-25, 1993, Ed. A. Menzione and A. Scribano, World Scientific, p. 493 (1993). A. Fassò at al, Specialists’ Meeting on Shielding Aspects of Accelerators, Targets and Irradiation Facilities. Arlington, Texas, April 28-29, 1994. NEA/OECD doc. p. 287 (1995).

[3] M.R. Bhat, Evaluated Nuclear Structure Data File (ENSDF), Nuclear Data for Science and Tech- nology, page 817, ed. S.M. Qaim, Springer Verlag, Berlin, Germany (1992).

[4] Kanji Tasaka, DCHAIN2: A computer Code for Calculation of Transmutation of Nuclides,JAERI- M 8727 (1980) 170p.

[5] H. Bateman, The solution of a system of differential equations occurring in the theory of radioactive transformations, Mathematical Proceedings of the Cambridge Philosophical Society: 15, (1910) pp. 423-427.

[6] E. Rutherford, Radio-activity, 2nd ed. Cambridge, University Press (1905), pp. 330-337.

[7] B.R. Kinsey, et al., The NUDAT/PCNUDAT Program for Nuclear Data, paper submitted to the 9th International Symposium of Capture Gamma-Ray Spectroscopy and Related Topics, Budapest, Hungary, October 1996. Data extracted from the NUDAT database version 2.22 (31.12.1996), CD- ROM.

[8] P.A. Aarnio, Radionuclide Library for Chain Decay Calculations, CMS Internal Note, CMS IN 1998/008.

[9] P.A. Aarnio, Decay Radiation Libraries for Assessing Consequences of Activation of the CMS De- tector, CMS Internal Note, CMS IN 1998/006.

[10] CMS Tracker TDR, CERN/LHCC 98-6, pp.A21-A22.

[11] P.A. Aarnio, M. Huhtinen Background in MB1 Due to Radioactive Cooling Fluids from the Inner Tracker, CMS Internal Note, CMS IN 1997/035.

[12] to be published.

22 A Tools

When solving complex problems involving accelerator production and decay of radionuclides we use FLUKA to calculate the source term and DETRA to obtain the time dependence of the nuclear abundan- cies. Several calculational steps are usually needed.

1. FLUKA simulation of particle ﬂux spectra.

2. FLUKA simulation of residual nuclei production using the particle ﬂux spectra obtained in the previous step as an input.

3. DETRA decay chain calculations to obtain the activities at the times of interest.

4. Conversion of the activities to decay radiation spectra.

5. Transporting the decay radiation using FLUKA to obtain its effects in the area of interest.

Below we brieﬂy describe some simple tools to facilitate some of the above steps.

From FLUKA output to DETRA input. In order to save space FLUKA reports the residual nuclide production in a cumbersome format. A simple program, RESNUC, has been made to do the conversion. The user has to set the parameters deﬁning the FLUKA residual nuclide output and RESNUC converts the output to DETRA input. Only the production rates are set. Other DETRA commands must be manually edited. The use of the program is straightforward and has been commented in the source code. Currently FLUKA produces nuclides in their ground states only. RESNUC divides the ground state production rate between the known metastable states[8] using equal weights. This approximation is not correct and should be checked individually for important nuclides with dominating abundances.

Selecting, sorting and converting the quantities of interest. Depending on problem the DETRA output can be quite large. It may contain several cases, i.e. many CALCULATE commands, each having thou- sands of time steps and hundreds of nuclides. Program SELECT allows the user to select the quantities of interest based on several criteria.

1. Radiation – either gamma, X-ray, beta, atomic electron, or alpha radiation.

2. Nuclides – user deﬁned number of most important nuclides at a user given time and belonging to a user given CALCULATE-case. The importance criterion can be the activity of the nuclide [Bq], its radiation energy [MeV/s], or the number of radiation quanta emitted per second. The output will be sorted according to this criterion.

3. Case – the number of the CALCULATE-command deﬁning the case to be output.

4. Time – the output start and end times can be selected.

5. Quantity – either activity [Bq], radiation energy [MeV/s], or the number of quanta per second can be output.

6. Offset – time offset can be given to select the time zero.

The selection is output to a ﬁle as an array where each selected nuclide has its own column. In addition there are always the columns giving the time, the total, the rest, and the CALCULATE-command number. The total is the sum over all nuclides. The rest is the selected nuclides subtracted from the total. The

23 headings and selection criteria are output at the end of the ﬁle making its use as input easier for programs like PAW, for example. SELECT interrogates the required parameters from the user but it also generates a log ﬁle which can be redirected to standard input if the user wants to repeat the selection.

From activities to decay radiation spectra. For this purpose there are two programs. RADLIST converts the DETRA activities to lists of radiation energies and intensities or energy emitted per second for each nuclide. The user can select which DETRA times and calculations are converted. RADLIST supports gamma, X-ray, beta, atomic electron, and alpha radiation based on decay radiation libraries [9]. RADBINS can be used to further convert the radiation lists to radiation spectra. The program performs only simple binning of the line spectra without any instrumental effects. Currently the beta spectra are not properly processed and only the average energies are reproduced as line spectra. The programs are interactive and their use should be obvious.

B Compiling DETRA

DETRA and DETRALIB are written in Fortran 90. They have been tested under IBM AIX version 4.1.5 and IBM AIX XL Fortran compiler version 3.02. During the testing the language level ﬂag has been set to ‘90pure’, which stands for pure ISO standard Fortran 90 without obsolescent features. In AIX DETRA and DETRALIB can be compiled and loaded using the commands xlf90 detra.f xlf90 detralib.f

DETRA and DETRALIB contain some 3300 and 1300 lines of code, respectively. The programs have not been ported to other operating systems. However, since they are fully conformant to the ISO Fortran 90 standard the porting should pose no problems as long as a proper compiler is available.