LA-7483-MS Informal Report (!3 ● ClC-l 4 REPORT COLLECTION REPRODUCTION COPY The Application of a Library of Processed ENDF/B-lV Fission-Product Aggregate Decay Data in the Calculation of Decay-Energy Spectra .- 15 > .- C#) Z > .- L%% LOSALAMOS SCIENTIFIC LABORATORY Post Office Box 1663 Los Alamos. New Mexico 87545
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Decay Data in the Calculation of Decay-Energy Spectra
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LA-7483-MSInformal Report
(!3●ClC-l 4 REPORT COLLECTION
REPRODUCTIONCOPY
The Application of a Library of
Processed ENDF/B-lV Fission-Product Aggregate
Decay Data in the Calculation of
Decay-Energy Spectra
.-
15>
.-C#)
Z>.-
L%%LOSALAMOS SCIENTIFIC LABORATORYPost Office Box 1663 Los Alamos. New Mexico 87545
This rrv.rt was prcswcd as an .ccount 01 work smmsorcdby the Un,ted States GOVCrI)!llC”l. Ne,lh,, the U“,ted SUtcs“or the Umtrd SI.trs Drp. rlmrnt of Enerm’. nor .nY of Ih.lr●mployees. mn ●ny of thmr contr.. t.rs. s. bcontr.ctorx ortheir .mplov. es. makes ..Y wamantv. cxvrcss or tmphrd. orassume. any legal habdtty or responsobd$ty for the accuracy.completeness. ?. usefulness of an. reform. tlon. .IWMr.t.*.product. or Prxess dmclo$ed. or repccwnti that Ifs usc wculdm.t lnfrmse mtvatelv owned twhts.
UN ITEO STATESDEPARTMENT OF ENERGY
CONTRACT W-740 S-ENG. 36
LA-7483-MSInformal Report
UC-34Cissued: September 1978
The Application of a Library of
Processed ENDF/B4V Fission=Product Aggregate
Decay Data in the Calculation of
Decay-Energy Spectra
R. J. LaBauveT. R. EnglandD. C. George
M. G. Stamatelatos*
‘Science “Applications,Inc.,P.O.Box 2351, La Jolla,CA 92037.-
ABOUT THIS REPORT
This official electronic version was created by scanning the best available paper or microfiche copy of the original report at a 300 dpi resolution. Original color illustrations appear as black and white images. For additional information or comments, contact: Library Without Walls Project Los Alamos National Laboratory Research Library Los Alamos, NM 87544 Phone: (505)667-4448 E-mail: [email protected]
THE APPLICATION OF A LIBRARY OF PROCESSED ENDF/B-IVFISSION-PRODUCT AGGREGATE DECAY DATA IN THE CALCULATION
OF DECAY-ENERGY SPECTRA
by
R. J. LaBauve, T. R. England, D. c. George, and M. Go stamatelato~
ABSTRACT
Results from summation calculations by the CINDER-10code and ENDF/B-IV decay, cross-section, and yield data
for fission pulses have been incorporated into an ENDF/B-type format. The organization and content of this basicfine-group source-term library is described. In addition,two codes are described that provide pulse functions asfits to a user-specified multigrouping of the fine-grouplibrary. These can be readily used, essentially as Green’sfunctions, to produce the spectra following any specific
reactor power history. A particular set of fitted betaand gamma spectra having wide utility is described. Ab-sorption effects are incorporated.
I. INTRODUCTION
The ENDF/B-IV fission-product files contain neutron cross sections, decay
constants, decay energies, and other decay data for 824 important fission prod-
ucts . They also contain fission yields for these fission products produced by
one or more fission-neutron energies (14 MeV, fast, and thermal fission) of232Th 233U 235
six important nuclides:u 238U 239PU and 241
9 9 s > 9 Pu . Also, spec-
tral data (i.e., beta end-point energies and intensities, gamma-line energies
and intensities) exist for the most important decay-heat contributors among
the 824 nuclides. In ENDF/B-IV, beta-spectral data exist for 163 fission prod-
ucts, and gamma-spectral data exist for 172 nuclides (nuclides emitting both
beta and gamma radiation are included separately in both types of radiation
counts). The contents of the ENDF/B-IV fission-product file are detailed in
Refs. 1-3 and summarized in Table I.
1
In recent years, great emphasis has been placed on obtaining experimental
and computational information on delayed energy release at short cooling times
for nuclear reactor safety studies of the hypothetical loss-of-coolant-accident
(LOCA). There is, however, also interest in long cooling times. A computer4,5
code system has been developed at the Los Alamos Scientific Laboratory (LASL)
that uses the ENDF/B-IV fission-product data to calculate cumulative delayed
beta and gamma spectra on arbitrary energy grids for arbitrary irradiation his-
tories and cooling times. This code system is shown in Fig. 1.
It can be noted in the figure that the basic ENDF/B-IV fission-product data
library is accessed along three paths. The center path, after the preparation
of an input library, proceeds via the CINDER-10 code. CINDER-10 is the latest
and most versatile version of CINDER, a well-known fission-product and depletion
code. The most recent documentation on CINDER is Ref. 4 (for Version 7), but
the additional features of Version 10 are discussed in Ref. 6. CINDER-10 calcu-
lates fission-product and actinide concentrations, activities, gaseous contents,
energy releases, effective group absorption cross sections, etc. for any fis-
sionable nuclide mixture irradiated in arbitrary neutron fluxes for arbitrary
intervals of time followed by arbitrary cooling times. The neutron cross sec-
tions7’8 used in CINDER-10 are generated by spectrum collapse of multigroup data9
generated with the NJOY code. This input path to CINDER-10 is the lower path
shown in the figure. Spectrum collapse is achieved with the TOAFEW8code.
The ENDF/B-IV data is also accessed by the FPDCYS code along the upper pro-
cessing path shown in Fig. 1. This code is used to generate multigroup beta and
gamma spectra for individual nuclides for which spectral data exist on the
ENDF/B-IV file. FPDCYS incorporates four options for calculating beta spectra
and two options for calculating gamma spectra. The differences among the four
beta-spectrum options are mainly in the ways in which the Fermi function F(Z,W)
is represented and calculated. The first of the two gamma-spectrum options
consists of incorporating the unbroadened lines weighted by their intensities
into an arbitrary number of energy groups. Alternatively, gamma lines are
broadened according to detector resolutions before multigrouping in the second
option.
The output of FPDCYS and the output of CINDER-10 are input to the FPSPEC
code. Actually, only a small portion of the CINDER-10 output is utilized,
namely, fission-product activities and total decay energies at the instant of
time when corresponding spectra are sought. FPSPEC combines the individual
2
TABLE I
SUMMARY OF ENDF/B-IV FISSION-PRODUCT DATA FILE CONTENT
824 Nuclides (total)181 Have differential cross sections180 Have individual ~ and y “lines” (spectral data
consisting of energies and intensities)712 Are unstable and each has an average ~, y, and a
energy and branching fraction10 Yield sets for 6 fissionable nuclides (%10 000 yields)
(%310 000 Data entries required in ENDF/B-IV)
— —.
—. \ \()1.54 cm ——_ ..__
{/4—.-
LFPOCYS
I:SCCY
I!__7KEY
m
0
LIBRARIFX
3~=:a*f=.&-@c(E)
23
[1ST:OADNJOY
&+6 ~NORMAL
~&!~}Nc- ;—ACGRffiATE kNU&IIUMC#TA~ : -—
1+3FITPuIs
4
0;Iu-rur:
FUNCTIONS.
(AN55’!!)STD )
:EEEEI
I
Fig. 1.LASL nuclide processing codes and libraries,
3
spectra from FPDCYS and the nuclide activities from CINDER-10 to generate aggre-
gate fission-product spectra for each irradiation and shutdown time. Note that
CINDER-10 also incorporates a spectral subroutine capable of utilizing the mul-
tigroup data produced by the FPDCYS code. Plots of sample output from FPSPEC
are shown in Figs. 2-5. In these illustrations, the calculated spectra are com-
pared with the LASL experiment cited in Ref. 10.
As indicated above, spectral data are not available for all 824 fission
products in ENDF/B-IV, and missing spectra must be approximately constructed.
This is done for a particular nuclide by assuming that the shape of the beta
(or gamma) spectrum for the nuclide is approximated by the spectrum shape of the
aggregate 181 nuclides from a pulse after a cooling time approximately equal to
the half-life of the nuclide in question. This shape is then normalized to the
average beta- (gamma-) decay energy of the nuclide. Figures 6 and 7, respectively,139
compare the gamma spectra of Cs with those constructed for a hypothetical nu-
clide having the same half-life and average gamma- and beta-decay energies as139CS
The nuclide139
. Cs is a relatively important nuclide in the 0.1-s cooling
time bin for 20 000-h thermal irradiation of235
U. However, it should be noted
that such constructed individual .spect~a are used only in the aggregate.
The libraries used in the CINDER-10 and auxiliary codes FPDCYS and FPSPEC
are extensive and the codes are designed to use the libraries for any specified
irradiation history. However, for many users the scope of application is un-
necessary and aggregate results, rather than the detailed nuclide-by-nuclide
output, are needed. The purpose of this report and the associated codes described
is to eliminate the need for extensive summation code calculations for a wide
range of problems. The summation codes and libraries have been used to produce
multigroup beta and gamma spectra vs time following fission pulses, including
the components of
of applications.
We have used
gregate libraries
which can be used
we have
the spectra due to halogens and noble gases for a wide range
the summation codes and libraries to produce secondary ag-
and pulse functions shown as “additional output” in Fig. 1>
directly or incorporated into neutronics codes. In particular,
● Used the summation codes to produce beta- and gamma-temporal spectra in150 groups following fission pulses for each fuel and fission neutronenergy in ENDF/B-IV and stored the results in formats similar to ENDF/B.
These files delineate the noble gas and halogen spectra. Users canreadily collapse the results to other tnultigroup spectra.
4
MeV
Fig. 2.
Gamma spectrum 5.56-h irradiationof 235
U, 70-s cooling.
MeV
Fig. 4.Gamma spectrum, 5.56-h irradiationof 235
U, 388-s cooling.
MeV
Fig. 3.
Gamma spectrum, 5.56-h irradiation
of 235U 199-s cooling$ .
MeV
??ig. 5Gamma spectrum, 5.56-h irradiationof 235
U, 660-s cooling.
5
1 : ..... lPCS Camma Spwtrum— COnslruckt Spectrum I
NTS = number of cooling time steps given for a particular MT
NP = number of DE,E’ pairs given in a particular TABI record.
Other quantities are defined in ENDF-102. Note that interpolation along cooling-
time steps (the TAB2 records) is always set to zero, meaning that interpolation
is not recommended, and that interpolation is always set to one (histogram) for
the TAB1 records. A sample PEFPYD listing is given in Appendix A.
III. REDUCING AND FITTING THE PEFPYD DATA -- THE FITPULS CODE
In general, the data in the I?EFPYD library are too detailed along the energy
axis and not detailed enough along the cooling-time axis for application to de-
sign problems. The FITPULS code is designed to access the PEFPYD library, col-
lapse the 150 energy-group spectra into few groups (up to 25), and fit the re-
sulting spectra along the cooling-time axis with a linear combination of func-
tions of the type
n
fc(t) =x
a.e_xit1 (MeV/fiss/s) .
i=1
Note that there are two sets of parameters in Eq. (l), namely, the set of a andi
the set of Ai. FITPULS contains options allowing either a least-squares single-
parameter fit, that is, a ff.tof the a’s, given a set of ~’s, or a nonlinear
least-squares two-parameter fit (a simultaneous fit of both the a’s and 1’s).
The first option is described in detail in Ref. 12, and the second uses the non-
linear least-squares STEPIT routine described in Ref. 13.
In both fitting routines, comparisons between calculated and original val-
ues are made for every data point. This, however, is not sufficient to guaran-
tee a good fit,because the function may oscillate wildly between data points. A
subroutine called FINECHK detects such oscillations by calculating the function
on a fine grid and printing out values differing more than 10% from those cal-
(1)
culated using a
this difference
negative values
simple semilog interpolation between points on either side. If
exceeds 100%, the point is additionally flagged by FINECHK; if
occur, the user is warned that a fit has not been achieved.
9
The percentage differences flagged are arbitrary and may be changed by the user.
Also , it is suggested that the user insert a plotting option at this point in
the code so that oscillations in the fit can be inspected visually. The LASL
CDC-7600 version of FITPULS contains the LASL plotting routines that compare
the fine-mesh points calculated in FINECHK with the original data. The FINECHK
routine will also flag those calculated points where slopes are ascending, thus
giving additional indications of possible problems with the functional fit.
Normally, the two–parameter nonlinear fitting routine STEPIT will run to
convergence at minimum chi–square, but an option is in FITPULS to stop the cal-
culation when an input maximum allowed percent deviation (DIFLIM) of the calcu-
lated values from all original values has been achieved. For efficiency of code
operation, it is suggested that the user set DIFLIM high in early passes and
tighten up as desired convergence is approached.
Although.,in principal.a two-parameter fit can be made from scratch, given
a reasonable set of parameters for a particular coarse group structure, a great
saving in total problem running time can be attained by first running single-
parameter fits. The code contains several options for selecting initial A’s,
removing duplicate A’s, making adjustments for resulting negative coefficients
with large values, etc. In fae-t, this and other fitting codes can only be run
effectively uith a rather Large mount of user interaction, as indicated by the
discussion in Sec. IV.
The FITPULS code also has an option for obtaining fitted pulse parameters
from data given for finite-irradiation times (IRAD=l option). This option is
particularly useful for reducing data from a number of different experiments
with different irradiation times to pulses for comparison purposes.
The technique of running FITPULS in the normal mode, for example, IRAD=O,
is provided in the example problem sequence below. A listing of the FITpULS
code is given in Appendix B and input specifications are listed in Table III.
Iv. FITPULS INPUT AND SAMPLE PROBLEM: A USEFUL FITTED SPECTRUM
The input specifications for the FITPULS code are shown in Table III. As
a sample problem, consider a mulfigroup collapse of the PEFPYD data for the-4
gamma spectra of fission products produced by a pulse (10 -s irradiation time)233
of thermal neutrons on U, (MAT1=1260, MF1=80, and MT1=803). Some of the
PEFPYD data used in this example are shown in Appendix
structure used for this problem is shown in Table IV.
10
A. The broad-group
Note that there are
TABLE III
FITPULS INPUT SPECIFICATIONS
Card No. Format Variable Comment
(Input for Subroutine CORSBIN)
1 6111 MAT1 MAT-No of desired fissioning nuclide.
MF1 MF desired (incident energy type).
MT1 MT desired (particle/photon data type).
2 6111 NE
3 6E11.4 EB(I)
(Input for Program FITPULS)
1 1216 NPUN
8A1O
1216
IRAD
NCORS
TITL(I)
IPROB
NTOTER
NPUN
No. of desired broad groups + 1.
Energy bounds in MeV, including lowerand upper bounds. Read low to highenergy.
Set NPUN = 7 here if rebinned datacards wanted; otherwise set to zero.
Set IRAD = O for regular pulse fit,set IRAD = 1 to reduce finite irradi-ation data to pulse.
Set NCORS = O to call subroutineCORSBIN. Set NCORS = 1 for no call,i.e., if input data is not to be re-binned, which is usually the casefor fitting experimental data (IRAD= 1).
80 character title, if TITLE(1) =SELECT subroutine SELECT is calledand this input goes here (see SELECTinput) . If TITLE(1) = DO NOT GO,program stops.
Problem No. Make negative if fitis to be made in segments. See con-ditional input below.
Option to read data from cards.Used for experimental data.See subroutine RUNTOTS.
Flag for punched output. Set equalto 7 if punched output is desired,equal to 20 if punched output notdesired. In general, punched outputis needed for subsequent runs.
11
TABLE III (cent)
Card No. Format Variable
3 (cont.) NSTEP
NFINL
6E12.5
1216
DIFLIM
RUNTIM
TMIN
GXMIN
KKN(I)
--- --- ___ --- --- --- _
FITPULS Conditional Input
--
Comment
Flag to call subroutine DHFIT, whichcalls the routine STEPIT, which per-forms a two-parameter fit. Routineusually not called until a couple ofpasses are made to get a coarse ad-justment of the parameters with a sin-gle fit. Set equal to zero if call toDHFIT is not desired, otherwise set to1. Also note below option for call-ing DHFIT by group.
Flag for option to read all parametersfor all groups from previous problem.Used when striving for final conver-gence. Set equal to 1 to activate,otherwise set equal to zero. See con-ditional input below.
Maximum per cent deviation allowed inSTEPIT. Set high on initial passes andtighten up as desired convergence isapproached.
Running time. Make fraction of secondless than time limit set on controlcard to get punched cards for subse-quent run.
:Minimum cooling time desired. If setto zero, code will choose minimum cool-ing time available on data file.
Maximum cooling time desired. If setto zero, code will choose maximum cool-ing time available on data file.
Minimum allowed value of decay energy.Set so fit is limited to about 15decades.
Flags for calling STEPIT routine (two-parameter fit) by group. If call fora particular group, say Group IG, isdesired, set KKN(IG) = IG. If call isnot desired, set KKN(IG) = zero.
--- -—- --- --- --- --- -
If NFINL = 1, cards ouput from previous problems are read here. These cards arethe a’s and A’s for ali groups,-and they-are in an ENDF-like format. If NFINL =1, no futher input is needed. Note, however, that a’s and A’s for particular
12
TABLE III (cent)
Card NO. Format Variable Comment
groups can be entered in subroutine PULSFIT, distinct from this option in thatthey need not be entered for every group, thus permitting a mixture of options.
IPROB set negative allows the data for the groups to be fitted in several seg-ments.
KKN is set negative for a particular group if a call to subroutine TRMSEE is de–sired. (See code listing in Appendix-B.)
5 1216 NSEG
NS (I)
(Input for subroutine PULSFIT)
1 1216 LWT
NOK
KTRM
IPRT
--- --- --- --- --— --- ---
NOK, KTRM specifications, NOK = O, KTRM =
2 3(11X,E11.4) ALAMDA(K)
Number of segments + 1
Breakpoints of segments
Weight function desired in single pa-rameter fit.
If LWT = O, Weight function = 1
If LWT = 1, Weight function = l/FX
If LWT = 2, Weight function = l/FX2
If LWT = 3, Weight function = l/FX1.5
Flag for parameter selection
Flag for parameter selection.
NOK, KTRM combination determines theoption by which the initial X’s forthe group are selected. See below.
Flag for print option. Set equal to1 for complete print, otherwise setto zero.
--- --- --- --- --- --- -
number of A’s to be read in,
Read in (ALAMDA(K),K = 1, KTRM)
(Note from format that if cards fromprevious run are used, coefficientswill not be read.)
NOK = 1, KTRM = 1, ~’s calculated at every pair of cooling time-decay energypoints. (No input is necessary, and card No. 2 does not exist.)
TABLE III (cent)
Card No. Format Variable Comment
NOK = 1, KTRM = number of ~’s to be calculated by code.
2 1216 KCAL (L) Selects points between which A’s are tobe calculated. First point is alwaysselected by code. Read KCAL(L), L = 2,
KTRM
NOK = 2, KTRM = 1, cards in ENDF-like format from previous problem for this groupare read in here, and the subroutine returns immediately to main program. No fur-ther input is needed for this group. This option is used after a single-parameterfit has been made in a previous pass, and two-parameter fitting is now being donein STEPIT for this group. This is similar to the NFINL = 1 option in the mainprogram, except that the two-parameter fits are allowed on a group-by-group basis.
3 1216 IWANT
KCAL(L)
(Input for subroutine SELECT)
1 1216 ITS ,ITP
(Input for subroutine TRMSEE)
1 1216 MLT
2 16 L
E12.5 ALF(K,L)
E12.5 ALAM(K,L)
3 1216 LT
LTM(L)
Select A’s wanted by position number.If all are to be retained, as in afirst pass, set equal to zero.
Position numbers of A’s to be kept.Do not enter if IWANT = O.
Number of time steps desired, indexesof desired time steps. SELECT used ifone wishes to fit a subset of a partic-ular data file, and this input followsthe title code.
Number of parameters to be changed
Time step number of parameters to bechanged
New value of a
New value of ~
Number of terms to be removed
Term numbers of terms removed
14
TABLE IV
GROUP STRUCTURE USED FOR SAMPLE PROBLEM
Group No.
12345
6789
10
11
Lower EnergyBoundary (MeV)
0.100.400.901.351.80
2.202.603.004.005.00
6.00
Upper EnergyBoundary (MeV)
0.400,901.351.802.20
2.603.004.005.006.00
7.00
NOTE : There are essentially no data on PEFPYSfor E > 7.0 MeV
15
negligible gammas above 7 MeV, the upper bound of the last firoup. Also note
that in collapsing to the broad-group structure, the code changes the units of
the fission-product decay energy from MeV/fission to MeV/fission-s, the standard
units in use for pulse functions.
For the first pass, we make only a single-parameter fit12 and allow the code
to calculate initial A’s from semilog slopes between pairs of cooling-time and
gamma-energy (MeV/fission-s) points. This is done by setting the input as
The above expressions, which are developed more generally in Appendix D,
do not include the effects of neutron absorption by the fission products that
become important for high flux levels and long cooling times (Figs. 27-29).
There are two effects of absorption; namely, the flux level can reduce the den-
sity of directly yielded products in the fission pulse, significant for those
nuclides having large cross sections and large yields; and nuclide coupling in
stable and long-lived nuclides tends to build up the concentration of more un-
stable nuclides.
Positive effects are to be expected from shielded nuclides such as134CS
9136C~
9 148’’’Pm,148Pm, and 154Eu and, indeed, an examination of the CINDER-10
output of the problems illustrated in Figs. 27-29 reveals that the very large
26
::
:;::
o
-12
Time (s)
Fig. 27.Per cent deviation of decay heatingdue to neutron absorption (235u ir-radiation for 20 000 h, no depletion
($ = 1013).
:;::::::::::::
z
1---‘;::
a) 100uL ,Total ;
L
,’~:
E.’
...
t?~------ , la ;,.
–loo
Time (s)
Fig. 28Per cent deviation of decay heatingdue to neutron absorption (235u ir-radiation for 20 000 h, no depletion($ = 1014).
70
60- ●
/;.;.,
c 50- ::0 ::::.“.4 ::.? 40-
::::::
8::!:
30- ;
z8 20-
&2 lo- ;
.“
o-
-10
Time (s)Fig. 29.
Per cent deviation of decay heating dueto neutron absorption (239Pu irradi tionfor 20 000 h, no depletion ($ = 3101 ).
27
effect at cooling times near 108 s is due to neutron absorption in the stable133
nuclide Cs, which produces the shielded nuclfde134CS
‘fherefore, it is“ 133 134CS
readily calculated by use of the simple two-nuclide chain Cs(n,y) .
Other reactions can be handled in a similar manner, and a list of the more im-
portant fission products contributing to absorption effects are given in Table
Yi . General equations are developed in Appendix D for approximating the effects
of absorption with two-nuclide chains.
These equations for the computation of fission-product decay-energy spectra
were incorporated into a code CALENDA (calculated decay energy spectra with ab-— — — —
sorption), which is useful for applying the fits to PEFPYD library data to ob-
tain decay spectra after shutdown for reactor fuel irradiated for extended times
at variable power. The present version of the code is limited to (a) a histo-
gram representation of the power history, (b) expression of the neutron flux and
cross-section data in two energy groups (fast and thermal), and (c) inclusion of
the absorption effects of only those two-nuclide chains shown in Table VI flagged
with an *.
In order to test the CALENDA code and, indeed, the practicability of apply-
ing the pulse fits to finite irradiation problems, calculations were made for a
20 000-h irradiation of235
u fuel with thermal neutrons at constant fluxes of
104 n/cm2/s (that is,14 2
negligible absorption) and 10 n/cm /s. Fission product
decay beta and gamma spectra were obtained in the n-group structure for both
cases, and these were compared with results obtained directly with the CINDER-10
code,
The parameters for the beta pulse fits used in the CALENDA calculation are
those given in Appendix C. The fits for the beta spectra were to within 1% of
the CINDER-10 pulse data for all groups except group 11, which was within 1.5%.
The beta-spectra comparisons with CINDER-10 results for several groups,as well
as the sum over all groups,are shown in Table VII for the 20 000-h irradiation4 2
at a constant flux of 10 n/cm /s (no absorption) case. Note in Table VII that,
in general, agreement is remarkably good, even for cooling times less than 0.1 s,
the minimum time for which the pulse was fit. Also note, however, the relatively
large deviation for group 10 at 100 s. This is due to the fact that the PEFPYD
data for the pulse is only given at two points in each cooling-time decade, and
thus is insufficient for a more accurate description of the spectra.
To obtain parameters for the gamma fits for the235
U thermal pulse, CINDER-10
data given at six points per decade are used as input to FITPULS. Although the
28
TABLE VI
FISSION PRODUCTS IMPORTANT IN DETERMINATION OFNEUTRON ABSORPTION EFFECTS ON DECAY POWERa
NUCLIDE PRECURSOR(s) COMMENTS
*90Y *89y *90sr
100TC 99T:
lo4Rh 103RU
lo5Rh 105RU
116In
1151n
130I
J34c~
*135xe
136ca
*140La
142pr
*144pr
147Nd
148pm
*1481?nl149pm
*150Pm
151Sul
*153sm
154Eu
156EU
1291 130m1
*133C;
*1351
135xe 135c~
*140B;*139~~
141pr
*144ce,*143pr
146Nd
147 147pmNd,*
147 147Nd,* pm
147Nd 147pm 148pm 148~m
147Nd’147pm’148pm’148, ~m, *14gpm
150 ‘ ‘Sm
*152sm
153Eu
155EU
Degree of importance (small) de-pends on uncertain branching frac-tions. Can be ignored based onENDF/B-IV data.
Very important at all shutdowntimes.
Major negative effect
(n,y) branching from147pm
0.53.
(n,y) branching from147pm
0.47.
29
aOnly the listed precursors must be considered in determination of theneutron absorption effect, but cross sections must be included forall nuclides.*Nuclides included in two-nuclide chains in CALENDA code as of June 1978.
TABLE VII
BETA ENERGY RELEASED FROP1FISSION-PRODUC’~ DECAY AFTER20 000 h THERMAL IRRADIATION OF 235U
(% difference between CTNDER-10 and approximate method calculations)
Flux = 104nicm2-s
Coo15.ng Group 2 Group 4 Group 6 GSOUp 8 Group 1.0 Total‘lV.me(s)0.4--0.9I.!cv 1.35-1.8>leV 2.2-2.6NeV 3.0-4.0rev 5.0-6.0;!eV ,\llGr~ups
Figs. 27-29. The comparisons for groups 1 and 2 of the beta spectra are shown
in Figs. 30 and 31; and groups 1, 2, 4, 5, of the gamma spectra are shown in
Figs. 32-35, respectively. Figure 36 shows the comparison for the total beta
release summed over all groups, and, finally, Fig. 37 gives the same for the
gammas. In all figures, the solid curve is the approximate CALENDA calculation
and the open circles are from CINDER-10. The large peaks at about 108s cooling
time occurring in beta groups 1 and 2 and in gamma group 2 are due to the133
Cs (n,y)134
Cs reaction. The very large peaks in gamma groups 3 and 4 are also
due to this reaction, but the large percentage differences seen are again due to
differences in small numbers, as practically all of the gamma energy is con-
tained in group 2. This is evident from Fig. 37, which shows that the sum over
all the groups in this cooling-time domain is about the same as for group 2.135
Figure 32 for group 1 gammas shows the negative effects of the Xe.
A large deviation of about 400% near a cooling time of 106 s is seen in group
5 (Fig. 35) for the CINDER-10 calculation that is not accounted for in the approx-
imate method. The precursor of the missing reaction would probably have a half-
life of about 1.5 x 106 s, and the product nuclide would emit relatively strong
gammas in the energy region from 1.8 to 2.2 MeV. As indicated by the other
figures, additional reactions could be included to increase the accuracy of the
approximate method.
VI. suMMARY
The results of the pulse calculations (10-4
s irradiation time) from the
CINDER-10 code have been collected and organized into a library of aggregate
fission-product release-energy spectra on a fine multigroup energy mesh. These
data are given for cooling times from 10-4 13
to 10 s at two steps per time decade.
This library of processed ~NDF/B-IV~ission ~roduct ~ield and~ecay data (PEFPYD)
contains spectral data for all ten yield sets given in ENDF/B-IV and is organ-
ized in an ENDF-like format. The library can be obtained from the National
Nuclear Data Center (NNDC) at Brookhaven National Library (BNL).
A method has been developed for using the PEFPYD data in approximate cal-
culations of fission-product decay-energy spectra resulting from nuclear fuels
irradiated for a finite time. The pulse data is first collapsed to a coarse-
group structure and the decay-energy vs cooling-time data points of the resulting
broad groups are fit with a sum of exponential. This is done with the FITPULS
code, which contains a nonlinear least-squares routine for making the fits.
33
8; o 1
?/i
of)-
00d ,.
1
Perfor
I
~
3’ 103 105 107 10°Coolinc Time (s)
Fig. 30.
cent deviation due to absorptiongroup 1 betas.
Cooling Time (s)
Fig. 32.Per cent deviation due to absorptionfor group 1 gammas.
34
o
00 II
i~
10’ 103 log 107 1
I
)“Cooling Time (s)
Fig. 31.
Per cent deviation due to absorptionfor group 2 betas.
)
t’
Perfor
Coding Time (s)
Fig. 33.cent deviation due to absorptiongroup 2 gammas.
210’ 103 10’ 107 1~
Cooling Time (s)
Fig. 34.Per cent deviation due to absorptionfor group 4 gammas.
— A roximate Methodf?o C DER
~I
103 105 107 10’Cooling Time (s)
Fig. 36.Per cent deviation due to absorption,total betas.
o
0
0
Cooling Time (s)
Fig. 35.Per cent deviation due to absorptionfor group 5 gammas.
l—10
A roximateN’C DER
Method
H4w+w7&A%%L10’ 103 105 10’ 1
Cooling Time
Fig. 37.Per cent deviation due tototal gammas.
(s)
absorption,
35
It is also worth noticing that FITPULS has the option of reducing data for a
finite irradiation time to a pulse. This option, which is not discussed in this
report, is useful for comparing different experiments run with different irradi-
ation times.
Finally, broad-group spectra for finite irradiation times can be generated
by simply folding the irradiation time into the analytic fits for the pulse
spectra. This is done using the CALENDA code, which also has an option for in-
cluding the effects of neutron absorption by approximating the more important
reactions with two-nuclide chains. Given the fits for the pulse groups, the
CALENDA calculation is very rapid, and thus the method provides an inexpensive
way of calculating fission-product decay-energy spectra for a variety of prob-
lems. Such a set of useful pulse fits are provided in this report.
ACKNOWLEDGMENTS
The authors are exceedingly grateful to J. C. Lewellen of the Division of
Reactor Research and Technology of the United States Department of Energy for
suggesting the need for the library of aggregate fission product spectral data
described in this report; to F. Schmittroth and R. Schenter of Hanford Engineer-
ing Development Laboratory for suggesting the fitting techniques used in this
worlc; and to D. Graham Foster, Jr., of Los Alamos Scientific Laboratory for im-
proving the FPSPEC code.
REFERENCES
1.
2.
3.
4.
Fission-Product Decay Library of the Evaluated Data File, Version IV(ENDF/B-IV). [Available from and maintained by the National Nuclear DataCenter (NNDC) at the Brookhaven National Laboratory; these data were com-piled by a two-year task force chaired by R. E. Schenter.,]
T. R. England and R. E. Schenter, “ENDF/B-IV Fission-Product Files: Sum-mary of Major Nuclide Data,” Los Alamos Scientific Laboratory reportLA-6116-MS (ENDF-223) (October 1975).
C. W. Reich, R. G. Helmer, and M. H. Putnam, “Radioactive-Nuclide DecayData for ENDF/B,” Aerojet Nuclear Company report ANCR=1157 (ENDF-120)(August 1974).
T. R. England, R. Wilczynski, and N. L. Whittemore, “CINDER-7: An InterimReport for Users,” Los Alamos Scientific Laboratory report LA-5885-MS(April 1975).
36
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
M. G. Stamatelatos and T. R. England, “FPDCYS and FPSI’EC, Computer Programsfor Calculating Fission-Product Beta and Gamma Multigroup Spectra fromENDF/B IV Data,” Los Alamos Scientific Laboratory report LA-NUREG-6818-MS(May 1977).
T. R. England, M. G. Stamatelatos, and N. L. Whittemore in “Applied NuclearData Research and Development,” Los Alamos Scientific Laboratory reportLA-6472-PR (1976) p. 60, and “Applied Nuclear Data Research and Development,”Los Alamos Scientific Laboratory report LA-6266-PR (1976), p. 13.
W. B. Wilson, T. R. England, M. G. Stamatelatos, and R. J. LaBauve, “ENDF/B-IV Fission Product Cross Sections: Absorption Buildup in Thermal Reactors,”
Trans. Am. Nucl. Sot. ~, 592 (June 1977).
W. B. Wilson, T. R. England, and R. J. LaBauve, “Multigroup and Few-GroupCross Sections for ENDF/B-IV Fission Products; The TOAFEW Collapsing Codeand Data File of 154-Group Fission-Product Cross Sections,” Los AlamosScientific Laboratory report LA-7174-MS (March 1978).
R. E. MacFarlane and R. M. Boicourt, “NJOY: A Neutron and Photon CrossSection Processing System,” Trans. Am. Nucl. Sot. ~, 720 (November 1975).
E. Jurney, Los Alamos Scientific Laboratory, personal communication (1977).
D. Garber, C. Dunford , and S. Pearlstein, Eds. , “ENDF-102 Data Formats andProcedures for the Evaluated Nuclear Data File, ENDF,” Brookhaven NationalLaboratory report BNL-NCS-50496 (ENDF 102) (General Reactor Technology TID-4500) (October 1975).
R. J. LaBauve, T. R. England, M. G. Stamatelatos, and D. C. George, “Approx-imations to Summation Calculations of Delayed Energy and Spectra from FissionProducts,” Los Alamos Scientific Laboratory report LA-6684–MS (January 1977).
J. H. Burrill, Jr., “360-STEPIT, A User’s Manual,” Ohio University ResearchInstitute report TNP-1966-2 (May 1966).
M. Tobias, Central Electricity Generating Board, Berkeley Nuclear Labora-
tories> personal communication (1977).
T. R. England and M. G. Stamatelatos, “Comparisons of Calculated and Experi-mental Delayed Fission–Product Beta and Gamma Spectra from 235U ThermalFission,” Los Alamos Scientific Laboratory report LA-NuREG-6896-Ms (July 1977).
37
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IS TnaT ON THE INITIAL PASSES~SINL+LE PANAMtTER(ALF) FITS ARL FIT ?91
MADE FOX !HE SIJFCTRA ANO ON FINAL PASSES,PATS ANF MADE MORE FIT 301
ACCUMATL dy FITTING BOTH PA~AMETEKS. AN EACEPTInN IS THE CASL FIT 311
WHERL GOOU FITS H“AVE sR$VIOIJSLY BLEN OBTAIIUiD FUI? A PARTIcIJLAR FIT 321
GROUP STR~CTUKE FOR ONC NUCLIDE ANO FITS AKL OESTREO FOR ANU!HER FIT 331NUcLIUE IN THL SAME GRUUP STRUCTURE. THENQTHE FTRST SET OF FIT 341PARAMLILRS CAN BE APPLILO OIRECTLY TO THE ~LCOND PROBLEM, FIT 3’JI—
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A FINITE IHHAuIATION TIME TO A PULSE ANU ~dTAININ(3 A FIT FnM IHE FIT 381EQUIVALLN~ PULSE. FOR !HIS OPTION <ET IHAU=l. FIT jgl
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COMMON /PjLSDAT/ NOTQ ~tJ(Z5)S Gx(d5!400)9 NERG? ALF(2595u)Q ALAM(dFIT 431
1 595UI FIT 441COMMWN /PULSCAL/ A(5095U)$ fl150.1) FIT 451
COMMUN /P:LSOuT/ ALpH~(50)9 FXC(1OO)S PCT(1OO) FIT 461
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REAO (NINs180J DIFLIM,~UNT IMCTMTNrTMAX!tiAMIN FIT
FIT
DIFLLM = MAX* PEMCENT ~oINTwISE oLVIATIQN ALLo~En IN sTEPIT* Flr
USUALLY SET HISH ON INITIAL PASSE> ANO TTGHTENED UP IN FIT
S}JljSEQIJLNT PA>sEs. FIT
RUNTLM = kUNNING TIME. MAKE FRLCT, OF SECUNO LESS TtiAN THAI usEU FIT(JN CONTROL CARIJ To GET PUNCHED CAMI)S FOR SU13SEQUENT NUN. FIT
TMIN = LOWEST COOLING TIME OESIMEU. FIT
TMAX = HIbHESl COOLING !IME DESIRtD. FIT
GXMII~ = MINIMuM ALLOWED VALUE OF G~(IsK)* THIS SHOULD Rli ski. FIT
SO NO FIT IS ATTEHPTLO OVkf? MORE THAN ASnlJT 15 DLCAu~5, FIT
FIT
NP=I isP FIT
IF (1(1) .LE.000) T(l)=l.E-4 FITIF (I INP).LE.OOO) NP=Np-1 FIT
DO lU N=l?NP FIT
Tc(NJ=T(N) FIT
in CONTINUK FIT
NE=NtHG FIT
NqOS=NE+l FIT
EB(NuuS1=7.5 FITFIT
f)ol
+,11
621631/$41
651
6bl
671f!dl6);
71J1
?11721
731741751
761771
781791
801
811R21831
841851R61
Rrl
R81R91
~ol911921Q31941qs~
961Q71
961Qgl
1001
10111(121
10311041
105110611(171
108110911101
1111112111311141
115111011171
65
cc
cc
c
c
c
c
2n
3n
c,cc
c
c
c
40
%0
fif)
66
KKtU = tLAb FOR CALLING STLPIT BY tiROUPS=OtNO C4LLOXGUOUP NO.*STEpIT KOUTINE cALLtD?=NEGAllVE uNOUP NOO$TRMSEE ROUTiNtCALLFU, NOTE IF NsTEP =O,KKN NOT ACrIVITATFn,
READ (NlNst70) (KKN(K)JK=l$NEHG)
OIITPUI CARDS FRoM PREVIUUS PROBLEM ARE REAO HERE IF NFINL=l@
IF (l~PINLoNE.1) GO To 30READ ININ~150) (T1TL(I)s1=19R)
READ (NIN*?20) C1.,C2,NUL*NLU,NUL!NER6
DO ?U K=lJNENbREAD IN1N!?20) EM [K) .EB(K+l )~NuL!NULSNULSKl~G
1ALUE3 GKEATER THAN TEN PERCENT FoLLoI’f) FIN 1110
16n FOUMAI (311Ho FUNCTION RISING AT POINT NO..13) FIN 1]20~7n FCIRMMl (23HFINE CHECK ~UR GROUP NO. c12) FIN 1130
lRO FOQMAI (2bti FINF cHEcK FOR GROUP No, $12) FIN 1140lqn FORMAI (~ljti TIME IN S~CUNOS ) FIN ~l!Jo
2nn FOI?MAI (lUH F%(I) ) FIN l16n
Zlfl FORMAI (4 (?X014,2X$F10.3.~X, E1O.3]) FIN 11/0
2?n Ff)RMAl (1’~HO AT POINT NIIMHF.R ,13*13H FOIF = 9F6.3) FIN 118(3
23n FnQMAI 1221i0 CHLCK POl\~T NUMflE* s13) FIN 119024~ FOI?MMI (ldyo IHFHE ARE S13,30H NEb FXP vALdtS. FIT NO GOOD.) FIN l?IJO
2%() FOQMAI I?lHO ~IT ~ATHFR PUOR AT ,13,21H PnLNTso cHECK PLOT.) FIN l?ln
FNn FIN 1?20
cc
cccc
cc
ccc
c
ccc
SUHRUUTINL PULSFIT LKKF) @UL
PULTHIS ROUTINE PROVIDFS A SINGIE PARAMETER FIT FOR A CUM131NAT1UN PUL
OF LINLAR FUNCTIUNS OF THF FORM ‘- F-UL
PUL
rA(T)=SISUM+ALPHA (K) #txP(-ALAMDA(K)*T( l))tsUM OVtN K=l,~!~M, PuLTIIE NUqHER UF TERMS USED To UEPRESLNT FX. PUL
PULGIvEN A SLT O} ALA~DAS*(SLE COMMENTS BELOW) *THIS RnuTINE SOLvkS PuL
FOR MLHPAS. PUL
IT Is AUVISAHLE 10 MAKk A SINGL} PARAMLTLR IIT WITH PIILSFIT DEFOMLPULMAKING A lwU PARAMETER ~IT wITH SIFPITC VIJL
j>lJ L
COMMVN /pdLSIN/ ALAMDA(50)$ FX(,JOU)9 T(401)s KTKM, ITSP, IPRuB* PUL
1 NTNs NUU1 PUL
COMMUN /PdLSCAL/ A(50t50)? H(50c11 PUL
CfIMMUN /PdLSOUT/ ALPHA FXC(1OO)! PCT(1OO) PUL
cOMMUN /MANJI/ W(loo)j 11TL(8)* KTK(50)J NS(10)$ KKN(25)Q DIhLxM PUL
COMMuN /FINRAu/ IRAD, NCONs9 NADT(200)~ UELT(200) PUL
OIMENaIUN KCAL(71)
ITs=L
PULPUL
IPRO=KKF PUL
RFAD ININ$?30) LWTtNOKcKTRMtIPRT PUL~UL
LwT=nl FCN UESIQL[),=n~W=l~=l~w=l /FX*X2sW=l/FX*~2.z3$W=1/FX**l .s. PUL
NOK=PLA6 FrIR ALAMDA PfiU2MLTt’$~ SFLtCTIoN P(JL
UTRM=NO. OF ALA~UbS USt-U IN FIT. PULNUK=O?K TRN=KIRVQREAO (ALAMUd (K)*K=lsKTHM} s THTS OPTION ~ul.
U3EIJ FON INTTIAL II~~UT ~FOR LXAMPLE$TwLJ ALAMnA% PLR TIMk PUL
ut~Jl@ -- 1.0*o.%9~:l *o*05*Qoul *o*ofJ~*~rc* PULIYuKxl!KT~M=ItALAM~A~ CAL CULLTLI) FROM SLUPES AT EVERY pAl~ PIJL
ut P(JINTS, PUL
10203040
5060
7080
9n
100
110
1211
130140
150
16rI
170
180
190
?O(l?10?20?3n?40
?5n
?bfl?70?80?91)
3U0310
82
cc
c
cc
c
ccc
c
!~uK=l,KrRM=K.TU~.REAU IN (KCAL(Ll !L=2tKlKM)(lZIA) *WHICH SLLEC!>PULlnUSE POINTS Tn HE uSLO IN THL CALCULATION Or THE ALAMnA>. ou~
~ult. -- KLAL ALSO U>ELJ TO ELIMINATE UNWANTED PARAMETEQ% lN PUL
>UU$EUIJLNT QUNS. SLE HFLOW. PUL(wuK=~*KTi4M=l,REA[) KIRM(55X*T11) ANO H(A*]), A( AM[)A(K),K=l$KTRM PUL
(u(K,l)=ALPHA(K)) (b~ll.4) fNU~l PNEVIUIJ~ PROHLEM. TdIS u~lIoN PUL1> l.JSLn WHEN FTT FOR THIS G~(JUP IS NOW HEING DONE IN STtPIT. PUL
IPRT=ljPRINT A-MATQIX,=OSNO PRIATQ ~ul-
NOTE --- BYPASS IW~NT READ J3t-LOii If N~K=?o PUL
IF (NUK,NLo2) G(l TCI loPULPUL
RFA(I (NIN?7501 KTRv pUL
READ (N1N$?60) (tJ(K, l),ALAMDA(K)sK=l sKTNM) PUL
RFTuHN PULIII CONTINUL PUL
IF (NuK.LL,O) GO TO 60 PULc KTRM IS N(J OF INPIIT ALAMOAS,ITsP IS NO OF IIME STEPS USED IN ~IT. PUL-.cc L.OCJp IU CALCuLATL ALAMoA FOR SELECTED sTEP~.c
NOK=L ISP
DO 2U L=l$NOKKCALIL)=L
20 CnNTINuEIF (n(R!!,lin.11 GO TO 30
KCAL(lI=IREAD lNIN$730) (KCAL(L)SL=?OKIRM)
3n Cf2NTlNUL
IF (LoL~.aO) GO TO 40WI?TTL (IJOIJT*22(T) L
sTnP4n CONTLNUL
IF (RIRM.GT.?) KTRM=KTMM-1IF (nlk!M.cQ,lJ KTRq=NfTK-lAL4MJJJ!(l)=l/T(l)@lo5Do FU K=ItKTRM
KI=KLAL(K)K?=KLAL(K”l )TST=(1(K2)-T(K]))
IF (IST.EJ.O.0) 60 To 50ALAMuA(K.l )=ALOGIFX(K1 )/FX(K?))/(T(K2)-T(Kl) )
K+l).LE.o.) ALAMnA(K. l)=ALAMuA(K) /2.
1) .LF.ALAMLJA(2)) ALA*OA( I}=ALAMUA(?)*> ,
1) .6T. (ALAMUA(?I*?.)) ALAMOA(l)=ALAunfi (?)~2,,
KTKM)oGT.(1./T(ITSp) )) ALAMUA(K IRM)S1.<T(ITSP)
All CCJNTLNUL
RFAD .(N1N!Z40) (ALAMDA (K)$K=l$KTRM)
70 CONTINUL
cc SFLEGI ALAMOAS WANTFO.EO.G. LEAVF OUT ONE op. A ISJ~LTCATEO pAl~Oc IwANI=NO. OF nLAMOAS KCPTO SET=O IF ALL ARc KEp~.
c KcAL(L)=ALAMofIs wANTEr3~ NOTE - IwANT USUALLY SET TO ZEQO ON
c INITIAL RUNs,Al~i3 KCAL(L) NOT READ.
c ALSO NOTE “-- IHTS INPijT NLtDED FOR EACH SEUMENT.
DO 2UU K=l,KT~MWI?ITL (NOUT,31O) (A(K,L)~L=loKTRMlCoNTAINukCONT&NUk.WQTTL (NlJUTo320) 7SRr)RWRfTL (NOUTS31O) (f3(l(, l)oK=19KTQM)
S(JFIRUUTINE LSS SULVES THE SET OF LINEAR LQuATIoN’3 AAXR
CALL LSS (KTRM, I!50QA,B90$DET)
WQTTL (NOUTt3ZO) IPROq
pUL 1230pUL 1?41YpUL 1?>0pUL 1?60PUL 121il
PUL l?#O~lJL l?qf)
‘~L 1300
‘UL 1310~uL 1320PUL 1730
puL 1340‘UL 13b0PUL 1?60plJL 13?0UIJL 17A(I
~uL 1390puL 1400”
puL 1410FUL 1420
pUL 1430PUL 1640pUL 1450PUL 1460PUL 141OPUL 14d0
PUL 14~0PUL 1500
PLJL 151OPIJL 1~~~
PUL 1530P(JL ]540PUL lqbn
PuL 1560pUL 1%70PUL Iridn?~L l%~iT
PuL ]600PuL 1610
pUL 1620puL 1630
PUL 1640PUL 16’50
PUL 1660puL 1670PUL lhdl)
Pul- ]690
84
WPITC (NOJT~31()) (R(KC1)*K=lSKTPM) PUL 1700
L=I’I ~ul- j7iocALL 5LFFIT (lPROOL)RFTUnN
pUL 1720PUL 1730
c PUL ~740270 FOQMMr [1111,45H ONLY FIFTY TERMS ALLOWEO AN FIT. YOU WANT 13) pUL 1750ZIO FORMRI (ld16) PUL 1760240 FOPMAI (3[11X!E]1,4)) PUL 17?02%0 F’OQMAI (5>X!11~) UuL 17ijo
This appendix contains a listing (Table IC) of the parameters for 11 energy
group pulse fits for 5 ENDF/B-IV fission-yield sets. The parameters for the fits,233 235U
given in Table IC, are in an ENDF/B-like format and are for U thermal,238U fast 239
thermal, Pu thermal, and232
9 Th fast neutron incident energy for
both beta- and gamma-decay energy spectra.
In the ENDF-like format, columns 66-70 contain the “MAT-No.” that is used
to identify the target nucleus, The MAT-Nos. used in Table IC are the same
as those used in ENDl?/BT-IVand a~e as follows,
MAT No. Target Nucleus MAT No. Target Nucleus
1260233U
1264239PU
1261235U
1296232Th
1262238U
91
A zero in column 70 signifies the end of the
columns, 71 and 72, are used for the MF-No.,
incident neutron. Here only two numbers are
neutrons, and MF=81 denotes “fast” neutrons.
end of the data for that MF.
data for that MAT. The next two
which identifies the energy of the
used; MF=80 denotes “thermal”
A zero in column 72 signifies the
Columns 73-75 contain the MT-Nos. that are used here to identify whether the
parameters are for a fit for the beta decay energy spectrum (MT=802) or a fit
for the~amma decay energy spectrum (MT=803). A zero in column 75 signifies
the end of the data for that MT.
The data for a particular MAT, MF, MT combination are given in field widths
of 11 and are organized as follows.
1.
2.
3.
4.
The sixth field on the first data card (number 11 in this case) gives thenumber of broad-energy groups for which the pulse fits are given.
On the next card, the first two fields give the energy bounds for thegroup in MeV; 0.1 to 0.9 MeV for group 1, for example. The sixth fieldgives the group number; 1, 2, 3, 4, etc.
The first two fields on the third data card give the cooling-time range
in s over which the fit was made; 0.1 to 1 x 109 s for group 1, for ex-ample. The sixth field gives the number of pairs of parameters used inthe fit; 16 for group 1, for example.
Next follows a number of cards containing the a< and A. used for the pulseL J.
fit .16
Y a e-~~tFor example, for group 1, fc(t) = ~ , so six cards are needed to~’
contain the sixteen parameters needed for the fit.