decay data in the calculation of decay-energy spectra

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

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Thisworkwassupportedby theUS DepartmentofEnergy,DivisionofReactorResearchandTechnology.

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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

I!l!L1so

Energy (MeV)

139 Fig. 6.Cs gamma spectrum compared with

constructed spectrum.

Ener~ (MeV)

139Fig, 7.

Cs beta spectrum compared withconstructed spectrum,

D

● Collapsing and exponential fitting and folding codes are also described,

the latter being useful for generation of spectra following finite powerhistories.

● For immediate use, exponential fits to a particular few-group spectra are

provided in this report.

Because the spectra are based on fission pulses, the libraries have a gen-

eral utility. The exponential fits, for example, can be folded into any power

(fission) history that can be described analytically or by a histogram repre-

sentation. The effects of neutron absorption are also described and approxi-

mately accounted for in the methodology.

II. LIBRARY FOR PROCESSED ENDF/B AGGREGATE FISSION PRODUCT SPECTRA

Of particular interest is the application of the LASL code system to pro-

duce delayed beta- and gamma-spectral data on a fine energy grid (150 groups in

O.OS MeV steps from O to 7.5 MeV) for irradiation of the ENDF/B-IV fissionable-4

nuclides with very short pulses (typically 10 –s irradiation time; shorter

pulses do not alter the calculated spectra) of thermal, fast, and 14-MeV neu-

trons. The results can then be further processed into broad groups and fit with

6

n

z

-Aitfunctions of the type fc(t) = aie , as described in Sees. III and IV.

i=l

The fine-group results from the LJ4SLcode system are assembled into a single

library in an ENDF-like format.11

Definitions for the format for this processed

~NDF/B fission–product and energy-yield data (PEFPYD) library are as follows,— —

MAT : Mat-No. of target nucleus,

MF: File No., used to identifydefined as follows:

MF=80 - fission induced by



Fission nuclide and energygiven in Table II.

same as in ENDF/B.

energy type of incident neutron,

thermal neutrons

fast neutrons

high–energy (14-MeV) neutrons.

combinations available in ENDF/B-IV are

MT: Section number used to describe data contents of the section. MT num-bers are

MI’=801 -

MT=802 -

MT=803 -

MT=811 -

MT=812 -

MT=813 -

MT=821 -

MT=822 -

MT=823 -

MT=831 -

MT=832 -

MT=833 -

as follows:

delayed energy/fission for @- + y summed over all fissionproducts

delayed energy/fission for @- summed over all fission products

delayed energy/fission for Y summed over all fission products

delayed energy/fission for (3-+ Y summed over all gaseousfission products (halogens plus noble gases)

delayed energy/fission for f3-summed over all gaseous fissionproducts

delayed energy/fission for y summed over all gaseous fissionproducts

delayed energy/fission for y + ~- summed over the noble gasfission products

delayed energy/fission for @- summed over noble gas fissionproducts

delayed energy/fission for y summed over noble gas fissionproducts

delayed energy/fission for (3-+ y summed over halogenfission products

delayed energy/fission for $- summed over halogen fissionproducts

delayed energy/fission for y summed over halogen fissionproducts

Other MT-numbers can be defined as needed; for example, MT-numberscould be assigned to any of the above spectra summed over energy.

TABLE II

FISSION YIELD DATA IN ENDF/B-IV

Incident Neutron Energy TypeHigh Energy

Nuclide Thermal Fast (14 MeV)

232Th -- Yes No

233UYes No No

235UYes Yes Yes

238U -- Yes Yes

239PUYes Yes No

241PUYes No No

The data are given in a TAB2 record with tables of spectra (decay energy/

fission vs energy) given for a number of cooling times. Standard ENDF/B inter-

polation schemes between cooling times (TAB2 interpolation) are not recommended

and, in any case, would be of interest only for fission pulses. However, when

the pulse data are placed on a broad-group mesh and fitted with parameters as

described in Sees. 111 and IV, calculations for ~ irradiation-cooling time

combinations are possible, precluding the need for interpolation on the fine

grid. Histogram interpolation is assigned for TAB1 interpolations.

File 1 (MF=l) information is also included, giving some processing infor-

mation and a “dictionary” of the data to follow. The structure of MF=l is as

described in ENDF-102.11

The structure of a section containing the processed

data is

[MAT,MF,MT/ZA,AWR,O,O,O,O] HEAD

[MAT,MF,MT/O.O,O.O,O,O,l,NTS/TSint] TAB2

[MAT,MF,MT/O.O,TS1,O,O,l,NP/E’ ~nt/DE(E’,TS1)] TAB1

[MAT,MF,MT/0.0,TS2,0,0,1,NP/E’ ~nt/DE(E’,TS2)] TAB1

--- --- --— --- --— --- --- --- -

---- ---- ---- ---- ---- ---- -

[MAT,~,MT/O.O,TDNFS,O,O,l,NP/E’ ~nt/DE(E’,TSNTS)] TAB1

[MAT,MF,MT/O.O,O.O,O,O,O,O] SEND

where

8

TS = cooling time step in seconds

DE = decay energy in MeV/fission (MeV/s)/(fiss/s)

E’ = energy of particle ((3-)[photon (y)] in MeV

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

follows:

Input to Program FITPULS

Card 1 NPUN = 20IRAD = ONCORS = O

Input to Subroutine CORSBIN

Card 1 MAT1 = 1260M_Fl= 80MT1 = 803

Card 2 NE=12

Card 3 (group bounds from Table IV)

Input to Program FITPULS (cont.)

Card 2 IPROB = 1NTOTER = ONPUN=7NSTEP = ONFINL = O

Card 3 DIFLIM = 1.0 (not used in this run)RUNTIM = 29.9 (not needed in this run)TMIN = 0.1TMAX = 1.0E+9GxMAx = 1.OE-21

Card 4 KKN (K) = group numbers, although not used in this run as NSTEP = O

Input to subroutine PULSFIT

Card 1 LWT=lNOK = 1KTRM = 1IPRT = O

Card 2 IWANT = O

Repeat cards 1 and 2 for each energy group.

16

and

and

are

Examination of the results of pass No. 1 reveal that (a) groups 1, 2, 3,

10 appear converged and plots are smooth; (b) although groups 4, 5, 6, 7,

9 appear converged, plots show rather large reversals in shape; and (c) fits

not achieved by groups 8 and 11, as negative computed values occur between

fitted points. This run took 30 s on the CDC-7600 computer.

For the second pass, we keep the same input as pass No. 1, excepc we

set LWT = 3 for groups 4, 5, 6, 7, 8, and 9. As a result, all groups are appar-

ently fitted to within 1% except group 9, but plots are not smooth for groups

4, 5, 6, 7, 8, and 9 as illustrated in Fig. 8 for group 4. Note the apparent re-

verse of slope near a cooling time of 100 s, not indicated by the input data

points. Examination of the parameters for these indicates that there is a large

negative value of a in the sixth pair of parameters for each group. These pairs

of parameters are eliminated in the third pass. The running time for pass No. 2

was 25 s.

The input for pass No. 3 is the same as that for the second pass, except

for groups 4, 5, and 6 in subroutine PULSFIT. This is now as follows.

Card 1 LWT = 3NOK = 1KTRM = 1IPRT = O

Card 2 IWANT = 20

KCAL (J) = 1,2,3,4,5,7,8,9,10,11,12,13 ,14,15,16,17,18,19,20,21(note parameter number 6 is omitted)

The results of the third pass indicate that plots for groups 4, 5, 6, 7, and

9 are now smooth, as illustrated in Fig. 9 for group 4. The running time for pass

No. 3 was 31 s.

Smooth fits for groups 7, 8, and 9 were not obtained in the first three

runs, as shown in Fig. 10 for group 8. An additional run (run No. 4) was made

for these three groups in which the input was the same as that used in run No. 2

except that the option for selecting the points for the fit was used. This op-

tion is activated by setting the first characters of the title card to the word

“SELECT.” The input was set to remove the points at cooling-time steps of 0.5,

5, and 50 s. This effectively smoothed the fits for groups 7, 8, and 9, as

shown in Fig. 11 for group 8.

The FITPULS input for pass lJo. 5 is the same as for the previous passes

except NSTEP and NFINL are now both set to 1, and the card output from passes

No. 3 and 4 are added after card No. 3. No

suits of this run were that all groups were

running time. The fits extend to >30 yr of

additional input is needed. The re-

fitted to within 1% after 104 s of

cooling time. These aqe shown graph-

ically in Figs. 12-22, and a comparison of the parameters for group 4 from the

first run with those from the last run are given in Table V.

Note that these fits are not unique, and also that good fits can be ob-

tained by using smaller numbers of parameters. The subroutine TRMSEE can be

used to assist the user in reducing the number of parameters. It is called for

a particular group by making KKN the negative of the group number.

Parameters for selected incident energy-fissioning nuclide combinations

are given in Appendix C. The accuracy of these fits vary from about 2 to 5%,

and although closer fits could be obtained, the extra effort hardly seems worth-

while for ENDF/B-IV data. An indication of the accuracy of the ENDF/B-IV

fission-product decay and yield data can be obtained by comparison with another14

evaluated set, as is done in Figs. 23-26 , where the spectra are normalized to

the same total values. More important validations are the comparisons with ex-15

periment. The fits gjven in Appendix C are certainly as accurate as warranted

by the ENDF/B-IV data.

v. APPLICATION OF FITTED PULSE TO CALCULATION OF

EXTENDEII IRRADIATION

The fitted pulse can be folded with a reactor

DECAY-ENERGY SPECTRA AFTER

power history so that decay

spectra from irradiated fuel can be calculated as a function of cooling time.

Consider a reactor operated at variable power P(t’), O < t’ <T, for a time in-

terval T followed by a shutdown period t . In the following equations, givens

for a particular energy group,

t=

P(t’) =

K=

T =

t=s

H(t,T) =

L

time since fission pulse

power in watts at time t’

0.32042 X 1010

w-s/fission

total time at power

shutdown time of interest, measured from T, and

decay-energy release at time (T+ts) for some energy bin (MeV/s).

fc =x

ak<~kt , MeV/fiss-s,

k=l

(2)

18

o7’0 ~

410-1 101 103 105 107 109

Cooling Time (s)

Fig. 8.Fit for group 4 after second passthrough FITPULS.

Cooling Time (s)

Fig. 10.Fit for group 8 after second passthrough FITPULS.

04

‘c -.:?

\

Cooling Time (s)

Fig. 9.Fit for group 4 after final passthrough FITPULS.

Cooling Time (s)Fig. 11.

Fit for group 8 after final passthrough FITPULS.

19

\\

\

‘\“\

‘\\,‘.\

\

-- a

‘$’-hT’w-~10-1 10’ io3 105 m’ log

Cooling Time (s)

Fig. 12.Final fit for group 1.

Cooling Time (s)


1’

1

-a-.= I—‘\

‘\

\

\

‘,,bA107 1

Cooling Time (s)

3°

Fig. 13.Final fit fcr group 2.

N

Cooling Time (s)

Fig, 15.Final fit for group 4.

20

y L0

Cooling Time (s)


Cooling Time (s)


~ ~~19 10-’ 10’ 103 103 107 10’

Q

Final

Cooling Time (s)

Fj.g, 17.fit for group 6.

Cooling Time (s)


21

a7’ ~~

10-’ 10’ 103 105 m’ 1Cooling

Fig .

Time (s)

20.Final fit for group 9.

\

“~~v~,,,,,,q -,,,,5 ,-

10-’ 101 103 105Cooling Time (s)


\1=1

Cooling Time (s)

Fig. 22,Final fit for group 11.

22

●

o

.......

%.t %,

; ... . . . ...%

c ...-\ -/ ....>U -..

. ...E -...

. . .

~%,.

x %..

2 %0...

>? ...

— TOBIAS.\

t% ..O------- [NDF/B-IV ..

T03-0.0

r I , ,1.0 2-0 3.0 50 6.0 7.0 6

ENERG; ‘(f’lEv)

Fig. 23.

D

ENDF/B-IV beta-spectra comparison with UK Data File, all

fission products, betas (cooling time 0.1 s).

-]LIM i — TOBIAS-------ENDF/B-IV

-1

:--10 t I

- 0.0,

1,0 2.0 S*O 4.0 5.0 6-0 3

ENERGY (MEV)

Fig. 24.ENDF/B-IV gamma-spectra comparisons with UK Data File,

o

all fission products, gammas (cooling time 0.1 s) .

23

‘o 1-0.0

, r ,1,0 2.0 3.0 4.0 90 too o

F.tlERGY(MEV)

Fig. 25,

ENDF/B-IY beta-spectra comparison with UKall fission products, betas (cooling time

Data File,1000 S).

?o

1

?-1- 0.0

1’.1

— TfJ~IAs------- ENDF/B-IV

ENERGY (MEV)

Fig. 26.

ENDF/B-IV gamma-spectra comparison with UK Data File,all fission products, gammas (cooling time 1000 s),

24

TABLE V

COMPARISON OF GROUP 4 PARAMETERS AFTER FIRST PASSWITH GROUP 4 PAIL4METERS AFTER LAST PASS

Group 4 Parameters Group 4 ParametersAfter First Pass After Last Pass

a

3.912 X 10-2

-1.119 x 10-1

1.408 X 10-1

-1.006 X 10-1

6.292 X 10-2

-1.505 x 10-2

6.710 X 10-3

4.864 X 10-4

5.344 x 10-5

4.883 X 10-5

2.318 X 10-6

7.550 x 10-6

6.540 X 10-7

-1.522 X 10-8

1.141 x 10-7

5.353 x 10-7

-5.628 X 10-7

-6.003 X 10-12

6.433 X 10-12

-1.140 x 10-13

2.535 X 10-16

1.152

5.760 X 10-1

3.301 x 10-1

1.329 X 10-1

6.460 X 10-2

2.611 X 10-2

1.383 X 10-2

5.507 x 10-3

1.287 X 10-3

3.279 X 10-4

1.804 X 10-4

6.440 X 10-5

3.068 X 10-5

1.722 X 10-6

5.725 X 10-7

1.656 X 10-2

-2.081 X 10-2

2.335 X 10-2

-9.164 X 10-3

9.333 x 10-3

2.427 X 10-3

5.633 X 10-4

5.195 x 10-5

4.965 X 10-5

1.624 X 10-6

7.552 X 10-6

6.660 x 10-7

-1.297 X 10-8

8.819 X 10-9

6.395 X 10-7

1.596 X 10-7

6.215 X 10-7

-8.615 X 10

1.127 X 10-7

7.535 x 1;;

2.710 X 10-3

5.539 x 10-12

1.903 x 10-8

4.687 X 10-13

4.419 x 10-10

2.295 X 10-16

1.297

6.712 X 10-1

3.606 X 10-1

1.415 x 10-1

6.486 X 10-2

1.366 X 10-2

5.521 X 10-3

1.283 X 10-3

3.271 X 10-4

1.568 X 10-4

6.466 X 10-5

3.037 x 10-5

2.343 X 10-6

5.688 X 10-6

6.395 X 10-7

6.216 X 10-7

1.617 X 10-7

2.705 X 10-8

2.437 X 10-8

3.424 X 10-lo

25

H(t~,T) is given by

\

T

l{(t~,T)=P(t’)— fc(T+ts-t’)dt’

K@eV/s)

o

(3)

or

H(t~,T) =[w&e-Ak(T+ts-t’)dt, ‘Mev/s) c ‘4)

Assume, for example, that the power history can be approximated by J histograms

with a power of P. at irradiation time T .J j

Then,

‘(ts9T)=~!~ak ~JAk(T+t:t’)dT ‘“eV/s) (5)j =1 k=l JT

j-1

or

H(ts,T) =~~~}~e-Ak(T+ts-Tj)-e-’k(T+ts-Tj-l~ (MeV/s). (~)

j=l k=l k

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

1.0!$-041.OJ+O11.OK+OOI..OEKI11.OE-I-021.oE-l-031.0E4.041.OE-I-051.OE+-061.0)?+071.0K+081.OE-I-09

1.21.21.2S,41.92.11,92.22.92.7

- 6.9- 0.7

1.21.11.11.11.71.71.81.50.00.0

- 2.10.4

1.81.61.51.50.80.90.61.60.10.10.50.5

1.71.51.51.32.53.60.8

- 0.3- 0.3

0.01.13.5

2.82.52.43.5

11.54.11.20.50.63.1------

1.4

1.31.2

1.21.61.71.31.51.10.2

- 4.1- 0.1

fits could not be made as accurate as for the beta spectra fitted at two points

per decade as shown in Table VIII, the comparison with CINDER-10 for the 20 000-h4 2

irradiation at @ = 10 n/cm /s case is much better, as can be seen in Table IX.

In general, the deviation is less than that for the pulse fit for a particular

cooling time.

Comparison results for the case with a flux of 1014 2

n/cm /s, a case for

which the effects of neutron absorption are very significant, are shown in Table

X for the beta spectra and Table XI for the gamma spectra. As noted previously,

only the “two-chain” reactions flagged with an * in Table VI were included in

the CALENDA calculation. Note, however, from the total sum over the energy

groups that most of the important absorption effects have been included for both

the beta and gamma totals.

Some rather marked deviations of the approximate spectra from the CINDER-10

calculations can be seen for the individual beta and gamma spectra. For example,

group 10 at 107 s, may indicate an inconsistency in the way missing spectra were

constructed, but note that these large percentage differences are due to differ-

ences between very small numbers that contribute little to the total. Also note

significant differences in some of the other groups for several cooling times,

indicating a need for including additional two-nuclide chains.

It is interesting to compare spectral absorption effects graphically, that

is, as a per cent deviation from the case without absorption, as was done in

30

TABLE VIII

MAXIMUM PER CENT GAMMAS DIFFERENCE

Pulse 20 000-h Irr (104 flux)

GroupNo.

12345

6789

1011

Maximum% Dev

3.83.88.8

13.52.1

4.312.66.0

-2.1-2.44.7

CoolingTime

4.0+73.0+74.0+72.0+51.0+9

2. 0+71.5+51.5+51.0+74.0+52.0+2

Maximum% Dev

-2.02.0

-1.83.1

-3.5

2.43.52.23.61.33.2

CoolingTime

1.0+71.0+71.0+75.0+51.0+7

5.0+75.0+51.0+71.0+71.0+35.0+2

TABLE IX

GAMMA ENERGY RELEASED FROM FISSION PRODUC20 000-h THERMAL IRRADIATION OF

$3~:CAY AFTER

(% difference between CINDER-10 and approximate method calculations)

Flux = 104n/cn2-s

CoolingTime(s)

1.OE-041. OE-011. OJZ+OO1.OEW)l1. OE-1-021.0E+031. 0E+041. 0K+051. 0E+061. 0E+071.0E+081. 0E+09

Group 2

0.4-0.9 HeV

- 0.1- 0.1- 0.2- 0.3- 0.3- 0.2

0.00.40.62.0

- 0.6- 0.3

Group 4 Group 6 Group 8

1.35-1.8 bfeV 2.2-2.6 >leV 3.0-4.0 l!eV

0.10.1.0.10.20.20.40.91.72.3

- 1.3- 0.4

1.7

0.20.20.20.20.30.50.8

- 0.5- 1.9

1.11.40.6

0.40.40.40.50.70.91.40.3

- 0.82.2

- 0.40.5

Group 10

5.0-6.0 >!ev

0.30.30.40.60.91.3

- 0.60.4

- 0.71.1------

0.10.10.10.10.10.10.20.50.91.8

- 0.6- 0.3

TABLE X

BETA ENERGY RELEASED FROM FISSION PRODUCT DECAY AFTER20 000 h THERMAL IRRADIATION OF 235U


Flux = 1014n/cm2-s

Cooling Group 2 Group 4 Group 6 Group 8 Group 10 Total

U!Xi@ 0.4-0.9 }lCV 1.35-1.8 FfeV 2.2-2.6 MeV 3.0-4.0 XeV 5.0-6.0 MeV All ~rcu~s

1.OE-041.OE-011.OE+OO1.OE+O11.OE+021.0E+031.0E+041.0E+051.oE+061.0E+071.OE+081.0E+09

4.54.54.54.74.5

5.36.57.65.13.3

- 5.10.2

3.83.73.73.72.93.04.47.55.10.7

- 1.40.7

3.23.02.93.01.21.41.63.70.90.60.80.7

2.92.72.82.72.84.31.14.9

- 0.10.10.85.6

3.43.03.14.7

13.14.71.54.80.7

318.3------

TABLE XI

GAMMA ENERGY RELEASED FROM FISSION PRODUCT DECAY AFTER20 000 h THERMAL IRRADIATION OF 235U


Flux = 1014n/cm2-s

Cooling

~

1.OE-041.OE-011.OE+OO1. OE+O11. 0E+021. 0E+031,0K+041.0E+051.0E+061.0E+071. 0E+081.0E+09

Group 2

0.4-0.9 MeV

0.60.60.70.81.01.3L.51.22.11.0

- 0.62.0

Group 4

1. 35-1.8 NeV

0.60.60.70.81.01.42. 53.63.0

- 2.3- 2.817.1

Group 6 Group 8

2.2-2.6 WV 3.0-4.0 I+eV

0.40.40.50.50.61.02.15.54.0

13.3- 7.2

0.4

0.50.50.50.60.71.01.74.72.5

10.0- 8.7

0.1

Group 10

5.0-6.0 MeV

0.40.40.50.60.93.4

56.460.184.1

1161.4------

3.43.33.33.43.03.74.76.33.70.9

- 2.60.1

-rot&l

All ~YOUDS

0.90.91.01.11.52.13.13.76.11.0

- 0.82.0

32

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)


— 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)


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

SAMPLE PEPPYD-LIB LISTING

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3.n50flo*~.onooo+4.150C)13+&.3nUu0A4.4%ofJo*4.6n900*

fb.7G3(fD~40~noo(l+

S.f)sooo+

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l.oflooo->.5nf)oo-4.0no(Jf)-

5.5nOr)O-7cOnOOO-

a,5flooo-

l.Onooo*1.lFOOO*l*3nOOO*1.4%o(ffJ*

].6nOoo*1.7%000+le9nuoo+

?.OGOOO*?.?nooo+>.3q(JOO+

?.5nooo*2.6’if)(Jo+

?*Rnooo*>ooqoofJ*

>.lnuoo+?.?5000+1.40(Jf)fJ+

7.5%000+3.70000+3.R50fJ()+koOnOOO+

4.16UOO+

4.3n000*4.4%cf(J()*

4.6nOoo+fJ.7<lJ@n*&.~’looo+

o 3012yU0-2d1260ROtJU3 ~’3”fo ‘i.21?~l/-?dl?6nRuHu3 34380 5,52b~4-2<126nqo803 3439

0 4,?R0/*-22)260ROf3(f3 34400 5s5ft3n?-221 ?60Qd8u3 3441

0 5034d~0-221 ?6(fR~J8U3 3+42o 3.050 f6-221?6f)QOflU3 3443

0 1.51iZ/~-211?60808u3 3+44

o ?.oHf5~l-211 ?6:)Qti3u3 344sO l=04+~H-2L 176npi]8U3 34460 2. 788*2-2<l~f.n808U3 3447

0 107Z6’)0-2d) P60R\]3U3 3*4Hf) 3.34114-2d12AnHtfRu3 3+49o 1.”rldu9-231260R08u3 34S0o 1055iua-2~l ?6ff9uf3u3 3451

0 3c~R3+~-2~] ?6fJqOflU3 3+52o 1.9f)ddl-2317f>nqofJu3 3!53

o 3.43dd7-2~l?6nRuf3u3 2454

0 ?.395!J2-2Sl Z6nq08U3 34550 5.53ZO$J-2+1760R08U3 3+561) 5010114-3u12 flono9u3 3*57o s.9H~(3-291?6rIf?of3u3 3’458

0 fl.lU4/n-2Yi?6qRUtiU3 3+59o 1.442d9-231260QfJfJu3 3*b0

O 0.00ouO+ u12AOQu@U3 3461

0 COOUouO* U1760RCJ8U3 3+62o 4.09900-291z6rIRogu3 3463

0 ooOOUuO+ U1260Q08U3 3Q64O O.OnouO* ~12611ROf3U3 3~6,EI

126na0 ‘2 3+66

o U1Z60Q0811 3467

1 2+l?FioRof311 3+48

(1 u126nRof311 3*691 1511?60F70811 3970

n u]75r3ql181] 7Q711 1.544d9- 71?6090811 3q/21 9.37117- Rl?6nqof3il 34731 6,1JF1493- 81?6nRIJ~lll 3’~:41 2054~05- /l~6f)qc8J1 3+75

1 f30l4’+ul- Hl?6n801Jli 3’$76

1 6.0uO+2- Hl?6nQo!:!l 3*??

0 6.69+3M- 8126nnoflil 3*7Bo 90923U4- Ml?~qqoQll 34?9

o 2.03938- 7\?6f)’3ufJll 3480

0 10151317- 7126f’fqflf311 348:

0 8.6013-$2- fll?6ffQ0811 3*8Z() 1.225’?’5- flzfi(lnonll 3VH2

o u.6f$9LJ5- dl?~f)RuuJl 3484

0 8.7H8b6- M1?60RUR11 341150 9.~fft197- M]213~R0811 3’$H6

f) R.8?4u7- M1260R0811 3*H7

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o R.3?/d4- f$176f)R0811 34H9

o 9.2’-4~Z2- 61?60ROf311 3+90

() 7*481uo- 81P6(IRU011 3’+91o 7.79d+o- M126n80811 34920 605,)3d~- 81?6nn(lfJ!~ 349.3

0 6*610~~- M]p%oRuHll 3494

0 9.12745- 8126nR0811 3+95,e 5.33239- 81?60R(J811 3496() 4.3/093- f31?60~oflll 3*97O 3.~511R- 81260ROH11 3+98(1 3.399’Jo- f3126nf30f311 3499

0 3.030~l- 8126nROf3Ll 33000 z.~t?fb- 81260R0811 3501

0 3.151/7- 81?6nnof311 35020 ?.111’+6- 81?6081)S11 3~03o 2.17203- L31?60RV811 3304

63

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APPENDIX BLASL Identification

No. LP-0847

PROGRAM FITPULS (TAPE5tTApE60TApE70TAPE100 I,APE20) FIT 21PIT 31

THTS VUOGUAM fickEPTS FIbsION-PRnDUCT DATA (HFTA ANn GAMMA) IN FIT 41

{JNrT3 OC LKJ~~bY /~ISS[@CV *PIC}I +,1~ i+t}:N LNcHGY /jlNNE~) INT1> Pll$k FIr 5)

GROUFS(150)FOK A NLJMFjER 0} cOOLINb TIMt STCPS OERIyELJ AS FoLLUMS- Fli 01

YILLL) IjATA ~ROM ENDF UAS FIRsT PROCESSLIJ MY TtiF FPcYS cntyk TO FIT 71

SUFPLy fINE GRO(lP IN~:T FOR ThE FPSPEC coDEo FPsPKC ALSll FIT 81

REuuINE~ (JUTP(JT FRoM THL CIN~FR-lA COOL. FINALLY THE oUTPul 0~ FIT 91

FPswliC *AS PRoCESSED By THE FOTOELF CO(JE wHICH P(JTS THE DA!A IN FIT 101AN LN[)F-LIKE FURMAT ~rlICll IS THL INPUT DATA LIRRARY FoR TMIS FIT l\l

cOuL(FITPULs) . FIT 121FIT 131

FTTPULS RERINS THIS OA!A INTO A USER CHOSEN BROAII GROUP sTRUclUl?~ FIT 141(NOTC IHAT THIS RE81NN1NG CHANGES UNITS To LNEI?Gv/FISSION-SkGo~. FIT 151

ANO riTS !HE DATA FOR EACH GNOUP WITH A LINEAR COMRINATTON u~ FIT lbl

FUNCI1ONS AS POLLDWS -- FIT 171FIT 181

GXC(AOK) =sUM*ALF(L~K) ~(EXP(-ALA4(LSKl *T(I) )JSUM OVER L=19KTNM$ FIT ]91NHERE KTRV IS THE NUMBER OF PAIRS OF PARAMETERS (ALE* FIT ?01

ALAM) F(JR GROUP K. FIT 211FIT ?21

GXC(LQK)=~NERtiY(MEV)/FISS-SEC FoR GROUP K ANO COOLING TIME T~I)o FIT ?31MAX.ALLOWLO VALUL FOR KIRMIKTR(K))? NO. UF pARA~. PAINs/GP=50~. FIT 241

MAX.fiLLOHED NV, OF RRnAU GROUPS.NLRG=25. - FIT ?51MAX, ALLCJiEU No. OF IN~uT COOLING TIME sTEPScITSP=70. FIT ?61

FIT ??1

THIS LODE IS USUALLY RUN IN SEVERAL PASSES. TfiE REASON FOR !HIS” FIT ?81

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—

FIT 361

THE L.UUE A’LSO HAS AN op~ION OF ~EllUCING EXPERI~E~ITAL DATA FUM FIT 371

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

FIT 401

COMMUN /PULSIN/ ALA~10A(50)0 FA(?OO)Q T(40119 KTRM* IISPO IP~uBs fIT 611

1 NIN~ NOU! ~IT 421

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

COMhfUN /MANI/ WX(loo), ~ITL(8)o KTR(50)s NS[1O)J KKN(25), OJELIM FIT 471OIME!W>lON” TC(1OI)I* LXX(20)? TMN(2U)* TMJ((i?u) FIT 481

cOMMUri /ENDF/ MAT* MFs MTv RUNTIM* NPUN FIT 691COMMUN /TI?MOT/ TL(10)* LTM(50)o LT FIT 501

COMMUN /FINNAu/ IRnDo N~OMsO RAnT(200)9 UE~T(200) FIT 511

64

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cc

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:

ccc

cccc

cc

cc

c

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:ccc

cc

c

hJ:rd=!J

N(lUTZe

NnT=LuRFAD (NIN~170J NpIJN,IRAU~NcoRs

FITFIT

FIT

FITFIT

FIT

SET I’WJN=7 HERE IF RERLNNLD DATA CARDS WAN!EDoO~HERtiISE NPUN=20. FIT

IRAD=u$FEGULAR PULSE Fll REQUESTEUO =lsFIT FOR FINIYE IRRADIATION FIT

TIME DAIA wANTEO. FIT

NCOR=U$CALL CORSHIN* =l$NO CALL. FITFIT

IF (NLONS.LE.0) CALL CUHSMIN FITFIT

SEE LuWSdIN FUR MATIoMF1oMT1 INpUTo FIT

FIT

READ ININ*1501 (TITL(I)91=l~8) FITfl~

TITLGEL:I1llY c~ARAcTEQ (~{uLLERITH) IITLE. i’ii

FIr

TITL(I)=CMARACTFH FOR CALLING S!JBNOUTINE StLECT TO CHOOSE DA!A FIT

TO LIE UStD IN ~1,1. IF TITL(I)= SkLtCT ,SEL SU13ROul INE FIT

SLLECT F.OR INPu!. FITI; T’ITL(l)= DO NcT GOCPI+OGRAM STtJPS. dSFI) \/tfEN JUSr FIT

&~RINNED DATA U~slRED, FIT

FIT

IF (I ATI.(1).EQ.lLJH SELECT ) CALL sELECT FIT

WQITL INOUT!160) (T1TL(I)sI=1$8) FIT

IF (IITL(l)oEQ,IOH DO NOT Go) STOP FIT

REAO (NINs170) IPROF3,NTUTLN,NPUN*NSTEP*NF INL FITFIT

IPRDM = PHOBLkM NO. MAKE NEGATTvt IF FIT IS MADF IN SEGMENTS. FIT

NTOTLR . tLAG To DENOTL SPkC. OR TOTAL CALC.S=OQcODE REAoS SPkCT~ FIT

LATA FROM TAPE FILE.=l,cL)DF REAOS TOTAL’ nAYA FROM CARDSO FITNPIJN = F’LAG FUR pUNCIi,=/*PUNCH IILPtiAS ANO I-*MDASV=PO* NO PUN~n* - FITNSTEV = FLtIG TO CALL DHFIT RolJTINE WHICH FITs ROTH ALpHAS AfqU FIT

LA$lDAb,=U,ROUTIdE NOT C4LLE130=lsR0uT1NE fALLEn. 140U!INE FI1

U>(JALLY NoT CALLELI lJNTI!. A cOUPLL UP PAs<F5 ARE MAOt. Tu FIT

AtiJUSl TME PARAMETERS wITH THE SIN~LE FIT AL.UNEO SLL FITs~qROUT[h~. PuLs~IT wHERF POINTS AI?Ii SELLCTEn FOR FIT. FIT

NFINL ~ FLAG FOR NEAOING ALL PA~?AMATERS FRuM PREVInLJS PRORO1.LOS FIT

PuLSFLT WILL NO! BE CALLLU FOR ANY tiRO\jP,=O,NO EFFIEC!S=IO FIT

S:E READ sTATEM~NTS BELOW. FIT

FIT

IF (!*IQTEKOGTOO) CALL HUNTOTS FIT

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

READ (NIN*?20) TMN(K)9TMx (K) *NUL~NUL~NULsK IN(K)KTRw=KTN(~)

L?FAO (NINt?301 [A! F(K,L)* A1.A~(K.L)$L=l *KTRM)KKX=hNN(K)IF (RKX.LT,O) CALL TRMSLE (K*lI

IF (~KxcL1.0) CALL TRM>LE (K,o)

KTR!R)=KTNMCONTANUEGo Tu lUI)

cONTINUE

D(l 9U K=l$NERU

FIT ]IblFIT 1]91

FIT ]201

F1r 1?11

FIT 12?1

FIT ]?31

FIT 17!61

FIT 1251FIT 1?61FIT 1271

FIT I?81FIT 1?91FIT 1301FIT 1311

FIT 1321Fir :7JIFlr 1341FIT 1351FIT l?bl

FIT 1371FIT 1s81

FIT 1?91FIT 1401

FIT 1411.FIT 16<1

THIS PORTION UF ROUTINE ALLOWS FIT IN SEVERAL sEGMENTs. To FIT 1431

ACTIVATE? SET IP~o13 NE~ATIVE. NOTE - INPUr NEEOFfT FOR EA. GNUUP: FIT 1441

FIT 14>1

NSFG = NUMQER OF SEGMEl~!S ● 1 FIT 1461

NS = UMFAKPLIINTS OF SEGMENTS. FIT 1471

FIT 1681

NSEG=.Z FIT 1491

NS(l)=l FIT 1<01

IF (iPROB.LT.0) READ (N1N!170) tvSkG9(NS (Lxl*LX=2,NSEG) FIT 1511FIT 1%21

IF (IYsLG.LE.2) NS(2)=NP FIT 1531

NSEGl=NSEti-1LTRM-u

FIT 1541FIT 1%51

KTR(NI=O FIT 1%61

oO 6U N=lsMSEG1 FIT 1571

NIsNaIN) FIT 1581

N?xM=[N*l) FIT ]%91

ITP=I#d-Nl+l FIT )601

ITSP=U FIT 1611

IX=fl FIT ]621

IF (IMINoLT.Tc(ll) TMIN=TC~l) FIT 1631

IF (IMAX.LEOOOO) TWAX=T~lNp) FIT ]641

IF (lMAX.@ToTc(NP)) TMAX=TC(NP) FIT 1651

DO 4U I=l*ITp FIT l~bl

NN=NL*l-l FIT 1671

IF (uA(K?NN). LToGxMIN) GO To 40 FIT 1681

IF (1(.(NN).LT.TMIN) GO TO 40 FIT )691

IF (lL(NN}, .6T~TMAX) GO !0 40IX=IA*l

FIT 1701FIT 1711

ITSP=lA FIT 1721

LXX(N)=IX FIT 1731

FX(IAI=GX(KSNN) FIT 17+1

T(IX)=T~(~N) FIT 1751

CONTINUk FIT 1761

CALL PULS:IT (K) FIT 17?1

IF (N.NL.NsEG1) KTRM=KTHM-1 FIT 17Ml

00 5U J=l~KTRM FIr 1791

LTRM=LTRM+I FIT ]PO1

ALF(ficLTHM) =B(J,l) FIT lR1l

ALAM(K!l T~U)=ALAMDA(J) FIT IRt?l

IF (MLAML)A [KTRM).GT.ALAMOA (KTRM-1)) KTI?M=KIW4-1 FIT lR31

CONTLNLJL FIT lR$l

KTI?(n) =KT4(K)*KTRM FIT 1R51

CONTAmUk FIT 1861

KTRM=RTK(K) FIT lfi71

70

Fln9fl

iO o

110

12(!

13n

14n

c15n

160

T)o 7V J=lvKTRM

ALAMuA(Jl =ALAk4fK0J)

f3(J*lJ=AL~(K9:)CONTLNUk

WPITL (NOUTtlyO) KQEE3(K)oEH(K*1)OCI BU J=l?KTRM

WRITL (NouT,200) J~ALAMOAtJ)9B(JSl)CONTiNUEcnNTl~Utcf)NTA,*dL

DO 1*u K=l,NE~GIX=()

IF (IMIv.L1.TCI1)) TMIN=TC(l)

IF IIMAXOLF.O.0). TMAX=TCfNp)IF (iMAx.6T.TC(NP)) TMAX=TC(NP)00 1A(J I=~,N~

IF (oA(K~l)oL!oGXMINl Go TO 110IF (lL(l}~LT, !MIN) GO !U 110

.IF (IL(I) oGTo!MAX) 60 !0 11OIx=IA*lITSP=lALxx(nl=IxFX(IA)=GX(I($I)

T(:x)=TP(~)CONTLNULKTRM=KTK(K)DO l<U L=l.KTHMALAMUA(L) =4LAM(K!L)

R(L9A)=AL~(KSL)

CT)NTANULKKl=~hN(fo

KKz=-NKN(K)IF (hK2.EdOK) KK1=K

IF (R.NE.KI(l) GO TO 130IF (NsTLP!LEto) 60 TO 130WRITt (NOtJTtlbo) ,(TITL(I)oI=1?8)tiRITL (NOUT,21O) KKTRM=KTR (A)

IF (“srt~.~1.o) CALL PHIIT (K)KTRlm)=KT4M

CONT&NUE ,CALL ?INECHK (K)CONTINUEIF (INPuN,~Oo7) CALL PCHUUT (LXX)STOP

FORM*I (8Alo)

FORM*I (1H191OX$8A1O)

FIT ]nt41

FIT 1R91FIT IC)OIFIT 1011

FIT 1921

FIT 1931

FIT 1941F’Ir ]951fry lob~

T 1.)/1

Fil 1Q81

FIT 1c191

FIT 2001FIT ZO1lFIT 2021

FIT 2(I31FIT 2041FIT 2051FIT 2061FIT 2071FIT 20dlFIT 2091

FIT 2101FIT 2111FIT 2121FIT 2)31FIT 2141FIT 2151

FIT 2]bl

FIT 2171FIT 2181

FIT 2191

FIT 2?01

FIT 2?11FIT 2?21

FIT 2?31FIT 2241

FIT 2?51FIT 2261FIT 2?71FIT 278]FIT 2?91FIT ?701FIT 2311FIT ?321

FIT 2331FIT 2341

FIT 23S1

170 FORM~l (l<Ib) FIT ?3bl

190 FORMAT (b~12.5) FIT 2371lqo FoRMAl (JHI024H RESULTS FOP GRoUP NO. ●13911H E-LoWER = ,lPk12.~~FIT 2381

1 17H MEV. E-uPPLR = ,1PE12.5*5H MEV.) FIT 2391

2nfi FoRMAI [lt10*6H J =.13.9~ ALAMDA =tlPE12.5s$H 9 =OIPEIP.5) FIT 2601

21n FOR~Ml (lfloo3bH STEPIT HAS BEEN CALLEO FON GROUIJ .13) FIT 2411

i??n FOI?MAI (2L11.Q.4111 ,14*12$13,15) FIT 2421

230 FnRMMl (bCllo4) FIT 243]

Et.111 FIT ?441

SIJRRUUTIN: SELECT SliL

c SEL io

THIS ROUTINE USED FOR SELECTING A SUaSET OF THE nATA l’O BE FIT. SEL 20

: SEL 30

COMMUN /P~LSIN/ ALAMDA(50),9 FX(?OO)* T(401)0 KTRM9 ITSPO IPW13S SkL 40

1 NIN* NOUTCOMMUN /P~JLSDAT/

1 595U)

COMMUN /puLSC~L/

StL 50NoTt ~t5(Z5), GK(?59400)t NkRG, ALF(25,50), ALAM(dStL 60

SLL 70A(5fl,!5U)s R(5091) SEL 80

67

cOMM~l~ /M~NI/ wx(100)c TITl_(fI), IPT(50)* N~(lO)c KKN(?5), I)lFLIM SLL 90

c SLL ]00

READ tNIN*%rv) ITS, (IPT(A)SI=l,ITS) SkL 110

c SkL 120

c ITs=,*u OF TIML %TFPS DLSI~tO. ;, 1, 1 30

c IPT=LNULXLS OF [)LsI!?ED TIMt STEPS. -IL. ]*I)

c SLL 1s0

00 .?u K=lsNERb SLL ) 60

on IU I=l*ITS SkL ]70

Il=Irl (I) SLL ]80

ALAMuJV(I)=T(I~) SEL 190

A(K,I]=GXIK!IL) SFL ?00

10 CONT$NuL2(I COAJTJNUE

SkL ?10SEL 220

00 4U K=I!NERb StL ?30

cvfr 3U I=lsITSGx(K,IJzA(KsI)

SEL ?40SLL ?50

T(I)=ALAMdA(x) SEL ?611

30 CONTINUE SLL ?70

WI?TTL (NoUTS60) ([~T(I) sGX(KtI),I=191TS) SEL F80

4n CONTLNUE SEL ?90

ITsP=lTS stL 300

RETU~N SEL 310

c SLL 320

%0 FORMAl (1216) SLL 33o

60 FoRMMI (4H 1=,13,3H T=P1E12.5?4H GX=?1E1205) SL1. 360

END SLL Sbrl

slJRRuUTINt TRMSEL (LKQKKX) TRM

lUM

; THIS NOUTANE IS CALLED IF KKN IS SET NEGAT1.VE F~Q A GROUP. TRM

c ROUTANk PH1N7~ OUT TERM 13Y TERM CALCULATIUN OF Fx FOR Ust IN ruki

c ADJUSIING FITTING pARAMLTLRS. THIS DONL FJY CHANGING ANI)/OR TRM

c REI.IOVINti parameter%. THM

c rw

COMMUN /MANI/ WX(IOO)O TITL(8)s KTR(50), NS(Io)o KKN(?S), OLP.LIM TKM

COMMWN /P!LSIN? ALAMOA(50)9 Fx(?oO)i T(401)0 K!RMQ ITSP~ IpKuw* TRM

j NTN~ NOU1 rHM

C(IMMUN /P<LSOAT/ NDTs kB(25)3 Gx(Z5~400)S NtRG, ALF(2%*50), ALAM(drRM1 695(JI rRM

COMMUN /PdLScAL/ TRM(5u*50)s TP~T 150) rnM

CQMMUN /FINNAU/ IRAD. NCOHS* RAOT(200)* 0ELT(200) TI?M

COMMUN /TkMOT/ TL(10), LTM(50)0 LT TRM

c T’RM

K=LK TKM

KTRM=fiTN(KJ TRM

TLll)=lOtlORIGiNAL D TRM

TI. (2)=1oHAQAMATRS FTL(3Js1OHJQ GKOIJP

TKM

IF (hhX.E2.0) G(l TO 10

Tl?Mrnfl

WRITt (NOJT*lbO) (TL{I)*I=l~3)oK lKM

wmTc (NOST*1OO) (L,AL~(KSL) gALAM(KOL) OL=lSKIRw) rHM

In IF (KKX.GT,O) GO TO qO rnM

DO 40 I=l\ITSP lNM

Fx(I)=O. rnM

Do 3U L=l$KTRM TUM

TR14(LoL) =ALF(K)LJ *EXP(-ALAM (KtL)*T(Il) rRM

IF ($KAO.LE,O) GO TO 20 lb.:

IF (lKM(I~L).LT.O.) TRM(I!LI=O. rk,l

COFF=AL~ (~tL1/DELTfI)/A~AMfKqL) o*2 1)+.,

xPolaJ,-EAP (-ALAM(K,L)*RADT (I)) TN:t

XP02=1.-EAP (-ALAMIK*L) *UELT(I)) TN:’

XPn3=CXP (-ALAM(KSL)* (T(I)-DELT(I)/z.)) TR*I

20 CONTJNUk ru~.

IF (JMAussQol) TRM(I*L) =COFF*KPOleXP02*xp03 rhs

LTwfLl=L rl...

FX(Il=FX (I)*TNt-V(I,L)

1020304950

60

7080

1::110120]3014rl)50160

170lBO

190?00?10?2n730

?40?50

?bll?70?80

?Yo

300710?Z()

330?40350760?rfl

“+60

68

30 CONTLN~L60 CONTLI’+UL

K]=l

50 K2=KA**

IF (A<*GToKTRM) K2=KTRMWI?TTC (NOLJT*lIO) (LTM(L),lL=KI*K?)[)0 su 1=1oTTSPWQITC (NOUT$180) T(l),ti~(Ko I)*(TRM(ItL) *L=K1$K2)

60 .CONTJIMUEK]=Kz*lI!= (N1oLEoKTRM) GO TO 50DO RU I=IoITSP

PCTDbP=U.IF (oA(K$l)oLti.001 GO TO 70PCTDJP= fGA(K$~)-FX(I)) /6x(KoI)*)OOo

7fi cn~7iwuF.WRTTt (NoLlT!190) I*T(I)?GX (K, I),FA(I)sPCTDIF

8n CONTLNUL

IF (NNX.LL.0) RETIJRN00 cQNrllrnuL

RITAD IN1N9?101 MLTIF (MLT.Ld.0) GO TO 110

DO luu MM=I*MLTREA13 (NTN~200) t.t ALF(KsL)sALAM(K*L)

100 CONTANUL

lln CONTANU~RFAD ININ!21O) LT*(LTM (JJ)cJJ=I,LT)

cc MLT=NU OF 9ARAMETEQS TO BE CHANGEI)

c LT = NUMHllR OF TLRMS TO BE’ REM~vEU,c = “l~UM NOs. OF TLRMS TO ME REMOVED.I.TM(L) -

cIF (LI.EQ.0) RETURNLTx=uLL=lKTRM=KTR(K)

DO lJU L=l,Kl~M

IF (LIM(LL).Ew.L) GO TO 120LTX=LIX*l

TRM(IcLTX)=ALP (K*L)TRM(C,LTX ).=ALAM(K,L)Go TV 130

1?0 coNTll\uELL=l L*A

13n COMTANUEKTQM=LTX

Dn 1*u L=l,KTKMALF(~tL)=!RM(l,L)ALAM(N!I )=TRM(?*L)

14n CONTINUL

TLol=JOHHEVIsETI PATL(2I=1OHRAME!ERS FKTR(A)=KT*M

TL(3)~lUHUR GNouP

WRITL (NOUT*150) (TL(r)~I=l~3)~K

WRITL (NOUTslbO) (L*AL~(K~L)*ALAM(K~L) ?L=lsKTRM)RETURN

c15~ FoRM*I (]H]05xs3A10s13) rlisl 9?9

160 FORMWI (i?in.3H L=.X3.10H 4LF(K~L)=*lPEl 106JilH ALAM(KsL)=~ lPcli.*l TRM 9M()

17n F(31?MMI (1~0*.2bH COUL TIME Gx*5115) TKH Q)()

lHn FORM*I (1P7E13.S)TNP ln~()

lq” FOQMMI (lfl(l,sfi ~=,1303tl T=,1E]2.5,4H 13X=Q1L1?,594H FX=,1E1205*8H ‘rH” ~fll(l

lc7nIr=01ti2.5) T}{ ; . ,1

zoo FOQMAI (Ib,2El?.~) m,! ;1 1(1

21o FORMA! (ldIb) T’Krl 11140

(IND TWM 1050

69

cc

cc

c

cccccc

cc

c

c

c

c

ccc

cc

cc

70

SURRUUTIN~ COHSF!IN

RoIJTJNk F(JI?MS cOARSE GRUUPS FROM t’

FOR4A1 .

COMMuN. /P~LsIN/ Ao(50)r Fx1200), T1 unul

C(Mcow 10

NE QHOUP OATA IN ENoF-LIKL Col{ Z()

cow 30C(JR 40

401)s KCO ITSP, IPRoP, NINo Cl)u 50COH 00

“coiWWN /PULSOAT/ NDF, ~tf(25), Gf\(25,400)s NERG, Fo(200), 0UM(70)J CON 70

] 107(50) con no

COMMUN /ENr)F/ MATi~ MFIS MTIc R,JNTIMS NPUN Coti 90

READ lNW~ C(JU lUORFAD (NIN*90) MA1l*MF1$MT1 C(JU ]10

con lZO

MATI=MA1 No, OF FISSIONING NUCLII)L ULSIR~O. CON 130MF1=IJAT TYPE VES1QED9MfL=t)n=F.p.DATA FOR TflkRMAL PULSE$MF1=M1= Coti ]40

- POP. L)ATA FOR FAST ~ULSF*t’TC. C(lii 150

MTI=l YPk OF F.P. WANTEO*MT1 =801=L)ATA FOR BETA- Pt.us GAMMA*M11=802=COK 160WATA FOR GAMMA 0NLY$MT1=R03=OATA FOR HcTA- OM~y,ETCO

SEARLn kNOF TAPE F09 DkSIHtD OATA.

10 CONTINUEREAD [NDFsIoO) (An lI)t 1=1 S7)SMATtMFfMTSNSEUIF (mtsEQ,-0) MF=ROIF (IIAr.EfJ,-1) WRITE (NUUTOI1O1 MAT,NDf

IF (mA1.Ed’.-l) SlopIF IMAT.LI,MAT1) 00 TO lo

IF (mfi!.Ed.MATr) 60 Tn 20

WRITt (NO~TSllO) MATl,NuFSTOP

?(I CONTLNUE

IF (MF.LT:MF1) GO TO 10IF (~FoEtJ~wFl) GO To 30WQTTE (NOUTSldf)l MFloNU~STOP

>0 CONJTLNUEIF (M).LT~MTl) GO TO 10

IF (ml~fiQ:MTIJ (W TO 40wPITC (NOUTSISO) MTIsNuPSTOP

40 CONTLNUFREqO (WF!140) NTRl!ArI (NrJF~qO) NICHT

ITsP=rJT

RFAD (NIN$90) NE

REAO (NINs1501 (kR(N)CN=l!NE)

NE*NU OF dQOAU (iHC)UPS PLUS ONE,

E13=ENLnGY qO~JNOS INcLIJu~NG UPPLw ANO LOwER BOUNOS IN MEWNEzNt-1NEI?G=NE

DO 7U IT=ltNTREA13 (NUFS1601 T(IT),KL

RFItD (Nf)Fsoo) NICHTRFAD (N:IF$}50) (Lil(K) !FA(K)~K=l.Kt)

I(E1=RE*l

Eo(KcL) =(Eo(KL)-Lo(KE-1 ))*to(KE)

THE PuLLING Ll)Op CHANGLs UNITS TO MEV/SEC, THE TIME OUI?ATIONOF Tnt PuLsE 1S AsSUMEU TO BE 1.E-4 SEC.

DO 5U K=l!KEFX(K)=Ff(K)/1.E-4

%0 cONTiNUkCALL HLHrN

00 6U [L=L,NE

C(JN 170

CUH Ido

CUM I 90C(JH ?00

cr)N 710Cu}t ?20Cuu ?30

clJn ?40CON 250C(JH 76oCllu 7T0Cl)ti ZdoCOH 791’ICON 100

c(lt4 310Clllt ?.?0cut< 130C(JR 7*OCUN ?50Cuu ?60C1.lu 370Cuti ?80l;llH 390-f)~ 600,; ,~ l.inc{ .20:(.1/ 630

CON L40CIJU 450CON f,boc(lt4 6 ff)

con 480

CON 490con %00Cl)u 510CUK 520CUH G.30

Coti %+0COR %50C(JU Gfll’)CUM %;(I

cUK Gdo

COR %90

con ~ooCON AloC(3H f*I?o

COK (.30c@w 640CI)N 65(ICO}) f.b(lCON A70

COH 680

wRITc (NP~N09U) NF,NT

WRITt [NPJAJ*lYo) IE,ER(IEW12rTL (NP~N*190) (IT9T(LT

80 CCJNTINU~QEWINU 10RETURN

c9q FORM*I [6111)

lno FORMAI i6:10$A6~14s12,1J!15)

11o FORM*I (1H1s13H SOR2Y! MAT z ,14s13H NOT UN TAPE ,13)

lzo FO?~~Al (lm*14H SORRY* MF = 014c13H NO! ON TApE ~13)13n FORMMI (lrnlol~H SORQY* MT = s14s13H NOT 01~ TAPE ,13)lQO FORM*I (55J, Iii)1%0 FORM*1 (6E11c4)A60 FORMHI (l Jx, lLL1.4Q33XQ111)

CcieCuuCOHC(IUCcuCURCoucud

codCLJRCOHCUMccl+

CORCUMCO04

C(JUCON

C(,c1 $,

~(!.

Coti170 FnRMAl (1~0,1/H ENERGY 13iN NO. ,13,6H PROM ,lPEl~05,FlH MEV !U ,~#CUN

I E12.b*bH MLV.) Ctid

180-FORMMI (If’i Sl~H TTME <TLP = ,13,16H cOOLINW TIME = ,lPE12.5*61i FA cOti

1= *1VC12.’3) cod190 FORM*I (z(Ill??E1104)) Coti

END Coti

SiJnRWUT[NE I?Et51N Rk+

c tikdc RnlJTLNk F()$I BNOAIJ GRoup BINNING. BOUNOA~ILs 00 hlnT HAVE TO *Es

c COINl,il)t WITH FINE GRoUP BOUNDAQItS. WPITlkN dY GRAHAM Ntl-1

c FOSTt-Ktl.ASLo 1975.c CH&NUtU OCTC 1977, LASLt II. GEOi+Gt

cCOMMUN /PdLSIN/ Ao[50)scOMMuti /P~~SUAT/ N(jT,’

1)SLIM1=SUM2=0.NUj=Iiu+l

NXI=WA*l

DO lu I~J=ltNUl10 V[lUI=O*

DO %U Ix=i,NxSUM1=SUM1*Y(IA)

.?() CONTINUtC FIND THt. FIRST RIN

~o 3U lX=A,NX

If (AIIx).EQ,U(il) GO.3n IF (AI IA).GT,UI})) GO

WRTTL INOUT!11O)

PFTU~N

.~l) V(1) =7 (IX-I)*(XIXX)-U50 IX=IA*l

DO 8U IIJ=A,NU6n IF (A IIX)OGTOUIIU+l))

V(IU)=V (IU)+Y(IX-l)IF (JAoGToNX) GO TO 9(Ix=IA+l

‘GO Tu 60

u

TU soru 40

.i))/(x[Ix)-xfIx-1))

GO TO 70

7n V(llJl=V IIU)*y(IX-I)*(U fIu*l)-X(TX-l))/(X(I~) -X(1X-I))

f3n CONT41’JIJE90 IF (A(NxII .GT.u(NIJII) wHITE (NO11TS120) x(NAl)cU(NIJl)

0(3 luu [u=IsNUAno suM2*buM2*v(Iu\

h9(l

70071072073fl

7*O750

7bd

77n

Tt30790

noonloP~fJfi~~

~4(1

630fsboh7Q

F80A’lo

fjoo~lo92(IQ30

940~;o060

1020

304(I5 [j

0070d [)

9rJ

100

1101.?0

1 30]40150

lbO

170180190

?00.10-do23(I

?40

?50

760?70zdo

?90

.700

7107203307$0

-450360

EURD=UOOOA k:

IF (Am~(S~M2-~UMl ),GT.EKRDe5UM1 ) WRITE (NOuTQ130) SUM1OSUM2 #t *

RETU~lW *L .

c I’/: .

llo Ff3RMMl (29H CANT FIND FIRST ENEuGY BOUND) *t *Lao FoRM-1 (1/Ho*oo** LAST UAIUt-4Elo.3t271i EXTENUS REYoNO ENo OF tiRI[)kl~ts~

SIJRRUUTINL RUNTOTS I{IJ ~,

c d....

c RouTINE To REAo TOTALSO I.E.oGX(KsI) sUMMtO OVER FNERGY. Nu’w

c UuvCONMuN /pLJLsIN/ ALAMDA(~o)s Fx(?OO)s T(401)* K~RM, ITsPO IpRuU~ NIJN

I NINs NOU~ HUQ

cOMMUN /PJLSI)AT/ NOTt tM(25)~ Gx(Z50400)$ NERG* ALF(25$50)* ALAM(2HUN

cccccc

c

10

2n

c3n40

50

Ec

E

1-5,5UI -CCIMMUN /PULSCAL/ A(5fI,50)c R(50,1)

COWMuN /PJLSOuT/ ALPHA[50)s FXC(lUO)o PCT(1OOJCOMMUN /FINRAU/ IRAD, NCONSC RAI~T(200)s 0ELT(200)

THIS subroutine IS llSEU FOR I?EAI)ING EXpERII~kNTAL OATA.lJIJFOKIUNA!ELY! THIS cAN ~t RFCEIVLD IN A VAKIEly OF FORMATS*~u

THIS HOIJTINL MuST 8E CONTINUALLY cHANGED To ACCOMAI)ATE wHATkvtnFORMMI TH> DAIA IS IN. THE FUNNY LOOKING >lATEMFNTs t3ELOW AKL

FOR NcAr)INS SOME ORNL uATA IN Tfit WHITTEMOXt JUNK FOHMAT.

DIMENsION HOR(F3)

RFAD ININ$301 NPrNE,NH~

On lU L=l~NtiORFAD ININ$50) (HUR(I),I=IS8)

un17c (NO~T*50) (HOR(I)*I=108)

CONTANUEDO 2U 1=1*NP

IF (LoLE.’4EI kB(I)=o.oRIZAO [NINt40) (GA(K,I)oK=1O*15)WRITL (NO~T!40) (GX(K*1)CK=1OO1%)RAnT(l)=Gx(looI)TII)=GX (11*II+GX(12*I )z~.o

DZLT~l~=100

GX(191) =(UX(l~SI)+OX(14 lI))*6x(lo*I)/GX(12QI)CONTINUITSP=NP

NF=lNERGniI?FTUNN

FORMA rFoRMMIFOI?MA1

ENn

sUqRUUTIN~ LIHFIT (K)

RoUTLNE wITH SUaNOUTINES FUNK ANO STEPI? PLRFORM TWO FJARAMEILN

FITSCI.E. ROTI+ ALF AND fiLAM. ADAPTATION OF STEPTT CODE BY

M.Go=l AMA]ALA!Os~LASLC ly76~FOR THIS APPLIcATION,

CO~MUN /PbJLSIN/ ALAMDA(50)Q oH(?oO)* 1(100)9 NERSQ PERDIF(1OU)B

lfl20

3040

50607080

90100

110

1201301+0

150160170

1s0l%

?00

?10?20?30740

250?60

??0?80?90aoo

310720330

7*O~>o360310

380

--490400

l+ur” 410

Dlil 60

1 CALL[lI_JO)o W(1OO), KTqM, ITSP9 IPROB9 NINC NOUTC(IMMWJ /PUI_SDAT/ Nf)T. LB(25), HOLD(25,1OO)Q NV~ NTRACF, MASK170)$ DH~ 80

1 X(7U)S X~iiX(70)s XMIN(fO)s bELTAai70i* 0ELMIN(70)s DuMMY* MAf~IX$~H~ 90? EQR(tO*7U), ctiISo, RSAV(1OO)* VEL(70)* TRLAL(70), xsAVE(70~9 CHI(DHf_ 100

72

3 70)9 sEC3W(Z,?), 0LDvCC(70), %ALVO170), ~OSC[70,1!3), cri10SL(i5),0H~ 1104 PllMlbafJ)t NERG, ALF(?ZS50)* ALAM(.fcj?50) ~MF 120

COMMUN /M~’JI/ TC(1OO), TITL(13)9 KTR(50)9 N~(10)9 KKN(?5)Q Dl~LIM Dtif 130

It-l

2fl

4fl

50

C(IMVVN /El~nF/ MAT, MF~ Ml+ RuNTJMs NPUNCOMMUN /FINNAu/ IRAc)~ NCO~Ss RAnT(200)9 DELT(200)DIMEIYsIUN TTLIIO)* XLL (10)9 YLL(10)

Do lU KK=l.KTNMIr=2*l!K-1IJ=2VhK

X(III=ALF(K9KK)

X( rJ)=ALA~,(K*KI()cONTINULNV=2*RTRMIF (NVOLE.70) GO TO 20

WRITL ( 10dT*l~o) NVRFT(IKNCf)NTINuEon 30 I=l~r.Jvt2XMIN(l)=X(I)*~tll!= (A(I) .LT.O.) XMIN(I)ZIIO.~X~I~

XMAX(1I=X(T)*1O.IF (A(I).~~.o,) XMAX(I)=X(I)OO.I

xMIN(L*l )=o.5@x(I*l)XM4X11*1) =Z.0*X(I01)CONTINUk

wI?lTc [NOuT~lbn)

MWTTL [NO~T*llO) (X(1) 91=19NV)tiQrTc (NOJT,l#jO)

WRITC (NO~T91(fI) (XMIN(l)OI=]SNV]WRITr. (N05JT*191))

WRITC (NOLJT~170) !XMAX(I)*I=l~NV)00 4u I=lIN’JOFLTAA(l)=O. l*X(I)

DFI.MA”lw( I)=0.00I0XII)

MASK(l)=O

NERS=ITSPDO 5U l=19NERbQW=l*

W(Il=ll./DH(Il)**Q4CHS(J=U.

Dn luu I=lcNE~sCALC(l)=OOD(Y 9U J=1?NV9Z

IF (lRAD.LF.0) GO TO 60COFF=A(J)/DELj(I)/XIJ* !)**?

XPOl=j O-EAP(-X(J*l)@RAOT (I))X002=I.-EAP (-X(J+l)@OEL 1(1))

XP(13~txP (-X(J*l)*(T(I) -qELT(I)/?.))CALC:l)=C~Lc(I) @COFF*X~UI*XP02@XP03GO TU 90

TEMPaAIJ*l)*T(I)IF (*Bs(TCMP)*LF.6oo) tiU TO 70)(Pn=u.

GO TU 80

XPCI=A( Jl*&XP(-TEMp)

CALCI1) =CALCII)*XPOCONTINUL

PEROl?f l)=1000*(cALC( 1)/DH(I)-l.)CHSO=LHSQ* (CALC(I)-OH( I) }O(CALC (I)-O~(I))~*f I)@w{T)

WRTTC ~NOUT9200) CHSQ

WQITL (NoIJT~210)wRITc (NO~T02dO)WRITL (NO~T$~~O) fT(I)*uH (I )ocALC(I)9PERDIF (I)~I=lQNERS)CALL >TEPITDfl ~du I=loNENs

CALC(J)=OODO lCU J=i.Nv92

IF (LMAO.LE.0) GO TO 110cfiFF=A(J)/nELT(I)/XIJ* 1)**?

OIIFDtiF

OtlFoHt

OItf~hi0}4,

oh+

DtihoHFOHF’

UHF[)HF

OHFI.)HFl]llfUHF

[)HF(IMP

UHFL)HFoiiF

,Itit)1$$

,;}.,

oh-OF’:”[)}..()),1(),,;

Llrll-I.)HFOHFOHF

i)HFOHFOtiF

L)tiiDHFOtiFol+FOHF

DtiF

OHFDtiFDtIFOHF

DHFotiFDHF

DMFOHFUHF

OHF

l.)tiFOMF

(IHF

UHFUHFllnF

I.)MFOliFOtiFoHF

OHFOHF

140150

160

170180

190?00

?10220

?30240750

26(Ipro

T80790

300~lo32n

7303403!Jll

3603103*n3+IJ

600410

420

430440450

460670480

490%00

510F20

530C+o~bo

<60570

%80~Yo600

610620630

640A50

660

670Ado690700

?10720730

740750760770

,780790

73

XPOI=I,-LAP [-X(J+l)ORAUT (I))xPcI?=I o-EAP(-A(J*I)*nEL~ (I))

XPn7=LXP (-X(J*I)*(T(I )-UELTII)/>,))

CALCtl) =CALC(I )+COFF*XPUl*XPOi?~XP03GO TU 120

11o TFMP=A(J+l)*T(I)

IF (MBs[TL~P) .GT.601’)) Gu TO 120CALC{l) =CALC(l).X(J)UEXP (-TEMP)

120 CONTiNUk

DO lJu l=l,NEKs13o PFQDIF( I)=IOO,*(CALC (1)/Dti(I]-l.)

WRITL (NuUTt2Zo)

WQITL (NOUTS230) (T (I) JUH(I),CALC (I) OPERDIF (I)oIs19NERS)

ENcOUt. (3f,d40,TTL(I)) KENCOUL (20,i?5U,XLL)EI.cOUL (2u,?6U,YLL)

c CALL kLOTY (TsPERnIFS-l!LRS~l~o,n$ U.sl,~J.~[TL$34,XLLo20 .YLLtdO)r)rl 1*u J=L,RTRM

I1=?VJ-IIJ=2*J

ALF(~tJ)=A(II)AL4MIF*J)=X(IJ)

lLCI COFJTINULR:~,j~~

c15n FORMAI [24H0 TOO MANY TERMS. NV =,13)

16n FfIRMAl (lHo*lOX, 1RMINITIAL Parameters,/)~70 FOQHMI IIH v20x,6E12.4)lqo FORM*T (l? ?1OX,1?HLOWLR LIMITS,/)

190 FORMAI (ld *lUX,12HuPPtR LIMITS,/)

ZOO FOI?MMI (1A 010X,36HCHI-SC)UARE WITH 0RIGIN4L GUESSES IS OEI?.*)21o Ff)RMMl (l~iooloX.31HCOMPARlSON WITH INITIAL UUESSFS,/)

Z?n FORMMI [lmo,//,QAslHT,4oXo2HDHsQX* 4HCALC$?As6HOtRD~F*/)~?IY FOJ?MMI (1M ,4F1?04)200 FORNMl (32HPENcENT I)EVLATION Fok GROUP NO. *12)

2%(I FOQMUI (2UH TIME IN SkCONoS )

260 FrlRqMl (21JH PE~CF.VT DkVIATION )EIJn

SIIRRUUTINL FUNK

COMMUN /PyLSIN/.ALAMDA(bO)9 DH(?OO)* T(1oo)9 NERs, PEROIF(lUU)O

1 CAL~(lfIO), N(lOo)O KTXM~ ITSPo IPROBQ NIN! NOUT

L)HFoUF

OHFon~DnF

OUFonfl)hF

l)hFlltiF@tiF

DHF

DHFDhF

oHFUHF

onFL)HF

OHFUHFDHF-

On+I)t\fL)rlF

UHF

I)tiF

OtiFUHF

OtIFOh.

l)t-.Dt-

l)bohDH~OHI

Dt+tDHF

FUN

FUNFUN

fioi)

$I1OrldoR3fiR40

850A60R70

R&o890

000

Qlo

920930

940qfJo

9609ro

Q80q+o

lnOO

ln10lr12n1030

10401(350

lf1601070

lntio\f)90

1]00

11101~20l\31)1140

1150]]60Ilrl)

1020

COMMUN /PJLSOAT/ NOT, E~(i?5), HOLU(25,1OO)? NV* NTRACE, MASK(70)$ FUN 30I X(7u)* XMAX(fO)$ XMIN[tO), DELTA, 0ELMIN(7fI). UUM4Y. MAIRIAsFUN 40

? FRR1/0t70)* ~HIb13, f3SAv(100)o VEC(70)S TRlAL(70) ;-X5AVE171))P LH1[FUN3 70)s xECDNU(~,2), 0LDVEC(70), qALVO(70), AOSC(70,1S), cH[05L115), FUN4 f111Mlb86)t NEKG, ALF(2~*sO), ALAM[d5$so) FuN

COWMVN /M~NI/ Tclloo), lITLIR)$ KrR{SO)s N>(10)! KKN(?%)s oI~LIM FUN

cfJMMUN. /LNr)F/ MAl# MF, MT* TLIMIIs NPtiN F UN

COI.iMUN /FINRAU/ IRAIJ, NCOKS, N41)1(zcIo)$ UELT(200)

DIMEI!SIUN LxX(20)

FUNFUN

CHISU=OO FUN

00 5W I=l!NER~ FUN

CALC(i~=O. FUN

Do 4U J=lsNV~L FuN

IF (lKALI.LE.01 GO To 10 FUN

Cf)FF=X(J)/nt’LT(I)/X(J. 1)**? FuNxPol=I.-EAP (-x(J*1)*RAI)? (1)) FUN

XP02=1.-EXP [-X(J+I)*DELT (1)) FUN

XP03=cXP (-X(J+l)*(T(l )-qELT(I)/~.)) FUNCALCl I)=CALC (I) +COFF*xPOl*XP02.xP03

GO TU 40

FUN

FUN

10 TEMp=A(J+l)*T(I) FUN

IF [MMS(TLMp) .LE.AOO,) bo To 20 FUN

XPI-)=U.GO TU 30

FUNFUN

?() XPO=A(J) *~XP(-TEMP) FUN

5060

70ao9n

:;:

].20

130140

)s0]60

170180

190

?00?10

220230?40

?50?60?ro I

74

7(Y6~

50

6n

70

RO

c9fi

100lln1201701.4 IYlsn

CALC(l) =CA~C(l)+xPOCONTINUL

RShV(l) =($ALC(!)-nH(I ))*W(I)

CHIs~=~riI?O+BS&V (I)*RSAV (1)DTFMMA=(J.

nn 6U 1=19wEH3

PEQDiFI I)=lOo.*(CALC( 1TSTLIM=QB~(PERCJl~ (1))IF (uIF~AX.LTOTSTLIM)

cONTLNuFIF (UIFMAXOLEODIFLIM)

CALL StrO~D (TYM)IF (ILIMI1.LT.1) TLIM’IF (IYM.LIOTLIMIT) GO

WNTTLw~lTCwh’~TL

4nTT~WRITLWIJITCW12TTL

DO 7U fih=l,NERGLXX(RA)=IISPcONTINuk

)/DH(I)-l.)

LJIFMAX=TSTLIM

ctllSQ=I,E-2U

T=2Y9.51(J 80

FUNFUNFIJN

FUNFUN

FUN

FUNF lJii

FUN

FUNFuNFUNFUN

FUNF(JN

CALL P~HOUT (LXX)

CfJt4T1NuERETUKN

FOI?MAI (1P4E11.4).

FIIN

(X(I),lnl,NV) FIJN

X(1)$ I=l!NV) Fb!,

FIJN

fUN

(T (I )*OHIII *CALC(I)SPEKD: F(I) ~I=IsNERs) FUNFUN

FUNFUNFUN

FUNFUNFurl

FUN

l=OQ&41+1 (lHotllJx,22HTIME LIMIT W8S REACHEO,/) FUI,

FORM*1 (lti C1UX,30HTHE LAST SET OF PARAMLTLKS wA%s/) FU’

FflRMAl (lrn *20X,6E1?,4) FU:

Ff)QMAl (lrn sIUX;ZQ*COMPARIsON ,AT expiration TIME*z) FUP

FOQMMl (lIIn,//,qA,lIiT,lUX*?HDHS Qxs4HC4Lct7A06HpLPnIF*/ ) Fu’

FnF?MMl (17 *4L]?04) FU-...

ENIT P v 4’:

suRRuuTIN~ STkPIl ST?

cOMMUN /P~LSIN/ ALAMOA(5O)* DH(i?OU)O T(1OO)S NERs9 PEROIF(IOU)t ST.’

1 CALb(100),t w(100)s KT~Mt ITSPS I~ROBq NIN* NOUT ST:

COMMVN /P!LSOAT/ NDTS ~8(25)s HnLD(25,100)0 NVS NJTRACES MASK(/o)S sT’

1’ X(7U)0 XMAX(70)$ XMIN(/0)9 DELTAA(70)~ UELMIN(70)0 UUMMY, MATKIA*STt.

2 ERQ(10*7U)* CH15Q* RSAV(1OO)* VEL(70)0 TR1AL(70), XsAVE(70)S cHI(sTt

3 70)$ >EC~NDl~,2), OLOvLC(?O)S SALV0(70)t Aosc(7n,15), cH105$(15)*sTL

4 I_N1M(b~6)9 NERG, ALF(?~$50)t ALAM(dS150) sTE

cOMMuN /M~NI/ ox (1001* TITL(~)~ KTR~50)* Nb(lO)Q KKN(~5)* OIFLIM STE

NVMAA=?o STE

MfTsouL=15 STE

Ku=& STt

RATIU=lo.o STE

COL7N=0099 sTF.

Nc0MF=5 STE

ACK=C.O STE

SIGNL~=ZoLfI STE

HIJGE=l.L3(IF (Nv) 1J1O*131O*1O

SIESTL

in NACTIV=O STt

Do 9U 1=1*NV STE

IF (MAsK(I)) 90~~0990 STF.

2n IF (ucLTAK(I)) ISO~30s60 STC.

an IF (A(I)) 509409’30 ST;

40 DELTAA(I)=O.O1 STt

GO TU 60 ST:

%n OELT*A( T)=O.OAOX(II sTr.

6(I IF (AMAx(I)-XMIN(I)) 7Ut70S8TI ST:

70 XMAXIJ)=HJGE STF

XMTN(ll=-~\JGE sT{.

?s0~+o

?00310320

330740

350q6rJ

3703do

390400

410

420430440

*bI-l460

470460490

500

<10fii?o

%3054n%50

660

%70~8n<90~oo

fin620

630

1020

30

405060

70M(Y90

100110

120130140150160!70]80

‘]YG

?00?10220?30

?+0

250?60770?80p90

75

an NAcTIv=NA~TIV*I ST? 300X(l) =AMAXl(XMIN (I)tAMINi (XMAX(I)SA(I)))

9f) cCIVTLNUk

s:, 31rl51. 720

Cf)MPAK=O.(1 ~.. 330IF (txACTIV-1) lolJ,l~o,lZO ~..

100 00 llu J=l,NV340

,3,. 350lln MASK(J)=•

Go Tu 10

ST, 360

l?fI A=NAbl Iv

s’ 3’105 3H0

sljq=z.U/(A-l.u) s 391)

P=?.u*(loO/SQRT(A)/(1 .0-0050*SUA)-1.0) s 600c0wpA~=4M1V1 (0999,4Rs(~ 10-( l.-c[)LIN) **sU8)0(l. ●PO(~o-coLIN) )))

130 Ch[tL PUN4

s.: 610ST.

NF=l620

IF (l~V) 131O*13]O,14O

Srt 6.30

140 DO l=Ju I=l,NVSTL f440

150 Dx(I)=ULLIAX(l)SIL 4s0

STL 460cHIol_u=.HlslJ SIE 4?0Noscsu STL 680

160 NCIPb=O SIL 690

N71P=U STf %00

c MAIN UU LOOP FOR cYcLING THROUGH THE VARIAtiLESO Slc Slo

c FIRSI TRIAL SrEP WITH EACH VARIABLF IS sEpARATf-. STt. 520

17fI NAC~=U STL %.30

no q*u I=l,NV STL %40

OLnVCC(I)=VEC(I) STE %50

VFC(l)=O.U sTk %bo

TQTAL1l)=U.O sT~ 6“ro

IF (MASK(1)) 180t190*150 Slk %dll

18fI VEC(l)=-O.O STL 590

GO TU 940 STL 600

l_90 NACK=NACK*l STL 610

XSAVL(l)=X(I) sTE 620

IF (31GNIFoABS (DX(I))-A8s(X (I))) 3400340$2U0 STt 630

ZOO x( I)=ASAVL(l)+DX(I) srt. ~60

NFLAu=l STL 650

IF (A([)-XqIN(I)) 22O,21OO21O Srt 660

210.IF (A(II-AF4AX(I)) 230,230,220

2?(I NFLAu=NFLAG*3

Slf. 610STL 680

GO TU 2*O ~T~ 690

230 CALL PUNK SIL 700

NF=N* ● 1 SIF. 710cHTML=C&41SCJ STL 720

IF (Lti15(A-cHIULD) 3Er0,240,250

24n NFLAe=NFLAG*l

Sit. 730511. 7*O

250 X( T)=ASAVL(l)-DX(I) STL

IF (x(I)-A41N(I)) 350,2bOt260

750STL 760

2A0 IF (A(l)-ALiAX(I)) 270$~70S3S0 STL 770

270 CALL PUNK STt 7M0NF=NP+l STL 790

IF (LnISQ-CHIULD) 370,280$290280 NFLAu=NFL~G+l

Sit. R(IOSTt. nlo

29n IF (NFLAG-3) 300t340t350 STC R20

300 TRIAL II) =~X(1~*O*5* (CH15Q-CHIME) /(CHIME-2.0*CHIUI.D*CMISO) 5Tt R30

IF (INIAL( [)) 3100350,310 STE R40

310 VECI1)=lRL.4L(l)/AHS(Dx (1))X( T) =ASAVE( l)*TI?IAL(I)

STL R5frSTL n60

CAl_L FUNKNF=Nrol

STK R1OsTE Rflr-r

IF (WiIsti-fhIQLn) 320,-330s330 >Ik f19rl

3?0 CHIOLiJ=cHISd STE Q(IO

GO TU 360 Slt 910

33o T$?IAL(II=O.O sTL 920VFC(L)=OOO STL

GO TU 350

QAO

STE Q*O

34n VFC(II=-O.O STL 950

35o X(I)=ASAVL(l) STE. q60360 NCIRL=NCI~C*l STE q70

IF (NLIWC-NAClIV) 450s950s950 STL qdo370 nx(I)=-oX[l) STE Q90

76

c A LOnt N VALUE HAS 13EEN FOUNO.

38(Y NCTRL=O

DFL=uA(l)

39n CHTML=CI+IULDcHTOLU=~tiIS~

VFC(L]=IJEC(I)*OFL/ARS(oX (I))TPILL(l )=!RIAL(I)+oEL

13EL=MLK*l)~L

XSAvL(l)=R(I)X(I)=ASAVL[l)*OEL

IF (A(I)-XWIN(I}) 440J4UO$400400 IF (AI1)-AMAX(I)) 610.~10!460

410 CALL “FUNK

NF=NP+l

t{ENcE THIS VARIABLE WILI. cHANbk. Sfk If-1ooSTE 101o

srt 1020s’1 1030s-i 11-140‘> . lrlbflS.:. lnhoS L 1070

s.:. lfitll)s-l 109I)

S’F 1100\“l. llAOsnL 1)20fjl~ 1]30

IF (L.MISQ-CHIULD) 390s420$420 srE 1140

4%0 cINOCM=O.5/ACK*(ACK*~2*~HI ME- (ACK**2-1.0 )*CMIOLD-CHI SQ)/(AC~’’CHIYL\lL 1150

I -(ALK+100)*cHInL D+cHI?Q) ilk 1160

4.30

4R()490

51)fJ

510

520

590

591-Jbon

cc

x( T) =ASAVL( l)*CINDER~D!LCALL ?ONKNF=N?*l

IF (uni5(J-cHIuLD) +30,440s440CtiTOLU=CtilSti “TNIAL(l) =TQIAL( I)+C]NDLR6ULLVEc(AJ=ULc(I)*cTN[)EROo~\/ASs (OX(I))

G(T TU 450XII)=A5AVL(l)

IF (wLIP-I! 9309460,460IF (Ab5(vLC(I~)-ACKI 49u~470r47n

OK (I )=~LK~ABs(Ox(I))

VFC(ll=~EC(I)~ACKOLnVLLII)=OLDvEC(I)/ACKOn 4MU J=l,MO~OllL

ERQ(ltJ)=LRR(l,J)/ACKS(lvo=u.o

SUMV=U*ODO 5UU J=l,NV

stJMo=5u~o+oLovEc(J)~~2

SllMV=SUqV+VEC (J)0*2IF (suMu*51JMv) 930,930$510SIJMO=SUNT($UMU)SUMVZbtiNTISLJMV)COSINL=O,O

00 5.2u l=l,NvcnSINL=C051NE*OLOVEC (J)/SUMO*VEC (J)/SUMv

IF (NLIp-1) 9~O*530SS40IF (NACK-VACTIV) 930,5bu,560

IF (I~ACU-X4CT1V) 560s55U,550IF (NLIP-NcOMP) b60,570~570IF (~uSIN~-CUMPAR) 930,570$570SIMON SAY5, TAKE AS MANY blANT sTEPS AS PObS18LE...

NG~AIwl=O

NTRY=UNRETmY=Ll

KL=lNll$c=NOSC+’I

IF (NuSC-MOSQuE) 600,60U,580N05C=MOSQI+F00 5YU K=$,M05GUt

CHIO>L(K-l )=CmIOSC(K)DO 5YU J=l,NV

XOSC(J?K-l) =XOSC(J!K)ERR(J,K-l )=ERR(J*K)

00 6AU J=l,NvXOSC(J$NO>C)=A(J)ERR(J,NUsC)=vkc(J)/s(JMv

CHIO~L(NOSC) =~HIoLoIF (NUsC-J) b70!b20,620SEARUM F02 A ~RFVIOIJS SUCCESSFUL GIANT sTEP IN A

NFARLY ~AtiALLLL TO THE OIKLCTIO~J OF THE PRu~OSEil

!jl~. 1170

STL llan‘>TL 1190

Slt 1?00sTE 1?10>Tt. 1?20STt 12J(I

STR 1?40STE l?bOSTE ]?60STE 1?70Slk l?tJO

STE 1?90STL 1300STE lqlo

SIE 1?20STE 1330

STL 1340STE 1350

STL 1’360STF. 1370STL ln80sTE la%STE 1400STE 1610

STE 1420STE lt+~OSTE 1440sTE 1450STE 1460

sTt 1470STE 14d051E 1490STE 1~00

77

c IMMF.UIAILLY PNEV1OLIS oNL.

6,?0 Cf)x,cufil=rl.u00 6JU I=l,NV

630 COXCUM=LUACOM*ERR (J,NOSC) ‘EI+R(JoNUSC-l)tNAH=muSC-d

b6n NTRY=UDO 6ou K=KL*NAH

NRETKY=NAH-KCOSINL=O.U

00 62u J=l,NV650 COsINL=CObrNE*ERR (J,NosC) *ERR(J,K)

IF (buSIN~-COXCOM) 660J960t680660 CONTINUk670 cHIfJAK=CH1 (I)

Gfl TU 700

680 NTQY=LK1=I(*IDn 6VU J=A,NV

SALVUIJ)=IQIAL(J)6qo TQTAL(J) =(X(J)-Xl)SC(JSK ))/ACK

CHIRAK=CHIOLo* (CHICSC(K) -CHIOLO)/ACK700 Dn 7ZU J=l~NV

XSAVLIJ)=A(J)

TRIAL (J)=ACK*lRTAL(JIIF (IIASKIJ)) 720$710?720

710 X(J) =AMAX1(AMIN1 (X(J) +]NIAL (J) *XMAX(J))$XMIN (J))7?0 C@NTLNUk

CALL PUNKNF=Nr*l

IF (LmI’3Q-CHIULDl 730s1+0s740?30 cHIfjAK=CHIOLO

CHIOLU=CHISQNGIANl=NGI&NT*I”

GO TU 700/4fl IF (NRLIRY) 760$/60,750?%fT IF (l*QIANT] 81O*81OO76O?60 CTNDLH=O.5/ACKO(ACKe020\HIRAK- (AcKi$02-1 .o)eCHIOLn-cHIsQ~ /(ACK

1 {>CHibAK- (4CK*loO) t$cHIOLD*CHIsQ)

nO 7nu J=l,NVIF (MASK(J)) rRO*770t7BIjX(J) =AMAX1(AMTN1(XSAVE (J) +cINDER*TRIAL (J) tAMAX (J)) oXMIN(J))

CONTJNULCALL FUtJKNF=Nt ●1

IF fI.HISfJ-CHIULD) 890, 7Yo$790IF (NuIAN1) 8Jo,Bno,830IF (NIRv) 810s83U081n

oo R<(l J=l,NVTRTALtJ)=S,ALVUIJ)

X(J)=AS5VL(J)GO TU e50DO 8*U J=l,NVTRIAL(J)=lRIAL(J)/AcKX(J)=ASAVL(J)IF (NbInNl) Pb(I,M60,900IF (NHETRY! 8100870c6/,UIF (wINY) R80J9?098floNTI?Y=u

GO TU b70CMTDLU=CHIS(J

IF (NINY) 910*16U~Q10NOSC=U

GO TU 160NO%C=MAXO(NLJSC-I $0)cHI(ll=CtilnLD

CONTINUEANOTI!LR CYCLE THROUGH TtiE VAI?IAtlLkS IIAS tlEcN cOMPLETLD.

PRINI ANoIHER LINE OF IKACES.

NzTP=NLIP+I

“}. ]A9fl!“L )700

!t. 171!1:;IL 17.?0‘.TE 17JfI

>TL 1740STE 1750

STL 17b0

STE ]770STE 17R0STF. 1790

STL lfiOOSTk lR]O

STE ln20STL lp~o

STL 1940ST.E IR5fISTE 1A60Slk 1870

Slil 1R80SIE. IR90

STE 1900

STt. lqlo

STE IQ20STL Iqj(l

STE 19405TE lq50

STE 1960STL lQT(>

STE 1Q80STE 1990STE 2oO0STE ?nloSTE 2020STE ?oJOSTE 204fJSTE 2(I5O

STL ?060STE Z070

STli ~ntio

STE 2090STL ~100srE 211O

STE 21.20srk 2130STE Z140STE ?I!JO

STE ,?lbO

STL ?170STL 218(1STL 2190srE 7?00STE 2?1oSTE 2?20STE ?p30

STL ?240CTE z?50

sTL ?260

sTL ~?[oSTL 2~80

STE 2?%STL 2300

STE 23111

STL ?320SIL ~730

STL ??40

STL 23’JnSTL 2?60

SIE 2770

78

60 ru 170 STE ?a8nc A L!INIMUM HAS 13ELN FOUNV. PRINT THE REMAINING TRAcES. STE 2390

9%(I NOSC=U Srt 2400

I)o 90U I=1ONV STE 2610

c STFPLI 4.b *s* STEPIT WITH OSCILLATION SEAuCH9 MATR!X INVEKSIONt STE 2420

c AND AuToMAIIC STEP SIZE ADJUSTMENT. cr+hNVLtR d~STE 2.430

IF (AMAX1[VEC (I)*SIGN(1O 00VECI:)))I 970*96U~970

960 CONTANUE

GO TU 101U970 NGATL=l

DO luufl I=l*~vIF (MASK(1)) ifIflOC ’?f313Clo(l(l

Yf?o IF (HBblOA (I) l-AdS(OELMIN (1))1 IOO0,990999U

99fl NGAft=Olol)ll DX(Il=DX(l)/RATIO

IF (NbA7El 160,1bos101ij

1010 CHISu=C~~IOLOIF (LAHs(MATR1x-loO) -5O) 1020,1020s1310

IU?O IF (NACTIv-Nv) lJ)O~lOJu*l~10

c cnwPulk TME S!ANUARP EMHoRS ANO TtIE Correlations.

l(l?n F4C=m~T10@~} (MATRIX-10n )

ESUM=UOUor) lu% I=IJNVIF (utLMIN(I)), 1060s105091040

104o OX (I I=AQS(FACVOELMIN (I)IGO TU “106O

105n OX II i=A~S~FAc*Ux(I))

106o IF (uAII)) 1310*131001070107o XSAVL(!)=X(I)

DO lUMO J=],?X( I)=ASAVL(I)*DX(I)

CALL FUNKNF=NF*l

SECONIJ(l *J)=C~ISQIORO DX(II=-ux(T)

EQR(l~l )= (sEcONO (l, l)-20-O*CHIOL[)*SECOND (ltd))/Da(I)**2

1090 EsuM=csLJM”Ads (ER~(191) )OO lL<O I=?*NvIM=I-i

DO l~do J=l$IMDO 1110 K=],?X( I)=ASAV\(I)*OXII)DO lAuO L=],2

X(J) =ASAVE(J)*OX(J)CALL FUNKNF=N? +L

SFCONU(K*L)=CMtSUX(.j)=ASIV~(J)

11OO DX(J)=-UXIJ1

X(T)=ASAVE(I)

llln Ox(I)=-l~xiT)ERa(LtJ) =0,25*(SkCOND( l?l)-sEco~D(192)-sEcuNo (2sl)*s~c0u~(2*~))

1 /A13>lUX(i)@DX(J))ESUM=LSUM+ tI!3S(FRH(I,J) )

1120 EQR(~sl)=EQR(l,J)RRAAtiA=Adb(kSUM)/FLOAT (NV)**?

NGRAFL=[)

DO 1A*O I=lsNvDO 1$*U J=I,NVlF (LRK(I,J)) I140,1130t]140

ll?n NGRArt=l1160 ERR(i,J)=Et?R(I,J)/9RAAAK

OFT=A*U

DO 1130 J=lQNV1150 SALVUIJ)=l.O

r)o lAUU I=ItNv

BIGA~J=O.O

STL 2440

STt 2450

STE 2460STL 2470STE 2&ti0STE 7490>Tt ?<00

SIE ?510sTE 2F20sTL ?qJO

STE 2%*O

STE ?c,50STk z%60STE 2K?0

sTE z5un

STE z5Y0STL 2AU0sTk .2610STL ~620STE ~630

STE 2640

STE 26S0STE 26bf)

STE 2670STL ?680STE ?690

STL .?700STE 2710STL 2720STL 2730STE 2740

STL 2750

STE 276oSTE 27111STE ?780STL 2790STE 2800STE 2810STE 2F120STE 2R30

STE 2940

STt 2P50STE ZR60

STE ?s70STE 2RM0STE zR90

STt 29(JO

STE ?910STL 2Q2nSTE 2Q30

STE .?~40STE 2Q50STE 2Q60

STF. ?q7fTSlf- 29dl)STL Z9Y0

STt 3nOo

STE 3nl.o

srt 3n20STE 3n30STE 304fl

79

11601170

Ilqn

119(7

lc?n(’)

1210

13no

1310

c13201330134~13’50

00 1150 J=I$NV

IF l~ALvu~Jl) 116n.llRU*l160IF (AtIs[E~Q IJJJ))-RIGAJJ) l180tllHOOl170RIGAJ.J=Atib (ERR(J$J) )K=JC(3NTLNUE

IF (MIGAJJ) 1200s1190,1ZO0fJET=u*oGO TU 1310SALVVIK)=U.ODET=ucT*L~R(KoKI

TPIAL(K)=l.O/LNR(K,K)ERR(n*K)=uoo

XSAVLIK)=lOOM=K-iIF (MI 1240,1240c1210

DO IZSO J=l?MXSAVc(J)=tQR(K,J)TNIAL(J) =LQR(K,J)*TRIAL(K )

IF (sALvO~J)) ll~fi~1230s1220TQTALIJJ =-TRIAL(J)

ERP(~*Jl=UoOMZK+L

IF (M-NV) 125u,12SO01290DO lc@O J=M*NVXSAVC{J)=EQF?(J,K)

IF (=ALvO(J)) 119(I,1260?Iz70XSAVC(J)=-XSAVC(JITRIAL (J) =-ERR(J,K)*TRIAL (K)

ERR(JtK)=U.OfTfJ lJUO J=I,NV

on lJUU K=J,NVERR (R*J)=~F?R (K,J)+XSAVL[J) *TRIAL(K)

CALL kUI’JKWRITC (NOLJT?1320)

WRITC (NCJUTt1330) (X(I) $I=lSNV)

IF f~nlsIJ.GT*l.E-?o) WRITE 1NOuTs1340) CHI~QIF (LHIb(J.LE.l.F-?O) WKITE (NOUT,1350) OIFLIMRETUmN

FnRMAl (//,lX$d2H FINAL VALiJES (]F X(I))FOC?MMI (lX,lOti X = QloE1204/(lox*loEldo 4)}FORMMI (///lXs~3HFINAL VALUE OF CtiIsQ = .E15,8///)FORhi~l (1Mos1Y$,1OH OIFLIM = EIS.8)ENn

SURRuUTINL FINEcHK (I(L)

cc F?oUTlrit CALCULATES INTLHMLDIATE PIIINTS AND CHECK% FOM

c REASUNAtJL~kJLS5.

c

COMMuN /PuLsIN/ ALAMOA(50)$ FX(200)0 T(401)s KTRM, ITSP, IpHuHs1 NTN, NOUI

STE 3n5fJ

STE 3n60STk 3n70

STE SnHnslk 309n5TE 3100

STL 311nsTE ~12n

STE 3130STL’ 3140

sTL 3150STL 31b0sTt 31/0

STE 3100STE 3190STE 3700

STE 3?10

STE 3?20ST}. 3?3n

STt 3?40STL 3250STE s?bOSTE 3??0

STL 3?80STE 37%STL 33OOSTL 3110

sTL 33tnSTL 3330STE 3340

STL 33!J0

STE 3360STE 3770

Srfi 3380

STE 339oSlk 34oo

STE 3410sTt_ 3/,~()STE 343o

STE 34$(ISTE ~45n

STL 3460STt 34/0STE 348o

StE 349fl

FIN

FIN 10FIN zoFIN .3[1FIN 40FIN bll

FIN 60cnMMoN /P:LSUAT/ NOT, ktJ(25). Gf(Z5,400)t NERG. ALF(25,so), ALAM(<FIw

1 %,5U) FIN

COMMUN /PJLSCAL/ A(50*5u)~ B(so,l) Fl:l

COMMUN /MANI/ rc(100)t TITL(8)t KTR(50)$ N~~lO)! KKN(25)0 nltLIM FIN

CnMMUN /FJNRAD/ lRAo, NCOHSC RA13T(200)s DELT (200) FIN

DIMENsION FXp1700), TP(joo)c TI(lU)s XL[lU)~ YL(;o)s TPF(700) FIN

I(=KL FIN

ENCOIJk (27,170,TI (1)) K FIN

ENcoLJL (20,190,XL) FIN

ENCOIJL (1U,200,YL) FIN

XLI2)=1OH SECUNCIS, FIN

FXMIN=Pf (l) FIN

00 ~u I=I,TTSP FIN

IF (~A(I)oLT.FXMIN) FXMINI=FX(I) FIN

80

DFLF=u.~=~

TPll)=r(l)

OFC=I(lIrPK!TINUt.L=L+l

OELF=UELF+l.tJ

TP(L)=l)ELFoDECIF (I P(L) oGE.T(~rSP)) 6U TO 30

IF ((lP(L)/DEcl.6E.ltl.o) DELF=l.OIF (tlP(L)/OLC)obE,lO.U) Ukc=OEc*10.O

GO TU 20CONTANUL“NK=L

THIS LOOP IS TO INcREAst TIME MF.SH BY FACTuR OF FIVE.

NPF=lTPF(Nt’F)=TP(l)

no 5U N=2!NKIF (NPP.G?,700) GO TO 50

DELF=(IP(N)-TP(N-1))/5.000 4U NJrl,bNPF=NPt*lTPF(Nw)) =TPF(NPF-1) +oELF

CONTANULcONTAmukIF (ly~P.6r0700~ wRITE (NouT,140) TPF(NPFI

NK=NFt00 6U N=lsNK

TP(N)=T~F(N)CONTLNLJFWRITC (NOUT*lMO) KD(l 9V N=1oNK

FXD(N)=[I.

KTRM=nrN(*’1DO flu J=I*KTRM

IF ($~AI)oLCOo) Go TO 70XPO=HL~ !Ks J) *LxP(PALAM(K*J)*TP (N))

COFF=AL~ (K*J;/DFLT(N)/ALAM(K-J) **ZXPOl~L.-LAP (-ALhM(K.J! *HAUT(N))XP02=1.-!.AP (-ALAMIK!J) *OELT(N))

XPn3~tX;’ (-flLAM(K9J) 0(TiN)-@tLT(N)/2.))FXP(N)=FXP (N)+COFF*XPO1 *XP02*~03

GO TU 8nTEMP=ALAM (<,J)*TP(N)

IF (AtsS(T~<P) oGT.hOO) GU TO HoFxP(NI=I xP(N)*ALP(K,J) *~xP(-TEMP)CONTINUkcclNTINuLwRITL (N(JUT*21(I) (N.TP(N) sFXP(NISN=l $NK)

TEST ?u~ ~OINl OLVIATIUN ANn sL(IPL HEVEHSAI..

NFLG=uLFLG=UNKl=!wh-I

wRITE lNOUTo130) K

D(I lAU N=d,NKl

IF (FAP(N),GToo.1 GO TO 100

NFLG=mPLLjil

GO TU 110CONTINUEIF (F A~(N-l) OLE.O.O.OR.FXP (N*l).LL.o.o) 60 TO lln

AVLG=l ALOG(FXP (N-l) )*ALUG (PXp(N*l))l/2.oFAvG=LY.P(AVLGI

Ff)lF=AuS ((FXP(N)-FAVG) /~XPIN))

FIN 210FIN ?20FIN 730

FIN 740

FIN ?50FIN ?f,r)

FIN ?70FIN ?doFIN ?90

FIN 300FIN 310FIN 320FIN 330FIN 74f3

FIN 350FIN 7b0FIN q’/oFIN 381)

FIN 39t)

FIN 400FIN 410FIN /,20FIN 430FIN Jb40FIN 450FIN 460FIN &70FIN 480FIN 490FltJ 600FIN SloFIN 620;;; 530

%40FIN <50FIN 5boF I N 670FIN q8cJFIN 590FIN 600FIN hlf)FIN h~I)FIN 630~IPJ 640FIN f>!lfl

FIN 660FIN 670

FIN 680FIN h90FIN 701J

FIN 710PIN 720Flru 730

FIN 7+0FIN 750

FIN 76!lFIN 770FIN 780FIN 799FIN ROO

FIN 810

FIN R2nFIN R311

FIN 1340FIN U50FIN Rho

FIN R70FIN RbO

81

TF (ru IF. yF. l*o) lFLG=L~LG*l FIN R90IF (rul}. GE. l.o) wRITE lNUUT~23n) N FIN Qoo

IF [tul~.oF.O.1) WRITE (NOUT$22n) NsFDIF FIN 910

IF (r AP~N).GT.FXP (N-1)) UNITE (NOUT*1601 N FIN Qzo

lltl CONTLNuL FIN 930

IF (rAP(ll,LE.o.) NFLG=NFLG*l

IF (rAP(Nfi).LE.O.) NFL~=NFLG*l

FIN Q40

FIN Q’JO

IF (NFL(~.LO.01 GU TO l~u FIN 9ho

kI?TTL (IIOIJT*240) NFLG FIN QIO

.RF TURN FIN ~tio

120 CONTLNUL FIN ~91-1

IF (1-PL~.t-f3.0) Go TO 1~(1 FIN 1000WRITt .(NOIJTJ250) LFLG FIN Inlo

130 CONTiNULNK=l-NK

FIN I(I2O

FIN in-i(l

c CALL rLOTM (Tp9FXP9r~K$-l*O?O!Oan~ 10UolOOtTl s30sX1.S?OsYL9 10) FIN 10+0

cc CALL rLOT~ (T*FX$-ITSPo-10-19-47t0, $1.0* 1.UsTI*3n,xL~?o, YL*lU) FIN lnbl)

RETUIXN FIN ln60

c FIN ~fl?l)

140 FORMUI (ltioslMH PLOTS cuT AT TP =lpE12.59215H sEc 4S PoINl LIMIT KLFxN 1080

lACHEU.) FIN 109I)

150 FORMA! (33I+O POTNT OEvIATIoN cHrCK FoR 6NouPo13//%IH NOTE - uNLY vFIN llofi

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.

cRFAD iN1Nt?30) ;WANTO(KCAL IL)~L=lOXWANT)

IF (LwA(J1.LE.0)DO 8U L=l,IWANT

LL=KbAL(LlALPHAIL)=ALAMUA

f10 CONT~NULKTQM=lWANIDO 9U K=l;KTRM

ALAMuA(K)=ALPMAQfi C(3NTANUL

100 CONT$NUk

GO TO lUO

LL)

K)

3203,40

~4fJ

350

7tiojrl)

380

~’Jo

400

410

41?0.430440

450

4606?0480

PUL 49QPUL 500pUL 510PUL ‘G21-JPUL 63a?UL 640

PUL KboPUL 660PIJL %/0PIJL 6H0PuL 59iJP(JL 600PIJL 6A0P(JL 4>20PIIL 63nP(JL 640VuL 650PUL 6611PUL 610PUL 68nPUL 690@uL 700PUL ?10P~JL 7.?0PUL 730PUL 7404(11. 75(JP(JL 760#uL 710pUL 7dflP(JL 790

PUL RooPULPLJL

PULPUL

P(JL

pULPUL?UL

pUL9LJLPULPULPULPtJLpuL

Pul-PULPuLPUL

PUL

Alfl

820

R30

$40950

Q60RIOCJA()Ja90

QooQlo

920Q30’34095n960Q70

‘aJ30990

1000

83

c PUL 1010

c WRITC OUT INPUT FoR DEUUG PLIL ln~~

lfl?ITL .(NOUT~2~o) (K9ALAmDA (K) sKcAL(K)tKCAL (K*l)~KxltKTRW) #UL lo31)

WQITC (NO+Ts~80) (I tT(1)9Fx (1)91=1$1T5P) PuL ln+n

00 Iiu I=l,IT~P PUL 1050

W(I)=IO/FX(1)~02 PUL 1060

IF (LwT.Ea.3) w(I)=l./FX(I)**l.% PuL lnTO

IF ILwT.EQ.1) wII)=10/~.x(II PUL lodll

IF (LW1.LL.LJ) W(I)=l. PUL 1090

11o CONTINut puL 1100

WRITL (NL)dTs290] LwT puL 1110

c PUL 11~(1

c CALCULATk R MATRIx PUL 1~30

c PUL 1140

DO 1*u K=l,KTRM PUL 1150

H(Kol)=o. Pul_ 1]60

DO ~JU ?=l, ITSP PUL 1]711

IF (LRAU.LE.0) GO” TO 120 PUL IIdO

XDl=LAP (-~LAMnA(K)*RA13T [I)) puL 1190

XP?=LAP (-ALAMUA(K)*DELT (I)) PUL 1?00

XP3=LAP [-AL4MOA(K)*(T (J)-uELT(I)/2.)) PUL 1?10

B(Ktl)=Hlfi,,l)+FX(I)*l./DELT (l)/ilLAMDA(K)**Z*(l.-%PI )*(1.-XP2)*XPJ PUL 1?20

ll?(l

130140

c

cc

]%n

160

17nlRfi

l.q n

cc

c

20021f)

c

GO TU 130CONTIINuEB(Ksl]=H (K*l)+FX(!)*EXP (-ALAMOA (K)@T(I))*W(I)

CONTINULCONTIIWE

CALCuLATE A MATRIX

DO ]=u l=i,KTHM

no 13u J=1oKTNM

A(T,dJ=fl.cnNTl!vUtDO lYU K=l,KTRM

A1-hhi=ALAMu4(K)

00 10U L=l,KTRMDnlfu I=l,ITSP

IF (iHAU.LEOO) GO TO 16UXPf)l=l .-EAP(-ALAMITA(K) “HAIJT(I))XP02=l.-EX~ (-ALAMDA(L) dRAIJT(I))

XPn3=10-EAP (-ALAMOA(K)UUELT (1))XP04=l.-EXP [-ALAMDA(L)~UELT (1))XPO%=LXP (-ALAMOA(K)*(T (1)-DELT(I)/2.)1XPn6~LXp (-ALAMDA(L)*(T (1)-OELT(Tl/2.J)

COFF=l.10~LT (1)/ALAMl)A(K)**2.COFT=l./ULLT(l]/ALdML)A [L)**2A(KsL)=A (K*L)”CO~F*COFT*XPOl *XPn2*Xp03*XPOID*XP050XP06*W (I)GO TU 170

coNT1lwhA(K,I-) =LxP(-(ALAM*ALAMuA (L) )*T(I)) *W(I) *A(K9L)coWTAIVuk

cnNTLlvuECovTllvUt

PRyNl A MATRIX

IF (1PNT.L101) GO TO 210WI?ITL (NOUTS300) TPPOR

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

26(I Ff)RMAl (lP$Ell,/J) PUL 1790270 FO~MAl (3d K=913911H ALAMUA( K)=,1PE12.!509H KCAL(K)zo13,11H KLAL(K+~UL ]1300

ll)=*lJ)2Q0 FOQMAI (4d I=,T3,6H T(1)=,1PE12.5,7H FX(1)=,1PE12.5)290 FOQMAI [lIi0,6fi LwT=,13)300 FOL?MUI [16Ho A-MATRIX ~OR 916)~ln FOQMM1 (2X,1OL1?.3)

3?0 FORMAI (l~HO H-MATRIX ~ONO16)

ENfl

SUQRUUTIAI~ SELFIT (IP9L?)

c

c ROljTINt COkiPAl?ES FIT WL,lH ORIGIFJAL vAl,.UEs.

c

COMMuN /P~LsIN/ ALAMDAlbo)o Fx(?ou), T(401~c KTRM, ITSP, IP~UMt1 NINS PJUU1

10

20

3Q

6Q

50

CnMMul~ /P~LSCAL/ A(50950)0 R(SO,lIcOt-iMuN /PdLSOYT1 ALPHA(5o), FXCf100)9 pCT13kF(100)

cOMMUN /Fi,NRAu/ Il+&ll* NuoKS, ~Al)T(200)* DELT(200)nTMEw>Inh TII1O)* XL(lO)_O YL(IO)

00 3U l=]j ITSUPcT1.)lr([]=tl.FxC{A)=O.o() .2u K=I*uIRM

IF (l-ALAqnA (K) *TlT))CGT,300.) ALAMOA(K)=-3UoO0/TII)IF (AwAO.LE.0) GO To 10

Cr)FF=u(K)/DELl (T)/ALAMUA(K)6*2XPnl-l.-EAP (-&LAMOA(K) *HAUT(I))XPn2=l.-EkD (-ALAMDA(K) *UELT(I) )

XP03=LAJ (-4LAMOA(KI*(T (1)-OELT(I)/2.))

Fxc(L)=FXC (I )*CoPFQXPOl*XPO?*xpn3

GO TU i?flCONTLWUCFXC(l)=FXC (I)*B(K1OEXP (-ALAMOA(K)*T(I))

ALPt4AlA) =~(K)*EXP(-ALAMOA (lo*T(I))

cgNTINUEPCTI)JF (I)=(FX(I)-FXC (1))/FX(I)*

IF (LP.LE.n) SO TQ 30WRTTL (NOUT090) 1sT(I)WRITG (NO<T?1OO) (K~ALPHA(K)!K=’CONTANUL

IPRo=lPWRYTL (NoUTollO) IPRO

000

!I(TRM)

WR~TL (NOUT~120) (19T(I) sFX(I)~FXC (I} OpcTOIF (I)~I=IsITSP)Fl!-fs=u.

00 4U I=lsITSPl?Ms=NMs.PCTDl~ (1)*92

CONTINUL

RMs=nMs**o.5WP7TL (NOUT0130) RMS

AR1=u.AR?=U.00 su l=2tITSPAR1=AK1+ (FX(I)+FX(I-1) )/2.o*[T(I)-T(I-11)

AR7=MKZ* (FXCll)*fXClI-1) )/2.0*lT(l)-T(l-1))

cONTINUtAPOIP=(AtfA-AR2)/ARl*100 ●

WRITG (NoUTt1401 AR1$AK2$AR(’)IF

?uL 1810pUL 18/0PIJL 1R30PUL 1$740PuL )R51)

PUL lRtIOPUL ]F!7fl

SkE

SEt

SEE

SklzSEL

SEEStkSEEsk~SLk

sEt

~LESLLSLL

SLESEEskLSLEstE

SEE

skE

SEESEESEE

SEE

SEESEE

SEESEESEE

SlikSEE

SEE’SEESLE

SEESLESLE

SEESEESEE

SEESEESCESEESEESEESEE

10i?!)30

*O>060708090

\oo110l??,]130]4015016017rlltio190

?00

?10?20230?40

?50?60p70280

p90

300

310320330

340

350360

370

380390/+00

410420430

440450660

470

85

FXMIN=FX(l)DO 6U I=l!ITSPIF IPA(I).LE.O.0) FX(I)=FX(I-l)/lO.

IF (tA(7)*LT.FXMIN) FxM~N=Fx(I)60 CONTANUE

DO 7U I=lsITSP

IF (rAcfI).LT~FXMIN) Fx$(I)=FxMIN70 CONTANUC

ENCOUL (4U01509T1(1))

ENCOUL (lO*l~O*XL(l))ENCOIJL (IU,17u,YL(I))ITS=IISP

1s=1IF ([Px(l)/FX(ITSP-I))OLTO 1000) GO TO 80ITS=-lTSP

SEESLESEL

SLESEESEE

sEESEESEE

SLESELSEE

SEEskESEE

ISX-L80 CONTINUE

SEESEE

c CALL rLOTM (T$FX!ITS~ ISOO!OtOOtl.0 1. wTIt40JXLo10 tYL$10) SEE

c CALL PLuTM (T~FXCQITS*~=~-lS-47SO~ cl*oIc$TLoQooxLolO~yL~ 10)RFTUnN

SEESEE

c SEE

90 FrJRMAl (18H TERMS FOR STEP Q1398H T(I) = olpEl>.q) SEE

lrJO FnQM*l (6(2X? X302XtlPE10.3)) SLt

lln FORMAI (107H1 STLP NO. TIME uRIGINAL VALUE CUMPufSEE

IEO V$iLUE PENcENT oIFFtRLNCE IPRnt3=,131 SEE

l?n FORMMI (Ib,lP4FltJ.5) SLt

l?o FORMMI (?lHo RonT-M~AN-sQuARE=lPElz.5) SEE

l~fl FORM*I (lJHO (JRIG INT =,lPt1295$14ti FITTLU IN[= sIPE1?.5*15H p~~st-t

lCENT ul~F =sIPEl~.5) SEE

lso FORMNI (*OHCOMPANISON UF C)RIGIN4L WITH FITILO DATA.) SEE

16n FORMA1 (lOH~IME(SEC) )

170 FORMMI (lUII FxII) )

SEESLL

ENO SEE

SURRUUTINE PCNO(JT (LXX) PCH

c PcH

c ROIJ71NE PgNcHEs 01J7 PARAMETERS IN ENOF-LIKL FONt4A7. PCH

c PCH

COMMUN /pJLSIN,’”ALAMOA(sO)* FX(i?oo)* T(401JQ KTRM, IlsPt IPtiUtJt PCH

1 NTN, NOUi Pctl

COMMUN /PjLSDAT/ NOT, EtJ(25)s Gx(k!5~40010 l~tRG~ ALF(25s50)s ALAM(~pCH

680490

5(JO510%20%30540

550560

570qdllq90600610620h30640

650IS611A70680690700

710

72n730

74rJ

7507b07rlj

7do790

800

102030405060

1 5,5U)COMMUN /M~NI/ TC(1OO)S TITL(8)s KTR(50)t N~(lO)* KKN(?5)~ DIPLIMCOMMuN /EN~F/ MAT, MF9 MTs RUNTIM~ NPUNDIME1w310N F(10), P(lolt L(10), 1.XX(20)

NPuN= I

ClsnoNIII.=UNS:QU1WRITE (NOuT930) (TITL(I) cI=lt7),MAT~MF ,MToNSEQ

WRTTC INPUN03U) lTITL(l) ~I=lt7)QMAT*MFtNT cNSE12—NSE(lZNSEQ~IWRTTC (NodTs40,) CItCl,NuL9NULsN(11.*NERGoMATOMF~ M~,NSEU

WRTTL lNPyN~4U) CIoCloNUL*NULoNt)LsNERG*”MAT!MF oMl,NsEfJDO 2U K=1sKJER6NPX=nlR(K)LPX=LAX(K)

ER1=LL$(K)EB?=cu(K*l)CALL LX~P IEB1,F( l),P(l) *L(I))CALL cx~P (EBLsF(2) oP(2)cL(2))NSEQ=NSLQ*IwRIT= (NOLJTS50) ((F(I) sP(I)sL(I)91=l t2)sNULsNULsNULoK9MATOMF ~MT

1 ,NSLU)WRITL (NPLJAJt50) l(F(IIop(I )tLfI)oI=l t2)oNULsNULPN1JL0K.MATCMF SMT

1 ,NSLLJ)ERI=I (l)ER?=I ILPX)

PCH 7(’IPcli 80

PCH 90PCH 100PCH 110PcIi 120PCH 130PCH ~40PCH 150Pcti 160Pcu ~11)PcH 180PCH 190Pcu ?00PCH ?10Pet’i ?20PCH ?30

Pcn ?40

PCH ?50PCH ?60Pcfl ?70Pctl ?80PCH ?90PcH 300Pcti 310PCH “ai!oPCH -?30

86

C4LL LxFP (E8AcF( 1)!P(1),L(1)) PCH

CALL LAFP (EP2,F(2),P (2),L(?)) Qcn

NSEQZIXSLQ{!l PCH

WRTTS (flo~T,501 ((F (I) sP(I)~L(I)sI=l.2) *NUkONUL9NUL$NpX*MAT~M~ ~M! ‘CH

1 ,Nscb) Pcll

WRITt (NPJN,5U) ((F II )*P(Il,L (I )sI=l?21sNuk~NuL ~NIJLONpX*MAT $M~~M! ‘Cti

1 ,NSLUI Pcti

ALF(h.NPX*l)=O. Pcti

A1.F(n$N~’X*71=U. PCH

ALhMl~*’f9A*lj=f). P(..i

ALAw(r. ~NPA*2)=(). P1:4

J]=l Pet’i

10 CONTLIMUE Pcti

EI=ALF(K,JII Pcfi

E?=AI.?(KsJI*l) PcH

E2=AL? (~sJl+?) Pcli

FXI=ALAM(6,J11 PCH

Fx2=*LAM(fi,Jl*l) PCH

Fx7=MLAMIK*J1*2) Pcli

N5FQ=tiSkQ*I P(H

C&I.L LxPP !El*F(l)sP(l)*L(l)~ ~cH

CfiLL La?P (Fxl*F(2),P (210L(2)) Pzq

C.ALL Lx~P (EZ!F(3) tP[3J?L(3)) Pctf

CALL LxFP (FxZ~F(4),P (4J,L(4)) PCn

CALL CA~P fE3$F(b),P (5) *L(S)) Pct4

CALL LA~p (FX3?F(6),P (6) OL(61) Pcli

WQITG (.,oUTf6LJ) (F (I ),P(I)*L(I), I=1*6) 9MAT~f4Fo~10NSE~ Pcti

WRITL INPUN,60) (F (I )*P(I)9L(I)c1=I *6) tMAT*MFsM~,NSEQ iJctl

TF ((.)I*?).6E.NPx) GO TIJ .?n Pcti

J1sJI*3 PCH

GO TU 10 Pcl+

%0 coNTll~u~ Pcti

MllTl=u Pcu

MFI=u Pcli

MTI=U (JCH

MEND=-1 PCM

NSFQ~iNSEQ*l PCH

WR:TL (NOUT97U) MAT,MFcMT1?NSEQ PCH

WRITC (NPUN*70) MAT,MF$MT1*NSEQ Pcti

N5FQ=NSEQ+I Pcli

WRITL (NOUTt7U) MATsMF!~MTl*NSEIl Pcli

WRITL (NP$JNt70) MAT~MFloMTl~NSEO PCH

NSEQ=IWSEQ*l PCH

wPITc (NOUT*70) MATl~MFlsMTl$NSFQ Pcd

WRITL (NP<N*701 MATl~MFA ~MT1.$NSFLl PCn

WRTTC (NO:TS7U) MEVD PCH

WQITL (NiJLJNJ7fI] MENO PCH

STflp Pcn

c Pcti

30 FORM~l (b~lo*A6t14t12s13t15) Pcti

Go Fnt2MAT (l~2E11,5*4111, 1*o120 13s151 ?Cri

50 Fm?MMl (2 iF9.~,Al, 12),4111,14t I?s13s15) Pctl

AO Fflf?MAl (6(F8.59A1*I?)*1491?$13! 15) Pcti

7(I FORf4Al (bbX*I@,~2*13t15) Pcti

ENP PCH

i CONVLuT X FOR PUl~CdING *CAF

c X - PLOAT1N6 pOINT NUMtit-R = F*ln.O**N *cXF

c ~ - u~qg9~95 LE F LT 9e~9Y995 *cXF

c aAGN (HOLLERITII + ON -) OF EXPONENT cXF

c N- LApoNkf.JT *cXFr *OO**V****@V****ou*{}**@*@***O**O**************~***O***************CAF

10.?0304050

6070

af)

90

87

F=n*u cXF 100

5=5P CXF 110

N=n CXF 120

QF711mN CXF ]30

In N=ALuulo(ARs(X)) cxf- 14n

IF (a”S(X)C1.0) 40,~(),20 c.) l>n

20 FzX/AUOO~*N CA} 100

s=~P CXF 1 ?0

IF (KuslF)-9.999995) 70*30Q30 CAF Id,)

30 F=F/iUO’l c~f 190

N=N*l CXF ?00

60 Tu ~0 cXF ?11)

40 N=l-1~ CXF 220

F=x*lu,o*@N CXF ?.3fl

s=~M cAF ?411

IF (Abs(F)-9,999995) 7o050050 CXF 7’JO

%0 F=F/lu.O CXF 76a

N=N-l Cxt ?rll

IF (N] 60s60~~0

An S=sP

CA} ?Ao

CXF ?911

70 CIINTLNUt cXF 300

RETURN CXF 310

ENn cXF 720

SUnRUuTINL LSS (N,voIcA~RcO*DET) LSS

DIMEIxsIUN A(I$N), 9(I,M), COM2(%)? cOM3(5)! ()(N) LSS 10

f)OURLt <1* S2* 130TpR0 LSS Zn

nATA [CUM2(J)*J=1,5) /5(lHLSS SINGULAR SYSTtM. INPIIT DESTitOYtU. N l-ss 30

lx / LsS 40

DATA 1COM3(J19J=lt5) /bOHLSS CALLtO WIIH N.LE.O nR N.GT.1* NLSS bo

lx / Lbs bll

c LSS 7n

c LIQM~LJ IS ONLY FOR LASL STATISTICS L5s dn

c CALL LIBMSG (loH R LSS ) LS5 90

c 10 Pul NE~SAGL IN SYSTEM DAYFILF L5S 100

c Lss 110NM=N LSS 12n

IF (NN.LEOO.OR.NN.OTo I) GO TO 140

14MXM

I.ss 130LSS ~40

SN=l ● LSS 150

DO Qu J=lsNN Lss 160

L=J-1 LSS 1 ?n

IF fJ.(iu,NN) b(l TO 70 LSS 18nT=ABs(A(J$J)) LSS 190

M1=J Lss ?00

M?=J*J LSS 210

DO lU K=M2,NN LSS 720

x=AB3[A(K)J)] LSS ?30

IF (AoLC.T) GO TO 10 LSS ?40

Tzx LSS 250

Ml=l( LSS ?60

10 CONTINUE LSS ?70

IF (IIL.k(JoJ) GO To 40 LSS >do

DO 2U K=ltNN LSS ?90

T=A(JsK) LSS aon

A(J,A)=A(M1*KI LSS -+ln

2TI A(M1*K)=TsN=-3N

LSS 0320LSS ?30

IF (MM.LEoo) GO TO 40 LSS 340

DO 3U I(=l*MM L5S 750

T=R(JcK) LSS 360

R(Jsm)=N(MI!K)3n Fl(Ml!N)=T

LS5 ?70LSS 3d0

4TI IF (M(J*J).EfJoo.) GO lo 130 LSS 390

DO 6U K=M2,NN LSS Lon

!jI=rj.

S?=O*

LSS 410

IF (L.LU,U) GU TU 50

LSS 42nl-s\ /,Ju

SI=DUIPNO IL*A(J,lIII*A (loK) *I) Lh!j 44n

88

A(J~n)=(A(JtK)-Si)/A (J*J)

S?=OUIPRO iJ$A(K,l),ItA (loM2),l)A(K*N<)=A(K,Md) -S?IF (PIM,I-E.0) b~ TO qO

IF (M(J,J).EQ.o.) GO T() 130oo RU K=l!MM

S1=O*IF (L.LQ.U) Go TO Rfl

S1=OUIPRU I[.!A(J,l)*I*B (loK) $1)FJ(,J,R)=(h(:j*K)-Sl)/A( JSJ)cOAITJNU~

oFT=~(l*l)*SNIF (ULI.LJOOO) GO TO lJO

IF (N.EQ.1) GO 10 150

00 luu J=d,NNI)ET=uLT*A(J.*J)

IF (ULT.Ed.0,), 60 To 130r~ lMM.~(!.O\ 60 lo l~fI

M3=Nfw-1

On lcu J=l,MM

nfl 110 L=AtM3

M1=NN-LSIso.M2=Mi*iK=NN-f.lZ*l

sl=DuIPHO (KSA(M1OM2),I *B(M2,J)sI)B($41!JI =B(M1*J)-SI

CONT*NUtGn TU 1<0

cALL LAHR1 (I$COM?$N)

I)FT=U.

GO TU lboCAIJ. LABRT (1?COM3SN)

oET=U.RETURN

Lss 450LSS 660Lss 470

LSS 480LSS 490LSS %00LSS 610LS’i %20LSS 5.3 [)LsS 54nLss 550LSS %bl)(.ss 6?0LSS %Ro

LSS %90

LSS 600

LSS 610LSS 620LS5 630

LSS 640LSS 650LSS 660LSS 670LSS 680LSS 690LSS 700LSs 710LsS 720LSS 730

LSS 740L!JS 7’JO

Lss 700L5S 710LSS 780LSS 790

ENll LSS Roo

*iOT*DOTPRU* IS A FURTNAN CoMPArIHLE FUNcTION-TYPE SU@PROGUAM UOOT

*OUT

CALLTNO bE(JUENCL z = IJoTPRO(N, A, Ix, Y, IY) *DOT*L)OT

titlE:~-x- &No ‘Y- ARE RtAL VEcT(JUS EACU cONIAING *DOT

-tJ- ELEMENTS, DOT-lx- ANO ‘IY- ARE TtIE SPACINti fjETWFEtJ Sl)CCESSIvL 9L)UT

ELENENTS OF ‘X- AND ..Y- 14ESPEcTIVLLY *I)OT*OOT

THE RE~uLT NETuRNEU aY *UUTrRO* IS THE INNtR ~~OOuCT (XQY) OF !~k ’001

VkclnRS -X- ANQ ‘y-. I.E. THE suM OF A(I)*Y(T) FOR 1=1 lo N*OOTSbui

fHr QESULT IS 00UBLk P14EcTSION BLIT Wll I at U$E!~ AS 3L ;~-~ +00 r

PRECISION UNLESS ‘ouTPFIOO IS OLCLANCO TO dF nOIJ~lLE fl~ol

PNECISIIJN lN THE CALLING pROGRAMo *l)or*DO r

TtiE RESULT IS ZERIJ IF N = O* lNFINITL IF N .LT. O *DOT*uor

4151617181

91101

111121131141151101I?l181191?01?11?21

89

t40n

Tw(l

d],HU,FINISHUl)tin

kinonu.LuoP

UPPLR PRuD X*Y 00 r

Our

LOWtR PROI) XrY l)OT

FETcH NEXT ELEMENT up -x- IIOT

OUTFETcH NEXT ELEMENT OF -y- 00 r

Do r

DOTDo r

DO r

Do r

DO rUPPLR PROD XSY 00 T

00 TLOWLR PROD X*Y 00 r

FETcH NEXT ELEMENT UP -X- DO r

DOT

FETCH NExT ELEMENT OF -Y- UOT[)0 T

OUTDECREMENT LLEMENT COUNTER RI D(ll

DorourDOT

LOUP BACK IF MORE TIIAN P ELEMLNTS sTiLL 00TREMLIN TO HE PFi(lcESSFfj W 1

BRANcH IF LAST 2 LLtMENTS A~.I?LADY PQUGkSSLOOUr~El?(J oUT -IX- ANO -lY- F(JW ?NU SEr (lt 00 r

FETCHES TO ELIMINAIE ACrF’5S OIJT of HANGL DOTLOOP BACN TO PRUCES3 LAST ? ELEMENTS 01) r

90

Iwt.

t \J

zkf?~ LI3A ?MAb

CLI

IN~ MAT

mA6

LL’

tN()

ti~.tit,NunL$II,AU,[.:JOP

ti}!Hn, TN1’Ido

o

dn9Ru,f)oTPRo

11

i3fi9fllJ,DUTPR0

BR4NCW XL -tJ- .tu, 1t!QANcH lo PQOCF5S 2 LLFME?lTSHRANCH IL -tJ- .LT. O

IF N = II

NE5ULT = ZEROFIETUNN TO CALLING PROGRAM

IF N .LT. 1)RESULT = INFINIT&

NETUHN TO CALLING PROGRAM

SURRUUTINL LAURT (ISW,LEOLCINX)DIMENsION LHOL(5)Lf)GILAL PS9 T>

IF (( lSw.Lr200J .OFf. (ISW.GT.5) ) RFTURNGO TU (10?20,30,40950)9 ISWOATA NP /10/$ PS /.TRuEo/s 7s /.FALsE.Y

10 IF (rs.Al*O. (NP.Gl.o)) PRINT 60, LHOL*INXND=Nr-1

IF (1>) S1OPRFTUMN

?rI Ps=.~ALSE,

RFTuflN

30 Ps=. lliut.NP=INARETUm,y

4fI TS=. IKUL.RFTUNN

5n TS=.rALcE.RETUNN

c6n FORMAI (lt10c9x.5A10.3XS(J6)

ENn

ENO

I)(J T Q?l[)(JT OM1

l)uT Q91I)u! lol)l

I.)(JT lnll

IJllr 1021UOT 103IDOT 10+1IJUT 1051oo~ 1~~1

LA@

LAtJ 10

I-A9 20

Lit;; :0LAK +0LAB 50LA}) 60

LAH /0

I-AI) 80LAH 90

LAH 100LAH 110LAH 121)LAH ~30

LAH 140LAB 15(-IL&H 160LAH IinLAH lHOLAB 190LAH ?00LAH ?10LAB ?10

APPENDIX C

SOME USEFUL FITS OF THE PULSE DATA

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.

TABLE IC

USEFUL FITS IN 11 GROUPS

92

131415

:;18

;:2122.232425262728293ti

313233:~

3536373839Uti414243444546U7

48

495051

:;545s565?5859606162636U

6566

676869

;:7273747576777879

8P

93

@l82B3fln

858657&3fl

899k391929394959b979899

1 i) i!

lbl1U21V3ldold5lk16lk)71U8l;3y

110

illllI?11$llU

115Ilk

117ttfl1191(?0121122123!t?fl!2s

l~b12712812913L!

13113213315U

1551301371581 5Q1411

lU11Q2143]~a145146

14711J8JU9

94

15W

151152153

15415s

156157

1551s916L!

Ibllbi?

ib3lb4

lbslbb

12311

56709

1011

i2!314151617

1819l?ti

2122

Z32425c?b27282930313?33Jll

35

363738

39u f{

U142

Q3Ua(J5

,4b474849ikl

5152

95

535455565?5859

6k461b~b~b465b6b768697d7!72

737Ufs

7b

77787960818?8>8/J

858687888Q

909192939U9596

979899

ldtiltill~lz

1KY3lIOQ

1051 @6l’d71(18

lk49110111112113114115116117118119

123

96

li?i122123!2fJ12512b12112812913fi151132133!3413513b13713H! :0

14tilU11U214314U1451U6

1U71u81491s0!s11’,215315U1’55156157158159lb$flbl162163lbUlb5lbblb?108169i7tilrl1?2173

:3

:67a9lq111213141516

97

l,h’ill!-1~ 2.J(.3??- fl7,57.229-!2 7,93br!7-lk33,uJP@fd+ 2 a,Afi’d@$tn12b2blf!V2Qo.Jd”Arip-1 lo5.i,)pp+v a r’1 d 31?b?~!t)@(?l.vt~cda- I 1.ar3fdr+ q [4 l/l?b?~18?27,33?39- 2 1.71121+ a 3,?IJ98V- ; 3,76559- 1 \,2782Q- ; 9,176?7- 21r?bZ81aJ2U,5bld?W ~ 1.97:11~-2 IJ,333U7- IJU,6?7U7- 3 1.fJ3?55- U 1,55Q?5- 31Zb~918PZ4,45339- 5 u.77fi93- ~ 1,38751- s 1.8RQ38- Q 2.25079- 0 b,U0013- bl/bc?9!!j021.39J3i_J- ~ 2.Ii6j:!I- 5 I,u3!)u\- 7 7,4325,1- b 1,39$!110- 8 l,1477b,l- hl,?br?31si!2?*F2~11- 9 3.9~$v5- 7 1.6399?- 9 .2.318?;:- 7 7,151k~9-ik~5,79491- dlt?b~~lokl~?,h!!7ti5-lt) ,?eIlr1t>29- fl i3e9573’?-J~ 70/@b98..l$\ 0,Llfli)IIt4+ 4 o,4r)tIiA+ ill?hc?y~t!.l?IQsj;<;,$l+ J lc~.l(l”~lt+ ?

J ;1 d fJ1.?b?518n2~.?t.i,iti- 1 Ie;f,f,!,’kii 9 3 3),156JQ-

d 171?b~?lfi@2I l,7rI;tIfi+a 5,521’49- ~ 3,7!12qI- 1 1,71bu\!- 2 8,93:4?3- 21c?b?!!]’jJ2

5,j!ls17- 3 I,9?\I”/- 2 d,\h85Q- fiU,Q7515’W 3 l,5bbSU- ~ 1,~’i?R~- }l?b~~l)~t”ir?

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1

BASIC

This appendix

APPENDIX D

DECAY ENERGY FUNCTIONS, TERMINOLOGY, AND UNITS

summarizes the conceptual basis and equations of this report

for those readers who feel a need for additional detail.

I. BASIC EQUATIONS IN ABSENCE OF NEUTRON ABSORPTION EFFECTS

During the fission process, the fission products are generated directly by

fission and by decay or neutron absorption from precursors; absorption and de-

cay also transmutes each product. Summation codes account for all simultaneous

processes for each product, including the continuing decay following the end of

the fission interval.

The buildup and decay processes are continuous, and

sarily provide information at specified time intervals.

described in this report provide the aggregate summation

summation codes neces-

The basic libraries

spectra following a

fission pulse in a multigroup format at two or more points per time decade out

to 109 s (= 2.778 x 105 h = 31.7 yr). Collapsing the fine-energy multigroups

to a reduced, but still multigroup, set and fitting each group to a single func-

tional form results in a practical but still accurate

decay energies at any decay or “cooling” time. These

functions to provide the decay energies subsequent to

histories.

description of the group

can be used as Green’s

specified power (fission)

105

Because the fits are to pulse decay data, the functions do not account for

changes in the ensemble of products due to neutron absorption. Absorption al-

ters the decay energies, particularly at long (?104 s) cooling times. The ef-

fect depends on the magnitude and spectra of the neutron flux, and on the length

of the fission history. As described in Sec. V of this report, neutron absorp-

tion effects can be accounted for to good approximation in a relatively simple

way. In general, absorption decreases the density, hence also the decay energy

of some products, and increases the density of others. The net effect, the one

of interest here, tends to be an increase in decay energy, depending on the de-

cay group. At some long cooling times, the net effect is very significant but

is due to only a very few nuclides -- primarily the shielded nuclides, which are135

generated only by absorption, and certain nuclides associated with Xe and

148pm●

The following equations apply only in the &sence of neutron alx?orptwn;

for Zong irradiutwn times and Zarge neutron fkrxs, tkese must be correctedat

long cooling times as described in this appendix and in Sec. V. For s~mpl~c~ty

of exposition, subscripts for the decay group and fissioning nuclide are ignored;

group results must be summed over each fissioning nuclide. In addition, decay

energies can be presented in four related ways: (1) gamma (beta) energy/s,

(2) gammas (betas)/s, (3) gamma (beta) energy/fission, or (4) gammas (betas)/

fission. Presentation on a per fission basis is preferable in that it eliminates

the actual fission rate and is, for example, independent of the reactor ”power

level. However, this is only possible for a constant fission rate prior to cool-

ing. For broad multigrouping, it is also preferable to use the actual energy

release per second or per fission rather than multiplicities, because the latter

requires a specification of the average energy per group, which is not neces-

sarily well approximated by, for example, the

We have specified energies in MeV units.

and MeV/s is

MeV/fiss sMeV/s

fiss/s ‘

where the numerator is the

denominator

ffss :

106

is the fission

midpoint of the group energy.

The relation between MeV/fiss

(Dl)

energy release rate during the cooling time, and the

rate prior to cooling. Let

total number of fissions prior to the beginning of thecooling time,

S(t’) z

f(t) z

F(t,T) ~

H(t,T) ~

Then

ffss =

fission rate (per second) during the fission interval,

MeV/fiss/s at t seconds following a fission pulse (that is,the release rate”in MeV/s normalized to the numberduring the pulse),

MeV/fiss at t seconds following a constant fissionT seconds,

MeV/s at t seconds following a fission interval of

L

[S(t’)dt’

‘o

If S(t’) is a constant

H(t,T)F(t,T) = S

●

= s, then

MeV/s MeV

1=— .

Fiss/s FissL J

All quantities are computed on some arbitrary

Here we are using f(t), F(t,T), etc., to

by summation codes or approximations obtained

of fissions

rate of

T seconds.

(D2)

(D3)

al quantities can be accurately fitted to several

report we have used a linear sum of exponential

N

f(t) =x

a e-A~ti

i=l

(MeV/fiss/s)

unit volume.

represent quantities calculated

from fitted functions. The actu-

functional forms. In this

. (D4)

The number of exponential N required for a specified accuracy of fit depends

on the total time over which f(t) is to be used and on the energy group. How-

ever, the number is minimized by use of a nonlinear least-squares fit to the

xi parameters. For example,al ‘

an extreme case is the fit achieved to the

total beta plus gamma energy release rate over a very long cooling time -- 23

exponential have been used to achieve fits well within 1% out to > 300 000 yr.*

*The pulse fits are based on 6 points per time decade from 0.1-10 13 s and formthe bases of the 1978 ANS 5 decay-heat standard.

107

The multigroup spectra, which do not require such accuracy, use fewer exponen-

tial. In addition, when the pulse fits are folded into an extended fission

interval, the resulting accuracy is further improved.

In practice, the pulse values were obtained from the summation code using

10-4 s as the irradiation

Therefore, the pulse data

The use of 10-4

s for the

time. The code normally provides F(t,T) and H(t,T).

F(t,T)f(T) is obtained from f(t) = T = F(t,T) X 104.

-2pulse is arbitrary; for times < 10 s, the value of

f(t) does not change. This is to be expected because the fission product half-

lives are long compared to such short irradiation times.

The value of the fitted pulse function is that, once obtained, it can be

used to produce F(t,T), the energy release rate at t seconds following any

finite fission period T.

T

F(t,T) =

~

f(t+T-t’)dt’ (MeV/fiss) ,

0

or changing variables,

t+T

J

F(t,T) = f(t’)dt’ (MeV/fiss) .

t

Using the fitted form [Eq. (D4)], this results in

N

x AF(t,T) = se- t -A T

Ai (l-e i ) (MeV/fiss) .

ii-1

(D5)

(D6)

(D7)

Alternately, we can fit Eq. (D7) to the result of, for example, an experi-

ment following a finite fission time to generate the parameters a ,~ Ai for an

equivalent pulse function. This is useful, for example, when it is necessary

to combine the results of several experiments, all having different fission

intervals.

In the ANS 5.1 decay-heat standard, the concept of heating (or energy re-

lease rate) function following an infinite, constant fission rate (H) is used

and is derived from the pulse function. That is,

108

w

JF(t, m) = f(t’)dt’ .

0

(D8)

This is only possible in the absence of neutron absorption. Otherwise the mag-

nitude of F(t,T) would increase continuously with increases in T. However, the

concept is useful in that the F(t,w) function could, like the pulse function,

be used to generate the finite irradiation values. Thus,

t+T a m

J

F(t,T) = f(t’)dt’ =

~

f(t’)dt’ -J

f(t’)dt’

t t t+T

= F(t,m) - F(t+T,m) (MeV/fiss) . (D9)

The pulse function f(t) and the infinite irradiation function F(t,~) therefore

contain equivalent information. In this report, we have concentrated on gen-

erating the pulse function because users will likely find it easier to apply to

a variable fission history (although either function can be used). It should

be noted the the a., Ii parameters apply for either function, provided that the1

fit extends over a sufficiently long cooling time. Thus ,

Na

z .F(t,~) = -Ait~e

A.(MeV/fiss) .

1=1 =

However, for practical reasons, the ANS 5.1 decay-heat

13as infinity so that the factor (l-e-AiT) with T = 10

(D1O)

*

. .standard defines 1015 s

s multiplies the terms in

this expression; that is, Eq. (D7) is used for F(t,~) in the standard where13

T=1O S.

In the case of a variable fission rate S(t), it is necessary to use MeV/s

rather than MeV/fiss [i.e., H(t,T) rather than F(t,T)].

T

1

H(t,T) = S(t’) f(t+T-t’)dt’ (MeV/s) .

0

(Dll)

109

Alternatively, one can use the power level

T

H(t,T) =r

P(t’)— f(t+T-t’)dt’

K(MeV/s) ,

where

P(t) = power in watts at time t ,

and

K= watt-s/fiss (for 200 MeV/fiss, K

11 ● APPLICATION OF THE FUNCTIONAL FITS

The Variable f(t) is only a symbolic

-- of the summation code output. Most of

in the main text are useful only when the

simple functional form such as Eq. (D4).

(D12)

= 0.32042 X 10-10) .

representation of pulse library data

the expressions in this appendix and

pulse values are approximated by a

With Eq. (D4), the user can readily

construct expressions to be used in practice. For example, if P(t) can be de-

scribed by J histograms of constant power P over time intervals T+, theni.l J

-~ (T+t+j) -ak(T+&T )ek

-ej-1 1(MeV/s) ‘ ,(D13)

where T = O.0

If the power is constant over the entire period, then this reverts back to

Eq. (D7) multiplied by P/K.

The reader is reminded that -there is a set of coefficients (ai,Ai) for each

energy group and for each fissioning nuclide. Thus, the expression [Eq, (D13)]

should, strictly, have a group subscript, In addition, if there is more than

one fissioning nuclide, the P /K should be replaced by the fission rate for eachj

‘uclide Sjiand summed over k. In general, for each group g, Eq. (D13) becomes

110

111. ABSORPTION EFFECTS .

Absorption couples mass chains and changes the fission-product distribu-

tion. The effect is dependent on the type of reactor (neutron spectrum) and the

power history (magnitude of the neutron flux). Therefore > Precise Calculations

of F(t,T) or H(t,T) must resort to summation calculations; there is no single

tabulation of F(t,T) appropriate to all reactors and, of course, the pulse func-

tion does not incorporate the absorption effects.

Most of the absorption effect occurs in those nuclides near the line of

nuclide stability where half-lives are long and where most of the radioactive

nuclides that are shielded from precursor decay by stable nuclides are located.

Because these nuclides are not only generally long lived but also have rela-

tively small decay energies, their effect is primarily evident following a long

fission interval and long cooling times.

On a nuclide-by-nuclide basis, the neutron absorption, can increase or de-

crease nuclide concentrations, hence decay energy. Except for a few nuclides,

the net effect is small and positive. At some cooling times there is a very

large effect that can be identified as due to only a few specific nuclides. Be–

cause these are few in number and do not require explicit calculation of their

short-lived precursors, it is possible and practical to supplement the F(t,T)

and H(t,T) values with values from correction equations that account for the

modified spectra at long cooling times for any power history. This has been

done in the main text, and parameters appropriate to light water reactors are

included there. The equations apply to any type of reactor with appropriate

cross sections. For completeness, this apperidix summarizes the basis of the

corrections.

The most significant absorption effects do not require explicit calcula-

tions for short-lived precursors nor do the significant corrections require com-

putation of multiple captures in fission products. However, the corrections are

dependent on fluence, flux level, and neutron spectrum. While the net correc-

tion could be approximated with an empirical expression at short cooling times,

the long cooling times, where corrections are particularly important, require

aciurate knowledge of the behavior of specific nuclides. For this we have found

that two nuclides per chain for those few nuclides presented in the main text

are sufficient. In addition, a general solution of the differential equations

for two coupled nuclides can be programmed and used for all nuclide pairs.

Let N..N. = the density of the first and second nuclide, where 1 and 21- z

A=

A=

‘i =

Y~ =

E. =J

~.

[The

denote parameters of the respective nuclides (N is normallygiven in units of I/b-cm).mfOa@dE = absorption rate per unit density. If @ is given on

a group basis (two groups are adequate for thermal reactors), the.value of A is XgOg$g (S-l).

-1decay constant (s ).

fission rate from fuel i (fiss/s-b-cm).

nuclide yield per fission from fuel i.

energy per decay, where j denotes the 13or y decay group (MeV).

branching fraction for the type of coupling between nuclide 1 and 2.

nuclide subscript (1,2) has been omitted for simplicity of expres-

sion from A, yi, E. and a.]J

For simplicity in writing the general solution, let

fizA+A , (D15)

Y = Z.y.s.JJJ

(j denotes fissionable nuclide) .(D16)

y ❑ C@ or ct~ depending on the coupling, where the yields y. willbe cumulative, direct, or zero, depending on J

the nuclide. (D17)

With or without

the general solution

interval t are

neutron absorption (i.e., f3❑ 1 in absence of absorption),

for N1 and N2 during a constant fission rate for a time

-@lt-131t

Nl(t) = Y1 ‘l-e )131

+ N1(0)e 9 (D18)

112

r -fy

1N2(t) = ylY1 — -e-Bit

13;132 @@2-B1) + E2(132-B1)1

[

x -e-ez’+ylN1(0)62-61 [1

-(32’

-1-Y1 -e

+N (0)e-62t2 132 2

(D19)

For a constant fission rate over the entire fission interval, N,(O) = NO(0)

= O for fission products. If the power or fission history is variab~e, as w~uld

be the flux and (31~ values, these equations can still be applied by using At. in

place of t for eac~ histogram interval j and using N1(Atj_l) and N2(Atj_1) i:

Place of N1(0) and N2(0). In this case, N1 z(t) is the density at t = Atl -I-

At2+. ..+At..j

The associated group deca~ energies are thus ANEJ Y,@”

It is necessary to evaluate these equations during and following the fis-

sion interval. In addition, it is necessary to subtract the densities or ener-

gy emission rates that occur in the absence of neutron absorption because this

energy is included in the F(t,T) and H(t,T) functions. From a computer code

viewpoint, this is easiest to do by coding the

the results with and without a neutron flux —

result in any significant absorption effects.

We assume that the fission history can be

general equations and computing

or, rather$ a flux too small to

described by histogram intervals

during which Y and A are constant. In this case, N(0) = O for the first inter-

val and Eqs. (D18) and (D19) provide the basis for recursion equations for subse-

quent intervals j of duration At.. For example, Eq. (D18) becomesJ

(l-e-~ljAtj, +N -f3jAtj

Nl(tj) = N =Ylj lj B lj-le 9

lj(D20)

j = 1,2,3, .... J,

and similarly for N2(tj).

113

If J = the final fission interval, then at any shutdown time t

()e-A1t- LA2t -A2t

N2(t) =ct~N1 1 lJ A2 -Al + ‘2J e

.

(D21)

(D22)

Note that

(D22) ~8 zero,

if the coupling is by (n,y), tkn the f~pstte~ forN2(t)in@~

tkzt is,

-i2tN,.(t) = N2J e . (D23)

(In programming these equations, any potential problem with roundoff can

be avoided by replacing each beta or lambda with, for example, f3(l-10-9/t), where

t is the shutdown time or Atj

appearing in the exponentials. This permits Eqs.

(D20)-(D23) to be evaluated without resorting to a special set of equations, yet

the effects of the change in beta are too small to alter calculated densities.)

The energies are given by

MeV/s = ([N(t)-N(t)’] AE , (D24)

where N(t)’ is the density without neutron absorption.

If the fission rate

elide, the MeV/fiss from

MeV/fiss MeV/s=— =fissls

S is constant and applies to a single fissioning nu-

any nuclide is

[N(t)-N(t)’]~Es

. (D25)

* u.s. Government PRINTINooFFICE 19713-777-08W163

114

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