- Email: [email protected]
Numerical evaluation of the production of radionuclides in a nuclear reactor (Part II)
Appl. Radiat. lsot. Vol. 49, No. 4, pp. 383-395, 1998
Pergamon PII: S0969-8043(97)00288-1
© 1998ElsevierScienceLtd. All rights reserved Printedin Great Britain 0969-8043/98 $19.00+ 0.00
Numerical Evaluation of the Production of Radionuclides in a Nuclear Reactor (Part II) S A E D M I R Z A D E H *~ a n d P H I L L I P W A L S H 2 ~Nuclear Medicine Group, Life Sciences Division, Oak Ridge National Laboratory (ORNL), Oak Ridge, TN 37831-6229, USA 2Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803, U.S.A. (Received 21 April 1997; accepted 30 June 1997)
A computer program called LAURA has been developed to predict the production rates of any member of a nuclei network undergoing spontaneous decay and/or induced neutron transformation in a nuclear reactor. The theoretical bases for the development of LAURA were discussed in Part I. In particular, in Part I, we described how an expression based on the Rubinson (1949) approach is used to evaluate the depletion function. In this paper (Part II), we describe the full simulation of radionuclide production including the decomposition of a reaction network into independent linear chains, provisions for periodic reactor shutdown and restart, and implementation of an approximate solution given by Raykin and Shlyakhter (1989) to account for the effect of feedback due to alpha decay. Also included are some examples which demonstrate possible uses for LAURA. © 1998 Elsevier ScienceLtd. All rights reserved
Introduction
solution governed by the total depletion constants. In this form, the problem associated with singularity (arising from identical depletion constants) has been eliminated. For a specific problem, LAURA decomposes (1) into independent linear chains of the form (2) and for each time step it passes the depletion constants and current time value to a depletion factor subroutine. The depletion function has the following format (see Part I for the details):
An m-member reaction network of nuclei undergoing both spontaneous decay and nuclear transmutations is described in matrix form by (1)
dx/dt = Ax
where x is the concentration vector and A is the m x m reaction matrix (Raykin and Shlyakhter, 1989). It is known that equation (1) can be broken up into independent linear chains and treated as several separate problems of the type CI
N,
c2
, N2
c3
, N3
cn- t
,'"Nn_,
Dn(A~ . . . . . An)(t) = ~ MULT(n, i)t ~'~' ~e- a,,, (3)
cn
, Nn
,
where cn_ t is the formation constant 0.~- ~or ¢ G - z) of the nth species from (n - 1)th species of nuclei and is independent of time or the nuclide concentrations. The exact solutions to the corresponding differential equations are given by
o-,)
N n ( t ) = N ° C ~ = l c i " h n ( A 1. . . . .
A~)(t)
(2)
where Dn represents a depletion factor containing the time-dependent variables, and is the part of the *To whom correspondence should be addressed. E-mail: [email protected].
where each term in the summation is associated with the corresponding nuclide species, and where m ( n , i) is the number of depletion constants Ak such that i < k < n, Ak = A~ and m(n, n) = O. The general algorithm for obtaining MULT and m for the (n + 1)th depletion factor is given in Part I. In this paper we describe the general structure of the code LAURA, including how equation (1) is constructed and broken into independent linear chains and how the feedback effect is handled. Several illustrative examples, including the production of tssW, R5I and 252Cfin a reactor, are also given. In each case, graphic representations are employed to illustrate the construction of the reaction network by LAURA.
383
384
Saed Mirzadeh and Phillip Walsh
[ lpu'r I ACTIVITYSUBROUTINE, CALCULATECONCENTRATION VECTOR AT T, F
oiiv i- I Fig. 1. The general flow chart of LAURA.
Program Description
The general flowchart of L A U R A is shown in Fig. 1. Each section in the flowchart is briefly described. For each problem, L A U R A expects a simple input file containing all the necessary information such as neutron flux, irradiation time, etc. The input file, which is provided by the user, can be easily edited and tailored to a variety of specific problems to study the effect of each variable. This feature distinguishes L A U R A from other available codes that operate on large data files (e.g. code ORIGEN, Bell, 1973). In addition, provision for calculations for various irradiation times and fluxes, as well as the periodic shutdown of the reactor flux during refueling, have been provided in LAURA. L A U R A is written in F O R T R A N and is compiled on both VAX/VMS and MS/DOS operating systems.
so that specifying a target is all that is required from the user. Of course, there remains some limit on the actual length of a given reaction chain (see Part I). Currently we have only limited files for isolated sections of the Table of Nuclides. For multiple fluxes and time points, the irradiation times and fluxes are stored in the input file created specifically for each problem. The irradiation conditions include the minimum and maximum irradiation times and the number of intermediate time points for which the concentrations are desired, the minimum and maximum neutron flux values and the number of intermediate flux values, and the specific times of expected reactor outages and restarts. The link-list structure of the database means that each nuclide having an initial concentration must be specified by the user as a target (multiple targets are possible). At run time the user follows prompts to enter the number of target nuclides, then the filename and initial concentration for each target nuclide, the file containing the irradiation specifics (the input file), and finally the output file specifications. As the reaction network is built, the program assigns each nuclide a position in the concentration vector x, which serves as the nuclide's identification number or 'network integer'. The initial concentration of the target nuclide is stored in the first position of vector x. The first line of the target file is stored in the first positions of four vectors: S I G M A X (total crosssection), SIGMAI (total resonance integral), M A S S (atomic mass), and HALF-LIFE (half-life). The
Input phase The program L A U R A uses an input file containing the specifics of the particular problem. This information consists of the neutron flux values, irradiation times, reactor shutdowns, etc. The program prompts the user to enter the target nuclide symbols and constructs the reaction network, equation (1). The nuclide-specific reaction data are stored in a database. Given a target nuclide, L A U R A constructs a reaction network by following the allowed transformation routes linking the nuclides in the network. This database would presumably not be modified between problems, except to update when more current reaction data become available. The reaction data for a particular isotope are stored in a short file containing the isotope's total cross-section, total resonance integral, atomic weight, half-life, number of lst-order decay products, and number of lst-order neutron capture products. The information for each lst-order decay product includes the product's file name and the branching ratio leading to the decay product. For each l st-order neutron capture product, the required information consists of the product's file name and partial cross-section and resonance integral leading to its formation. Each nuclide in the database has a similar file so that a linked-list data structure is formed. In principle, the entire Table of Nuclides could be mapped in this way,
PEST OF PROGRAM
__~
NEWFLUX VALUEF
VALUET
t
~
~ T E NUCLIDE CONCI~TRATIONS AT T AND F
1 ATT, F
[
Fig. 2. Flow chart of the flux-time loop.
Numerical evaluation of radionuclides production II product files are opened and given successive positions in these vectors, then their product's files are opened, and so on. These four vectors, together with the concentration vector, are called the 'network array', A particular chain will terminate if there are no daughter products specified in a nuclide file. Also, a chain may be forced to terminate if the user-specified linear chain limit is reached. Additional targets are treated in a similar manner, with new nuclide species being assigned unused positions in x. If a target species already appears in x, its concentration is placed in the corresponding position and no further action is taken. The position a nuclide holds in the concentration vector will be used to identify that nuclide throughout the program. The reaction network is generated as the program constructs the concentration vector. Each reaction is stored (in the order it is encountered) in four vectors: P (the network integer for the parent nuclide), D (the network integer for capture product, or daughter nuclide), H O W (an integer which distinguishes a natural decay from neutron-induced reaction), and
I TIME,FLUXVALUES
SENTTO SUBROUTINE ]
[
PROGRAM
PERCENT (branching ratio, indicating fraction of transformation, or decay, of P to D).
Time-flux loops L A U R A can give the concentration vector for multiple time and flux values. The structure is simple---an outer flux loop picks up a flux between the minimum and maximum flux values (specified in the input file), and the inner time loop runs over all the time values between the minimum and maximum values. Control is passed to the activity subroutine for each time and flux value. The last task in the inner loop is to write the concentration vector and other information for that time and flux to an output file. The output file can be imported to a graphic routine such as Jandell's Sigma Plot for visual presentation. A flowchart of the time-flux loop is shown in Fig. 2.
Activity subroutine The activity subroutine uses the reaction vectors to calculate the concentration vector for a given time and flux. The depletion constants for all the nuclei in the network are calculated from At = L A M B D A ( i ) + f x SIGMAX(i), where f is the current flux value and L A M B D A is the decay constant. SIGMAX(i) can be replaced by the apparent capture cross-section which includes the contribution of the epithermal neutrons to the total reaction rates. The activity subroutine uses the initial concentrations provided by the user at the beginning of the simulation for every time step, except when the reactor is shut down (see Section 2.4). Any nuclide in the network array having an initial concentration has the term
Ni(t) = N°e- Air
I Fo~v FLAG= I
~
a=n+ 1 el(n) = DAUGRTER IN'I~GF.R, eclta t) = PERCENT
385
(4)
added to its position in x. If the network integer of this species also appears in P, then this species contributes to the concentrations of other nuclides in x and it is the starting point of a reaction tree. The effect on each subsequent nuclide is calculated one at a time. The current parent nuclide, say N , is relabeled as N~. Suppose the nuclide's network integer occurs in the j t h position of P. The integer at H O W ( j ) determines whether ct is simply PERCENT(j), for a decay process, or f x PERCENT(J) for a neutron capture process. The network integer in D(J) is the daughter product in the j t h reaction and is relabeled as Nv L A U R A performs a function call D2(A, A2)(t) and the term
ii-|
Nct(a)(t) = NIO~I gkDa(ACI(I), ..., Aca0~
Fig. 3. Flow chart of the activity subroutine. Time and flux values are sent to subprogram for a network of m nuclides. A flag marks the position in the P where a nuclide integer is found so that each branch is computed only once.
N~(t) = ctN~tD2(A,, A2)(t) is added to the daughter nuclide's position in x. If the daughter's network integer appears in P, the chain continues and the corresponding PERCENT value
386
Saed Mirzadeh and Phillip Walsh
I RESTOF PROGRAM RESETINITIAL CONCENTRA11ONSTO VALUF.SAT T-- 0 ]~ NEWTIME VALUE,T
-q
]
SHUTDOWN]
NEW VALUE,P.S
CONCENTRATIONSATF, T I
CONCENTRATIONS
ATF,RS
I
NEW RESTART 1 VALUE,RG
co cErerR^TioNs I ^rP-- 0. r
I P -SrOF [ I
I
ATF=0, TsRG Fig. 4. Flowchart of the reactor shutdown subroutine.
becomes c2 while the new daughter nuclide is relabeled N3, and the term
N3(t) = c,c2N°D3(A,, A2, A3)(t) is added to the position in x corresponding to N3, and so on. Equation (2) is used for a general linear chain of length n. A linear chain terminates when the last nuclide's network integer does not appear in
P.
Multiple branches result when a nuclide species occurs more than once in P. LAURA chooses the first such instance each time it occurs and explores this path until it terminates. Then alternate paths from the most recent branching are explored to completion, and so on. Thus, each nuclide species in x having an initial concentration results in a tree-structure reaction network whose branches are explored one at a time, with the equation (2) type contribution being added to the affected nuclides in x. The flowchart for the activity subprogram is depicted in Fig. 3.
Reactor outage LAURA accounts for the periodic shutdown of a reactor for refueling. During the input phase, the shutdown and restart times are read into vectors RS and RG, respectively. Thus, RS(1) is the time of the first outage and RG(I) is the time the first outage ends. If a new time value T occurs after RS(1), control is passed to the activity subroutine to obtain x at RS(1). If a time value T occurs between RS(1) and RG(I), control is again passed to the activity subroutine with x at RS(1) for initial concentrations, neutron flux set to zero and the time value of T - RS(1). The result is x evaluated at time T. If T occurs after RG(1), x is calculated for RG(1) - RS(1) with zero flux to give concentrations at RG(I), which are used as the initial concentrations in the activity routine with time value of T - RG(1) and current flux value. The same procedure is used for any numbers of outages occurring before time T. Since LAURA always uses the initial (T = 0) concentrations given by the user (as opposed to using the concentrations left over from the previous time point), the activity routine must be called for each
N u m e r i c a l e v a l u a t i o n o f r a d i o n u c l i d e s p r o d u c t i o n II
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11 11 / I 12 12
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Fig. 5. P r o d u c t i o n o f t"sW f r o m t ~ W w i t h b y - p r o d u c t s R e a n d O s isotopes: (a) r e a c t i o n d i a g r a m ; (b) r e a c t i o n n e t w o r k s t e m m i n g f r o m the t ~ W t a r g e t . N u m e r i c a l a s s i g n m e n t s , n e t w o r k integers, a r e given in p a r e n t h e s e s in (a).
Table 1. An example of input files for neutron irradiation of a ~ W target for production of tuW and byproducts, Os and Re radionuclides" Cross-section (barn) Network integer
Decay-constant (s-')
Reaction
Nuclide
Atomic wt. (g mol-~)
Thermal
Resonance
Effectiveb
Products
Type ~
1 2
m+W n7W
185.9544 186.9572
-0.8056 x 10-s
37+9 64
485 2760
5,730 x 101 1.744 x 102
3
~gBW
187.9585
0.116 x l0 -6
2
4 5
mTRe IsgW
186.9572 188.9619
-0.1050 x 10-:
76.4 2
6
rare
187.9581
0.1139 x 10-4
2
0
2
7
189Re
188.9593
0,8738 x 10-~
2
0
2
8 9 10 11 12
mOs ~9°R+ I~Os tg°Os ~9tOs
187.9559 189.9619 188.9582 189.9585 190.9630
-0.6418 x 10-+ --0.52 x 10-+
4.7 0 25
152 0 674
1.07 x 102 0 5.196 x 102
1 1 2 1 2 1 1 2 l 2 1 2 1 2 1
38.3
0
3.830 x 102
laTw maW tgTRe ~V mRe raRe ~°W ~Re t~Re mSOs Sg°Re I~Os tSgOs 19°Os ~Os ImOs ~9:Os mlr
13.1
"100% enriched ~ W target. bEffective cross-section = thermal + (1/25) × (resonance). °Reaction type = (1) for neutron capture, (2) for spontaneous decay.
0 300 0
30
2 8,840 x 102 2
1.430 x 102
1
I 2
Saed Mirzadeh and Phillip Walsh
388
Table 2. Reaction vectors for neutron irradiation of 100% enriched ~W target for production of ~ssW and byproducts, Os and Re radionuclides Reaction array (j)
Parent (P(j))
I
I
2 3 4 5 6 7 8 9 10 11 12 13 14 15
2 2 3 3 4 5 6 6 7 7 8 9 10 11
Reaction type (HOW(/'))
2 2 2 2 2
Daughter (DO'))
Percent (c~)~))
2 3 4 5 6 6 7 7 8 9 10 1o 11 11 12
ffSIGMA(I) ffSIGMA(2) LAMBDA(2) J'SIGMA(3) LAMBDA(3) f'SIGMA(4) LAMBDA(5) f'SIGMA(6) LAMBDA(6) f'SIGMA(7) LAMBDA(7) f'SIGMA(8) LAMBDA(9) f'SIGMA(10) f*SIGMA(I 1)
In this example only '"~W has an initial concentration, N~. The reaction diagram for this case is shown in Fig. 5a. P(j) and D(j) are the network integers given in the first column of Table 1. Reaction type 1 refers to neutron capture, and type 2 refers to spontaneous decay.
Table 3. Input files for neutron irradiation of natural Xe target for production of ~251 Cross-section (barn) Abundance (%)
Nuclide '24Xe '25Xe '251 i261 '26Xe mXe ~271 uq
0.096 ---0.090 --
r
I
i
I
0 1.130 x 1.333 x 6.170x 0 2.203 x
10 -5 10 -7 10 ? 10 -7
--
0
--
4.623 x 10 -4
Thermal 1.28 x 5.60 x 8.94 x 5.96x 4 0 6.2 0
Resonance
102 103 102 103
3.600 x 103 0 1.373 x 104 4.060x 104 3.8 x l0 t 0 1.47 x 102 0
Feedback and alpha decay T h e s o l u t i o n for the c o n c e n t r a t i o n o f the n t h nuclide w i t h feedback o n the ith (n > i) nuclide by the k t h (k > i) nuclide is n--I
N,(t) = N Ol~ c:tD,(x, . . . . . x.) j=l
+ NO I-I cjt
s h u t d o w n a n d r e s t a r t o c c u r r i n g before a specific time value to o b t a i n the c o n c e n t r a t i o n s at t h a t time. T h e flow c h a r t f o r this section is s h o w n in Fig. 4.
60
Decay-constant ). ( s - ~ )
j=~
c,'t
I=IL
'H'cmtT
,.=i
d
× o.+,(~_,+,)(x, . . . . . x . . { x , } , ..... 1.0
i
i
I
r
I
{x,},)
J
(b) "~ 0
0.8
40 0.6
OCt oO
'7
~
0.4
~
0.2
20
0.3
i
I
i
I
i
0.0
i
I
i
I
i
0.8
'
I
i
I
I
(d) ,~
0.6
0.2 0.4 O 0.1 0
O b~
t
0.0 0
I
200
i
I
400
i
0.2
i
0.0 600
0
I
200
i
I
i
400
Fig. 6. Production of tungsten-188 and by-products rhenium and osmium from tungsten-186 as a function of irradiation time: (a) ~88W; (b) total tungsten; (c) total rhenium; and (d) total osmium. Target 1.0 g of ~86W (100% enriched), q~th= 2.0 X 10~s n S-~ crn -2, th/epi = 25.
600
Numerical evaluation of radionuclides production II where xj = A d and c', refers to the feedback constant from species k to species i (Raykin and Shlyakhter, 1989). The first term is the 'unperturbed' l = 0 solution (feedback ignored). Each power of l accounts for an additional cycle along the feedback loop, so that each term is actually the solution to a linear chain like equation (2) with n + I(k - i + 1) nuclides. The analytical solution is reached as 1 approaches infinity. Since the depletion function discussed in Part I is valid for cases where some depletion constants are identical, the structure of the activity subroutine discussed above need not be modified to treat the feedback effect. In the input phase, LAURA can detect repeated nuclide species as it constructs the network. The nuclide is not given an additional position in the concentration vector, but the reaction arrays are built in the same manner as before until the user-specified linear chain limit is reached. Thus a certain number of feedback loops are carried in the reaction network, and some of the resulting linear chains have network integers which repeat at periodic intervals. As pointed out by Raykin and Shlyakhter (1989), a feedback contribution is usually a second-order effect and, in practice, only one feedback loop needs to be considered for a given chain. However, it is also pointed out that for longer time intervals more cycles may be needed as the bulk
389
of the initial concentration propagates down the chain.
Illustrative Examples The following examples illustrate the main feature of LAURA: Tungsten- 188 production
'88W is produced by double neutron capture of ~W. It is useful to predict production rates for '88W as well as for radioactive by-products such as radioisotopes of rhenium and osmium. Neutron cross-sections were taken from Neutron Cross Sections, Vol. 1 (Mughabghab et al., 1984). For this case, production of metastable states and hence of isomeric transitions is ignored, so that all isotopes represent ground states. Also, for the sake of simplicity, the partial and total cross-sections for parent nuclides are set equal in this example. A value of 2 barns was used for unknown cross-sections. The reaction diagram for this example is shown in Fig. 5a and the corresponding differential equations, equation (1), are: 1) d N i / d t = - A I N t 2) d N J d t = (a,dp)N, - A2N2
(a)
ol,)
o 2)
.. 126I(4)
t~Slo)
I
I
2
2
8
l 127I(7)
D 1:~|(8)
N(6%)
/\ 3 5
/ \
4
7
,_
I
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6
a4@
[~4(54%)
1"
5
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" 125i~2 ) 03)
Co)
127Xe . (6)
,. 126Xe(5)
l~Xe(t )
4
6
/\
\
7 5
I 8
\ 6
7
\ 8
\ 7 \
8 XENON-124
XENON-126
Fig. 7. Production of ~5I from natural xenon target: (a) reaction diagram; (b) reaction networks.
390
Saed Mirzadeh and Phillip Walsh I
'
I
I
I
18
'
~
'
I
I
'
I
[
16 2
~
102 14
E: •¢ 12
101
N
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,
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,
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~
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10-2~ 0
,
I
I~, 20
'
t
'
I 40
,
t
'
I
'
J
40
i
60
80
100
TIME (d)
i . t -6, 100 60 80
TIME (d)
Fig. 8. Production of ~25Ifrom natural xenon target as a function of time: (a) activity of ~25I;(b) specific activity of ~2~I; and (c) total radionuclide contamination. Target = 1 g of natural Xe, T~, = 28 d, Td~y = 56 d. Neutron flux (n s -~ cm-:): (1) 5 x 10~4, (2) 1 x 10~4, (3) 5 x 1013, (4) 1 x 10~3, (5) 5 x 10~2, and (6) 1 x 1012.
3) dN3/dt = (a2¢)N2 - A3N3 4) dN4/dt = 22N2 - A,N4 5) dNs/dt = (a3¢)N3 - AsN5 6) dN6/dt = (a,O)N4 + 23N3 - A6N6 7) dNr/dt = (a6¢)N6 + 25N5 - A7N7 8) dNs/dt = 26N6 - AsN8 9) dN9/dt = (0"7¢)N7 - A9N9 10) dN~o/dt = (trs¢)N8 + 27N7 - A~oNto l l) dN~/dt = (a~odP)N~o+ 29N9 - AHNH 12) dN~2/dt = (al~¢)N. -- A~N~2 where At = 2~ + a i ¢ . The nuclide file data are shown in Table 1. The integer identifiers are the network integers assigned by the program. For the target atom, the user enters " W 186", which becomes species 1, or Nz (the 1st position in x vector). The ~s6W capture product, J87W, is N2 and the first reaction reads "1, 1, 2" for P(1), H O W ( I ) , and D(I).
Tungsten-187 has a capture product and a decay product, N3 and N4. Reactions 2 and 3 are "2, 1, 3" and "2, 2, 4", respectively. The network is completed by continuing this process. Arbitrarily, the network terminates at mOs, 19~Ir and tsgW. This leaves twelve nuclides in the network array. The reaction vectors are shown in Table 2. In the P E R C E N T column, LAMBDA = In 2/t~/: and the cross-sections are replaced by their apparent crosssections, S I G M A , in this case obtained by dividing the resonance cross-section by the thermal to epithermal neutron ratio and adding this to the total cross-section. The thermal to epithermal ratio is 25 in this example. F o r this problem, only ~s6W is treated as a target for each time interval. Figure 5 shows the tree-structured diagram the activity subroutine uses to calculate x. In this case, there are eight independent linear chains, all of which end at nuclide 12. Thus, ~9~Os will have eight contributions from equation (2) (with n = 8) added to its position in x for a given time point. .To further illustrate, let c,~represent the constant of formation o f speciesj from species i and consider the
Numerical evaluation of radionuclides production II
(a)
391
248Cm(1)a!~ ~.27m(2) Y
--%*
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o4t
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. 252Cf(9 )
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3 /\ 45
3 /\ 45
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6
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8
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9
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1
1
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2
2
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3 /~ 45
3 /\ 45
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Fig. 9. Production of 252Cffrom 24~Cmtarget: (a) reaction diagram; (b) reaction networks.
392
Sa¢d Mirzadeh and Phillip Walsh 0
:.e
X
X x
XX
X X X X X
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Numerical evaluation of radionuclides production II n = 4 reaction in Fig. 5b. The following modifications to x result for each time point:
Ns(t) = Ns(t) + N°~c,2co3qsD4(A,, A2, A3, As)(t) N6(t) = N6(t) + N~tc,2c23c36D,(A,, A2, A3, A,)(t) Ns(t) = Ns(t) + N?cnc24c~D,(A,, A2, A4, A6)(t). The program was run with an input of 1.0 g of t86W (100% enriched), a neutron flux of 2 x 10~Sn s- ' c m - 2, a thermal to epithermal ratio of 25 and a total irradiation time of 600 days. The purpose of the long irradiation time was to illustrate the 'cascade' of atoms through the network. The results are shown in Fig. 6.
Iodine-125 production In this example, LAURA was used for the optimization of '251 production. This is the only example for which an earlier theoretical prediction was available (Martinho et al., 1984) and, in order to make a direct comparison, the input data were those used earlier by Martinho et al. These data, together with additional information for this case, are summarized in Table 3. The reactor shutdown feature of LAURA was also demonstrated in this example. The reaction network and reaction diagram are
1.6xlOq
'
I
'
I
'
shown in Fig. 7. The problem involves a 28 d irradiation of a 1 g natural xenon target, followed by a 56 d post-irradiation decay time. The recreated graphs are shown in Fig. 8. The activity of nq (mCi/g of Xe) as a function of time (up to 28 d irradiation followed by a 56 d decay period) is depicted in Fig. 8a. The specific activity of nsI and level of radionuclidic contamination are shown in Fig. 8b and 8c, respectively. Radionuclidic contamination is defined as the ratio of n6I + nq to nsI. Computations were performed for six neutron fluxes ranging from 1 x 10'2 to 5 x 10~4n s - ' cm -2, curves 1-6. For the sake of comparison, the contribution of the resonance integral to the overall reaction rates was neglected. Our results compare well with earlier computations for the time period, for which Martinho et al. (1984) has presented data.
Californium-252 production The following two examples illustrate the feedback effects involved in the production of 252Cf from an 243Am target. A sub-section of the reaction diagram for this problem, starting with 2~Cm and ending with 252Cf, is shown in Fig. 9a. There are a number of Qt-decay feedback loops in this network, but we have only considered the ~-decay of 252Cf to 245Cm in this
I
I
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9
I
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,
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,
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TIME Cv) Fig. 10. Production of 252Cf (a) from 2415Cm target as a function of time with 2~2CftomCm ~-decayfeedback and without feedback. Curves (b) and (c) shows the levels of 249Cfand depletion of 2~Cm, respectively. Target: 10 mg of 2'SCm (100%), T~,= T~y = 4 y, 4,,h= 2.0 x 10'~ n s -~ cm -2, th/epi = 0.
394
Saed Mirzadeh and Phillip Walsh
r~ 4 0xl0"3 ~
arawa,,
"°~
~ 8.0X10.4
""_5
Fig. 11. Production of 25:Cf(a) from 243Amtarget as a function of time and neutron flux. All the at-decay feedbacks were considered in this case. Curve (b) shows the levels of the intermediate radionuclide, 249Cf, under the same conditions. Target: 15 mg of 243Am(100%), T~rr= Tde=y= 4 y, ~b,h= 2.0 X 10 ~s n S- ~cm - 2, th/epi = 25.
illustrative example. Figure 9b shows the reaction network stemming from the 24SCm target during the irradiation. The input data are summarized in Table 4. The computations were performed assuming a target of 10 mg of 2~Cm, zero thermal to epithermal neutron ratio, and a neutron flux of 2 × 10~Sn s -j cm -2. The irradiation time was four years, followed by a four-year decay period. Figure 10 shows the results of computation with and without the feedback contribution for the same problem. As seen, the contribution from feedback of 252Cf to 2~Cm to the overall yield of 252Cfis insignificant and the two curves are not resolved (Fig. 10a). The yield of the intermediate radionuclide, 249Cf, and the depletion of 248Cm are shown in Fig. 10b and 10c respectively. The effect of ct-feedback is again very small, but the two curves are resolved. The linear chain limit was set to 25 nuclides in this example. The last example is the full problem of production of 252Cf from 243Am, including all major feedback loops. In this test case, 15 mg of 243Am was irradiated under the same conditions as above, except that multiple neutron fluxes are used and thermal to epithermal ratio is set to 25. The input data are also given in Table 4. The 3-D plots of yields of 249Cfand 252Cf as a function of time (4 y irradiation and 4 y decay) and neutron flux are shown in Fig. 11. In conclusion, this paper describes the general structure of the code L A U R A for calculation of the production rate of any member of a network of nuclei undergoing spontaneous decay and/or induced neutron transformation in a nuclear reactor. The theoretical bases for the development of L A U R A were discussed in Part I. In particular, in Part I, we described how an expression based on the Rubinson
(1949) approach is used to evaluate the depletion function. In this paper (Part II), we describe the full simulation of radionuclide production including the decomposition of a reaction network into independent linear chains, provisions for periodic reactor shutdown and restart, and implementation of an approximate solution given by Raykin and Shlyakhter (1989) to account for the effect of feedback due to alpha decay. Several illustrative examples, production of ~88W, J25I and 252Cf in a reactor, are also given. In each case, graphic representations are employed to illustrate the construction of the reaction network by LAURA.
Acknowledgements--The authors wish to thank students Laura D. McGinn and Donald L. Marsh for their valuable assistance in performing part of this work, and acknowledge their support by the Student Research Participation and Science and Engineering Research Programs. Phillip Walsh was supported by the Professional Internship Program. These programs were administered by the Oak Ridge Institute for Science and Education in collaboration with the ORNL Office of University and Science Education for the US Department of Energy. The authors also wish to acknowledge K. S. Brown for technical editing and formatting, Dr Russ Knapp, Jr, and C. W. Alexander for reviewing the article. This research was supported by the Office of Health and Environmental Research, US Department of Energy, under contract DE-AC05960R22464 with Lockheed Martin Energy Research Corporation.
References Bell, M. J. (1973) ORIGEN--The ORNL isotope generation and depletion code. Report ORNL-4628, Oak Ridge National Laboratory, Oak Ridge, TN.
Numerical evaluation of radionuclides production II Martinho, E., Anjos Neves, M. and Carmo, F. (1984) '2~I production: neutron irradiation planning. Int. J. Appl. Radiat. lsot. 35, 933. Mughabghab, S. F., Divadeenam, M. and Holden, N.E. (1984) Neutron Cross Sections, Vol. 1. Academic Press, New York.
395
Raykin, M. S. and Shlyakhter, A. I. (1989) Solution of nuclide burnup equations using transition probabilities. Nucl. Sci. Eng. 102, 54. Rubinson, W. (1949) The equations of radioactive transformation in a neutron flux. J. Chem. Phys. 17, 6. Siewers, H. (1978). Atomkernenergie 27, 30.
Pergamon PII: S0969-8043(97)00288-1
© 1998ElsevierScienceLtd. All rights reserved Printedin Great Britain 0969-8043/98 $19.00+ 0.00
Numerical Evaluation of the Production of Radionuclides in a Nuclear Reactor (Part II) S A E D M I R Z A D E H *~ a n d P H I L L I P W A L S H 2 ~Nuclear Medicine Group, Life Sciences Division, Oak Ridge National Laboratory (ORNL), Oak Ridge, TN 37831-6229, USA 2Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803, U.S.A. (Received 21 April 1997; accepted 30 June 1997)
A computer program called LAURA has been developed to predict the production rates of any member of a nuclei network undergoing spontaneous decay and/or induced neutron transformation in a nuclear reactor. The theoretical bases for the development of LAURA were discussed in Part I. In particular, in Part I, we described how an expression based on the Rubinson (1949) approach is used to evaluate the depletion function. In this paper (Part II), we describe the full simulation of radionuclide production including the decomposition of a reaction network into independent linear chains, provisions for periodic reactor shutdown and restart, and implementation of an approximate solution given by Raykin and Shlyakhter (1989) to account for the effect of feedback due to alpha decay. Also included are some examples which demonstrate possible uses for LAURA. © 1998 Elsevier ScienceLtd. All rights reserved
Introduction
solution governed by the total depletion constants. In this form, the problem associated with singularity (arising from identical depletion constants) has been eliminated. For a specific problem, LAURA decomposes (1) into independent linear chains of the form (2) and for each time step it passes the depletion constants and current time value to a depletion factor subroutine. The depletion function has the following format (see Part I for the details):
An m-member reaction network of nuclei undergoing both spontaneous decay and nuclear transmutations is described in matrix form by (1)
dx/dt = Ax
where x is the concentration vector and A is the m x m reaction matrix (Raykin and Shlyakhter, 1989). It is known that equation (1) can be broken up into independent linear chains and treated as several separate problems of the type CI
N,
c2
, N2
c3
, N3
cn- t
,'"Nn_,
Dn(A~ . . . . . An)(t) = ~ MULT(n, i)t ~'~' ~e- a,,, (3)
cn
, Nn
,
where cn_ t is the formation constant 0.~- ~or ¢ G - z) of the nth species from (n - 1)th species of nuclei and is independent of time or the nuclide concentrations. The exact solutions to the corresponding differential equations are given by
o-,)
N n ( t ) = N ° C ~ = l c i " h n ( A 1. . . . .
A~)(t)
(2)
where Dn represents a depletion factor containing the time-dependent variables, and is the part of the *To whom correspondence should be addressed. E-mail: [email protected].
where each term in the summation is associated with the corresponding nuclide species, and where m ( n , i) is the number of depletion constants Ak such that i < k < n, Ak = A~ and m(n, n) = O. The general algorithm for obtaining MULT and m for the (n + 1)th depletion factor is given in Part I. In this paper we describe the general structure of the code LAURA, including how equation (1) is constructed and broken into independent linear chains and how the feedback effect is handled. Several illustrative examples, including the production of tssW, R5I and 252Cfin a reactor, are also given. In each case, graphic representations are employed to illustrate the construction of the reaction network by LAURA.
383
384
Saed Mirzadeh and Phillip Walsh
[ lpu'r I ACTIVITYSUBROUTINE, CALCULATECONCENTRATION VECTOR AT T, F
oiiv i- I Fig. 1. The general flow chart of LAURA.
Program Description
The general flowchart of L A U R A is shown in Fig. 1. Each section in the flowchart is briefly described. For each problem, L A U R A expects a simple input file containing all the necessary information such as neutron flux, irradiation time, etc. The input file, which is provided by the user, can be easily edited and tailored to a variety of specific problems to study the effect of each variable. This feature distinguishes L A U R A from other available codes that operate on large data files (e.g. code ORIGEN, Bell, 1973). In addition, provision for calculations for various irradiation times and fluxes, as well as the periodic shutdown of the reactor flux during refueling, have been provided in LAURA. L A U R A is written in F O R T R A N and is compiled on both VAX/VMS and MS/DOS operating systems.
so that specifying a target is all that is required from the user. Of course, there remains some limit on the actual length of a given reaction chain (see Part I). Currently we have only limited files for isolated sections of the Table of Nuclides. For multiple fluxes and time points, the irradiation times and fluxes are stored in the input file created specifically for each problem. The irradiation conditions include the minimum and maximum irradiation times and the number of intermediate time points for which the concentrations are desired, the minimum and maximum neutron flux values and the number of intermediate flux values, and the specific times of expected reactor outages and restarts. The link-list structure of the database means that each nuclide having an initial concentration must be specified by the user as a target (multiple targets are possible). At run time the user follows prompts to enter the number of target nuclides, then the filename and initial concentration for each target nuclide, the file containing the irradiation specifics (the input file), and finally the output file specifications. As the reaction network is built, the program assigns each nuclide a position in the concentration vector x, which serves as the nuclide's identification number or 'network integer'. The initial concentration of the target nuclide is stored in the first position of vector x. The first line of the target file is stored in the first positions of four vectors: S I G M A X (total crosssection), SIGMAI (total resonance integral), M A S S (atomic mass), and HALF-LIFE (half-life). The
Input phase The program L A U R A uses an input file containing the specifics of the particular problem. This information consists of the neutron flux values, irradiation times, reactor shutdowns, etc. The program prompts the user to enter the target nuclide symbols and constructs the reaction network, equation (1). The nuclide-specific reaction data are stored in a database. Given a target nuclide, L A U R A constructs a reaction network by following the allowed transformation routes linking the nuclides in the network. This database would presumably not be modified between problems, except to update when more current reaction data become available. The reaction data for a particular isotope are stored in a short file containing the isotope's total cross-section, total resonance integral, atomic weight, half-life, number of lst-order decay products, and number of lst-order neutron capture products. The information for each lst-order decay product includes the product's file name and the branching ratio leading to the decay product. For each l st-order neutron capture product, the required information consists of the product's file name and partial cross-section and resonance integral leading to its formation. Each nuclide in the database has a similar file so that a linked-list data structure is formed. In principle, the entire Table of Nuclides could be mapped in this way,
PEST OF PROGRAM
__~
NEWFLUX VALUEF
VALUET
t
~
~ T E NUCLIDE CONCI~TRATIONS AT T AND F
1 ATT, F
[
Fig. 2. Flow chart of the flux-time loop.
Numerical evaluation of radionuclides production II product files are opened and given successive positions in these vectors, then their product's files are opened, and so on. These four vectors, together with the concentration vector, are called the 'network array', A particular chain will terminate if there are no daughter products specified in a nuclide file. Also, a chain may be forced to terminate if the user-specified linear chain limit is reached. Additional targets are treated in a similar manner, with new nuclide species being assigned unused positions in x. If a target species already appears in x, its concentration is placed in the corresponding position and no further action is taken. The position a nuclide holds in the concentration vector will be used to identify that nuclide throughout the program. The reaction network is generated as the program constructs the concentration vector. Each reaction is stored (in the order it is encountered) in four vectors: P (the network integer for the parent nuclide), D (the network integer for capture product, or daughter nuclide), H O W (an integer which distinguishes a natural decay from neutron-induced reaction), and
I TIME,FLUXVALUES
SENTTO SUBROUTINE ]
[
PROGRAM
PERCENT (branching ratio, indicating fraction of transformation, or decay, of P to D).
Time-flux loops L A U R A can give the concentration vector for multiple time and flux values. The structure is simple---an outer flux loop picks up a flux between the minimum and maximum flux values (specified in the input file), and the inner time loop runs over all the time values between the minimum and maximum values. Control is passed to the activity subroutine for each time and flux value. The last task in the inner loop is to write the concentration vector and other information for that time and flux to an output file. The output file can be imported to a graphic routine such as Jandell's Sigma Plot for visual presentation. A flowchart of the time-flux loop is shown in Fig. 2.
Activity subroutine The activity subroutine uses the reaction vectors to calculate the concentration vector for a given time and flux. The depletion constants for all the nuclei in the network are calculated from At = L A M B D A ( i ) + f x SIGMAX(i), where f is the current flux value and L A M B D A is the decay constant. SIGMAX(i) can be replaced by the apparent capture cross-section which includes the contribution of the epithermal neutrons to the total reaction rates. The activity subroutine uses the initial concentrations provided by the user at the beginning of the simulation for every time step, except when the reactor is shut down (see Section 2.4). Any nuclide in the network array having an initial concentration has the term
Ni(t) = N°e- Air
I Fo~v FLAG= I
~
a=n+ 1 el(n) = DAUGRTER IN'I~GF.R, eclta t) = PERCENT
385
(4)
added to its position in x. If the network integer of this species also appears in P, then this species contributes to the concentrations of other nuclides in x and it is the starting point of a reaction tree. The effect on each subsequent nuclide is calculated one at a time. The current parent nuclide, say N , is relabeled as N~. Suppose the nuclide's network integer occurs in the j t h position of P. The integer at H O W ( j ) determines whether ct is simply PERCENT(j), for a decay process, or f x PERCENT(J) for a neutron capture process. The network integer in D(J) is the daughter product in the j t h reaction and is relabeled as Nv L A U R A performs a function call D2(A, A2)(t) and the term
ii-|
Nct(a)(t) = NIO~I gkDa(ACI(I), ..., Aca0~
Fig. 3. Flow chart of the activity subroutine. Time and flux values are sent to subprogram for a network of m nuclides. A flag marks the position in the P where a nuclide integer is found so that each branch is computed only once.
N~(t) = ctN~tD2(A,, A2)(t) is added to the daughter nuclide's position in x. If the daughter's network integer appears in P, the chain continues and the corresponding PERCENT value
386
Saed Mirzadeh and Phillip Walsh
I RESTOF PROGRAM RESETINITIAL CONCENTRA11ONSTO VALUF.SAT T-- 0 ]~ NEWTIME VALUE,T
-q
]
SHUTDOWN]
NEW VALUE,P.S
CONCENTRATIONSATF, T I
CONCENTRATIONS
ATF,RS
I
NEW RESTART 1 VALUE,RG
co cErerR^TioNs I ^rP-- 0. r
I P -SrOF [ I
I
ATF=0, TsRG Fig. 4. Flowchart of the reactor shutdown subroutine.
becomes c2 while the new daughter nuclide is relabeled N3, and the term
N3(t) = c,c2N°D3(A,, A2, A3)(t) is added to the position in x corresponding to N3, and so on. Equation (2) is used for a general linear chain of length n. A linear chain terminates when the last nuclide's network integer does not appear in
P.
Multiple branches result when a nuclide species occurs more than once in P. LAURA chooses the first such instance each time it occurs and explores this path until it terminates. Then alternate paths from the most recent branching are explored to completion, and so on. Thus, each nuclide species in x having an initial concentration results in a tree-structure reaction network whose branches are explored one at a time, with the equation (2) type contribution being added to the affected nuclides in x. The flowchart for the activity subprogram is depicted in Fig. 3.
Reactor outage LAURA accounts for the periodic shutdown of a reactor for refueling. During the input phase, the shutdown and restart times are read into vectors RS and RG, respectively. Thus, RS(1) is the time of the first outage and RG(I) is the time the first outage ends. If a new time value T occurs after RS(1), control is passed to the activity subroutine to obtain x at RS(1). If a time value T occurs between RS(1) and RG(I), control is again passed to the activity subroutine with x at RS(1) for initial concentrations, neutron flux set to zero and the time value of T - RS(1). The result is x evaluated at time T. If T occurs after RG(1), x is calculated for RG(1) - RS(1) with zero flux to give concentrations at RG(I), which are used as the initial concentrations in the activity routine with time value of T - RG(1) and current flux value. The same procedure is used for any numbers of outages occurring before time T. Since LAURA always uses the initial (T = 0) concentrations given by the user (as opposed to using the concentrations left over from the previous time point), the activity routine must be called for each
N u m e r i c a l e v a l u a t i o n o f r a d i o n u c l i d e s p r o d u c t i o n II
o1+
(a)
P tST[W~2)
o2+
387
o3*
o+,:I,
188Re(6)
lSTRe(4)
o6+
07+
~ Ig9Re~)
•"- t~),.e(9)
/ 1agog(g)
o12+
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,189Os0o )
- 191Os(12)
tg°Os(tt)
I
kl2
(b)
l
1
2
2
I
/\
3
4
3
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4 5 6
I 8
12
6
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7
8 7
8 7
/\
/ /\
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10 9
10 10 9
10 10 9
/ II
7
6
5
\ ll
\
I / II 11
/
/
I /
\ II
\
12 12 12
\
11 11 / I 12 12
12
11 12
Fig. 5. P r o d u c t i o n o f t"sW f r o m t ~ W w i t h b y - p r o d u c t s R e a n d O s isotopes: (a) r e a c t i o n d i a g r a m ; (b) r e a c t i o n n e t w o r k s t e m m i n g f r o m the t ~ W t a r g e t . N u m e r i c a l a s s i g n m e n t s , n e t w o r k integers, a r e given in p a r e n t h e s e s in (a).
Table 1. An example of input files for neutron irradiation of a ~ W target for production of tuW and byproducts, Os and Re radionuclides" Cross-section (barn) Network integer
Decay-constant (s-')
Reaction
Nuclide
Atomic wt. (g mol-~)
Thermal
Resonance
Effectiveb
Products
Type ~
1 2
m+W n7W
185.9544 186.9572
-0.8056 x 10-s
37+9 64
485 2760
5,730 x 101 1.744 x 102
3
~gBW
187.9585
0.116 x l0 -6
2
4 5
mTRe IsgW
186.9572 188.9619
-0.1050 x 10-:
76.4 2
6
rare
187.9581
0.1139 x 10-4
2
0
2
7
189Re
188.9593
0,8738 x 10-~
2
0
2
8 9 10 11 12
mOs ~9°R+ I~Os tg°Os ~9tOs
187.9559 189.9619 188.9582 189.9585 190.9630
-0.6418 x 10-+ --0.52 x 10-+
4.7 0 25
152 0 674
1.07 x 102 0 5.196 x 102
1 1 2 1 2 1 1 2 l 2 1 2 1 2 1
38.3
0
3.830 x 102
laTw maW tgTRe ~V mRe raRe ~°W ~Re t~Re mSOs Sg°Re I~Os tSgOs 19°Os ~Os ImOs ~9:Os mlr
13.1
"100% enriched ~ W target. bEffective cross-section = thermal + (1/25) × (resonance). °Reaction type = (1) for neutron capture, (2) for spontaneous decay.
0 300 0
30
2 8,840 x 102 2
1.430 x 102
1
I 2
Saed Mirzadeh and Phillip Walsh
388
Table 2. Reaction vectors for neutron irradiation of 100% enriched ~W target for production of ~ssW and byproducts, Os and Re radionuclides Reaction array (j)
Parent (P(j))
I
I
2 3 4 5 6 7 8 9 10 11 12 13 14 15
2 2 3 3 4 5 6 6 7 7 8 9 10 11
Reaction type (HOW(/'))
2 2 2 2 2
Daughter (DO'))
Percent (c~)~))
2 3 4 5 6 6 7 7 8 9 10 1o 11 11 12
ffSIGMA(I) ffSIGMA(2) LAMBDA(2) J'SIGMA(3) LAMBDA(3) f'SIGMA(4) LAMBDA(5) f'SIGMA(6) LAMBDA(6) f'SIGMA(7) LAMBDA(7) f'SIGMA(8) LAMBDA(9) f'SIGMA(10) f*SIGMA(I 1)
In this example only '"~W has an initial concentration, N~. The reaction diagram for this case is shown in Fig. 5a. P(j) and D(j) are the network integers given in the first column of Table 1. Reaction type 1 refers to neutron capture, and type 2 refers to spontaneous decay.
Table 3. Input files for neutron irradiation of natural Xe target for production of ~251 Cross-section (barn) Abundance (%)
Nuclide '24Xe '25Xe '251 i261 '26Xe mXe ~271 uq
0.096 ---0.090 --
r
I
i
I
0 1.130 x 1.333 x 6.170x 0 2.203 x
10 -5 10 -7 10 ? 10 -7
--
0
--
4.623 x 10 -4
Thermal 1.28 x 5.60 x 8.94 x 5.96x 4 0 6.2 0
Resonance
102 103 102 103
3.600 x 103 0 1.373 x 104 4.060x 104 3.8 x l0 t 0 1.47 x 102 0
Feedback and alpha decay T h e s o l u t i o n for the c o n c e n t r a t i o n o f the n t h nuclide w i t h feedback o n the ith (n > i) nuclide by the k t h (k > i) nuclide is n--I
N,(t) = N Ol~ c:tD,(x, . . . . . x.) j=l
+ NO I-I cjt
s h u t d o w n a n d r e s t a r t o c c u r r i n g before a specific time value to o b t a i n the c o n c e n t r a t i o n s at t h a t time. T h e flow c h a r t f o r this section is s h o w n in Fig. 4.
60
Decay-constant ). ( s - ~ )
j=~
c,'t
I=IL
'H'cmtT
,.=i
d
× o.+,(~_,+,)(x, . . . . . x . . { x , } , ..... 1.0
i
i
I
r
I
{x,},)
J
(b) "~ 0
0.8
40 0.6
OCt oO
'7
~
0.4
~
0.2
20
0.3
i
I
i
I
i
0.0
i
I
i
I
i
0.8
'
I
i
I
I
(d) ,~
0.6
0.2 0.4 O 0.1 0
O b~
t
0.0 0
I
200
i
I
400
i
0.2
i
0.0 600
0
I
200
i
I
i
400
Fig. 6. Production of tungsten-188 and by-products rhenium and osmium from tungsten-186 as a function of irradiation time: (a) ~88W; (b) total tungsten; (c) total rhenium; and (d) total osmium. Target 1.0 g of ~86W (100% enriched), q~th= 2.0 X 10~s n S-~ crn -2, th/epi = 25.
600
Numerical evaluation of radionuclides production II where xj = A d and c', refers to the feedback constant from species k to species i (Raykin and Shlyakhter, 1989). The first term is the 'unperturbed' l = 0 solution (feedback ignored). Each power of l accounts for an additional cycle along the feedback loop, so that each term is actually the solution to a linear chain like equation (2) with n + I(k - i + 1) nuclides. The analytical solution is reached as 1 approaches infinity. Since the depletion function discussed in Part I is valid for cases where some depletion constants are identical, the structure of the activity subroutine discussed above need not be modified to treat the feedback effect. In the input phase, LAURA can detect repeated nuclide species as it constructs the network. The nuclide is not given an additional position in the concentration vector, but the reaction arrays are built in the same manner as before until the user-specified linear chain limit is reached. Thus a certain number of feedback loops are carried in the reaction network, and some of the resulting linear chains have network integers which repeat at periodic intervals. As pointed out by Raykin and Shlyakhter (1989), a feedback contribution is usually a second-order effect and, in practice, only one feedback loop needs to be considered for a given chain. However, it is also pointed out that for longer time intervals more cycles may be needed as the bulk
389
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Illustrative Examples The following examples illustrate the main feature of LAURA: Tungsten- 188 production
'88W is produced by double neutron capture of ~W. It is useful to predict production rates for '88W as well as for radioactive by-products such as radioisotopes of rhenium and osmium. Neutron cross-sections were taken from Neutron Cross Sections, Vol. 1 (Mughabghab et al., 1984). For this case, production of metastable states and hence of isomeric transitions is ignored, so that all isotopes represent ground states. Also, for the sake of simplicity, the partial and total cross-sections for parent nuclides are set equal in this example. A value of 2 barns was used for unknown cross-sections. The reaction diagram for this example is shown in Fig. 5a and the corresponding differential equations, equation (1), are: 1) d N i / d t = - A I N t 2) d N J d t = (a,dp)N, - A2N2
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3) dN3/dt = (a2¢)N2 - A3N3 4) dN4/dt = 22N2 - A,N4 5) dNs/dt = (a3¢)N3 - AsN5 6) dN6/dt = (a,O)N4 + 23N3 - A6N6 7) dNr/dt = (a6¢)N6 + 25N5 - A7N7 8) dNs/dt = 26N6 - AsN8 9) dN9/dt = (0"7¢)N7 - A9N9 10) dN~o/dt = (trs¢)N8 + 27N7 - A~oNto l l) dN~/dt = (a~odP)N~o+ 29N9 - AHNH 12) dN~2/dt = (al~¢)N. -- A~N~2 where At = 2~ + a i ¢ . The nuclide file data are shown in Table 1. The integer identifiers are the network integers assigned by the program. For the target atom, the user enters " W 186", which becomes species 1, or Nz (the 1st position in x vector). The ~s6W capture product, J87W, is N2 and the first reaction reads "1, 1, 2" for P(1), H O W ( I ) , and D(I).
Tungsten-187 has a capture product and a decay product, N3 and N4. Reactions 2 and 3 are "2, 1, 3" and "2, 2, 4", respectively. The network is completed by continuing this process. Arbitrarily, the network terminates at mOs, 19~Ir and tsgW. This leaves twelve nuclides in the network array. The reaction vectors are shown in Table 2. In the P E R C E N T column, LAMBDA = In 2/t~/: and the cross-sections are replaced by their apparent crosssections, S I G M A , in this case obtained by dividing the resonance cross-section by the thermal to epithermal neutron ratio and adding this to the total cross-section. The thermal to epithermal ratio is 25 in this example. F o r this problem, only ~s6W is treated as a target for each time interval. Figure 5 shows the tree-structured diagram the activity subroutine uses to calculate x. In this case, there are eight independent linear chains, all of which end at nuclide 12. Thus, ~9~Os will have eight contributions from equation (2) (with n = 8) added to its position in x for a given time point. .To further illustrate, let c,~represent the constant of formation o f speciesj from species i and consider the
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Ns(t) = Ns(t) + N°~c,2co3qsD4(A,, A2, A3, As)(t) N6(t) = N6(t) + N~tc,2c23c36D,(A,, A2, A3, A,)(t) Ns(t) = Ns(t) + N?cnc24c~D,(A,, A2, A4, A6)(t). The program was run with an input of 1.0 g of t86W (100% enriched), a neutron flux of 2 x 10~Sn s- ' c m - 2, a thermal to epithermal ratio of 25 and a total irradiation time of 600 days. The purpose of the long irradiation time was to illustrate the 'cascade' of atoms through the network. The results are shown in Fig. 6.
Iodine-125 production In this example, LAURA was used for the optimization of '251 production. This is the only example for which an earlier theoretical prediction was available (Martinho et al., 1984) and, in order to make a direct comparison, the input data were those used earlier by Martinho et al. These data, together with additional information for this case, are summarized in Table 3. The reactor shutdown feature of LAURA was also demonstrated in this example. The reaction network and reaction diagram are
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Californium-252 production The following two examples illustrate the feedback effects involved in the production of 252Cf from an 243Am target. A sub-section of the reaction diagram for this problem, starting with 2~Cm and ending with 252Cf, is shown in Fig. 9a. There are a number of Qt-decay feedback loops in this network, but we have only considered the ~-decay of 252Cf to 245Cm in this
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illustrative example. Figure 9b shows the reaction network stemming from the 24SCm target during the irradiation. The input data are summarized in Table 4. The computations were performed assuming a target of 10 mg of 2~Cm, zero thermal to epithermal neutron ratio, and a neutron flux of 2 × 10~Sn s -j cm -2. The irradiation time was four years, followed by a four-year decay period. Figure 10 shows the results of computation with and without the feedback contribution for the same problem. As seen, the contribution from feedback of 252Cf to 2~Cm to the overall yield of 252Cfis insignificant and the two curves are not resolved (Fig. 10a). The yield of the intermediate radionuclide, 249Cf, and the depletion of 248Cm are shown in Fig. 10b and 10c respectively. The effect of ct-feedback is again very small, but the two curves are resolved. The linear chain limit was set to 25 nuclides in this example. The last example is the full problem of production of 252Cf from 243Am, including all major feedback loops. In this test case, 15 mg of 243Am was irradiated under the same conditions as above, except that multiple neutron fluxes are used and thermal to epithermal ratio is set to 25. The input data are also given in Table 4. The 3-D plots of yields of 249Cfand 252Cf as a function of time (4 y irradiation and 4 y decay) and neutron flux are shown in Fig. 11. In conclusion, this paper describes the general structure of the code L A U R A for calculation of the production rate of any member of a network of nuclei undergoing spontaneous decay and/or induced neutron transformation in a nuclear reactor. The theoretical bases for the development of L A U R A were discussed in Part I. In particular, in Part I, we described how an expression based on the Rubinson
(1949) approach is used to evaluate the depletion function. In this paper (Part II), we describe the full simulation of radionuclide production including the decomposition of a reaction network into independent linear chains, provisions for periodic reactor shutdown and restart, and implementation of an approximate solution given by Raykin and Shlyakhter (1989) to account for the effect of feedback due to alpha decay. Several illustrative examples, production of ~88W, J25I and 252Cf in a reactor, are also given. In each case, graphic representations are employed to illustrate the construction of the reaction network by LAURA.
Acknowledgements--The authors wish to thank students Laura D. McGinn and Donald L. Marsh for their valuable assistance in performing part of this work, and acknowledge their support by the Student Research Participation and Science and Engineering Research Programs. Phillip Walsh was supported by the Professional Internship Program. These programs were administered by the Oak Ridge Institute for Science and Education in collaboration with the ORNL Office of University and Science Education for the US Department of Energy. The authors also wish to acknowledge K. S. Brown for technical editing and formatting, Dr Russ Knapp, Jr, and C. W. Alexander for reviewing the article. This research was supported by the Office of Health and Environmental Research, US Department of Energy, under contract DE-AC05960R22464 with Lockheed Martin Energy Research Corporation.
References Bell, M. J. (1973) ORIGEN--The ORNL isotope generation and depletion code. Report ORNL-4628, Oak Ridge National Laboratory, Oak Ridge, TN.
Numerical evaluation of radionuclides production II Martinho, E., Anjos Neves, M. and Carmo, F. (1984) '2~I production: neutron irradiation planning. Int. J. Appl. Radiat. lsot. 35, 933. Mughabghab, S. F., Divadeenam, M. and Holden, N.E. (1984) Neutron Cross Sections, Vol. 1. Academic Press, New York.
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Raykin, M. S. and Shlyakhter, A. I. (1989) Solution of nuclide burnup equations using transition probabilities. Nucl. Sci. Eng. 102, 54. Rubinson, W. (1949) The equations of radioactive transformation in a neutron flux. J. Chem. Phys. 17, 6. Siewers, H. (1978). Atomkernenergie 27, 30.