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PEACO
A Code lor Calculation of Group Constant
of Resonance Energy Region
in Heterogeneous Systems
December 1971
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legion
innvenous
heierogeaeou* systems.， T
This
code
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made
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Introduction
2
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cheme0
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'EACO白川
d
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Scheme
of
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code
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.
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S.
Numerics!
method
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calculating
neutron
distribution
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.
.
.
.
J.
Ullra-fine
group
equation
of
3
.2
.
l
c
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J
a
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no
!
巾
。刷削叩
n
叶
1
3.
2 C
Calculation
off.
sk)wing-do«m
source (Aft™)
:
1
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Calculation
offfiux
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一
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E.I¥CO
・
MCROSSaxle
4.
Resonance CfQ
cross
sections
in
PEACO-MCROSS
$
. Calculation
C
a
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c
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no
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offc
collusion
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Interpolation
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in
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12
.
5
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References
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MER1 1219
t.
Introduction
There are two ro;»in case» in the treatment of resonance absorption of neutron in inii'<-ai
reactors. The first is the uase where the resonances arc considered lo be theoretically wellseparated, that is, level-spacinjjs of resonances are much larger than Doppler width . this lr«itment has been applied to fertile materials ut neutron energies below nbuut 10 IteV awl HUM
been considered to be suitable for most thermal reactor*. The second case occurs when the
resonances are not well sepenued and'or when there are overlapping resonances due to ililferent materials. This case is important (or the fust and intermediate reactor*
Various methods are available for the 'irsl case and are well discusser!, for example, in
references (1) and (2). In this case, it is usually assumed that the flux recovers an asymptotic
l/E form below each resonance, that is. "the flux recovery between resonances' is assumed m
these methods for resonance absorption Obtaining an exact analytical solution lo the i-ijuiiiion*
for neutron slowing-down however is difficult even under the simple assumption of the I lux
recovery, and this difficulty has led to a number of approximate solutions to the problem
Examples wetl-kno'vn are the first-order solutions such as the narrow-resonance (NR) and
wide-resonance (WR) approximations improvements were subsequently made by iterating
them in the basic equations of neutron balance, including the intermediate resonance (IR>
approximation1' *'.
For heterogeneous systems, if the geometry is represented precisely, the special problem
is unnecessarily complicated for the calculation of resonance absorption. The assumption that
ihf neutron Dux is spatially constant within each sub-region of the lattice cell simplifies the
problem and reduces the transport problem to Uie determination of the first-flight collision
probabilities SOT the iuti and moderator regions The simplification was combined with the
selected use of the WR or NR approximation Especially for an analytical tteutment, the
approximation of representing the collision probability by a rational expression is quite useful
and the hrterogtiK-oa* systems were treated also with the IR approach using the rational
I*he aasmmtriJon of the spatially flux is the serious shortcoming on the analytical approach
•rid th* most difftcah to remove The coupled effects of the assumptions of the flat (lux and
of the fiux recovery msy introduce serious errors in seme situations7*'1", for examplo, when the
ab»orb« dimension become* very targe, or when a very heavy nucfide is mcKlerator. In practical
cues, further complications may arise from the non-uniform distributions of ternperajure in
fuel pin and/or absorber Aermty wstb fa-^f burnup The difficulty of removing these assumptions from »nai>'f.K»l method* re»utt* tn the u&e of the method* of directly and accurately
evaluating resonance capture in heterogeneous systems isach as RICM", RIPF RAFF' 0 ', RABBLE11', SDR1*' or ih<; Mont* Carlo torfe* 3 '• )
The calculating metbod of rcnonancr a'bsflrptioa «>d '"Joppleir coefficient of reactivity in
thermal reactor* is considered to be fairly Mlisfiictory, became the neutron spectrum does not
much deviate from l/E sind the resonance absorption almost cemn froai few resonance levels
of fertile materials (-HJ) at lower eaergvFk where «carxt« ret*m»nec panuneterit are obtained
and the assumption of flux recovery is cofmdererf to be suffkwfntly s-atisfjed
Moreover, th»f
ratio of fissile to fertile isotopes in gentraSly quit* small, hence the eostribution to the Dopplcr
effect from fissile isotopes are very arnait
-
PEACO • A Code for Olculatioo of Group Constants of
Resonance Energy Region in Heterogeneous Systems
1AFRT . 0 , o
JACKI I~U
la fast reactors which contain resonance absorbers in higher volume ratio than thermal
reactors, the neutron spectrum is entirely different and in much depleted due to resonance
absorption. In this case, the resonances at higher energies contribute most to the Doppler
effect, where may be the interference between neighboring resonances in the same isotope
Moreover, to make matters worse, the ratio of fissile to fertile isotopes will be jjeneraiiy
much larger and the interference effects between various resonance isotopes become quite
important. Hence the second cuse. where the resonances are not well *cpera*ed, is important
mainly for fast and intermediate reactors.
For fast reactor anlysis, the efferlive cross sections have often been calculated by ustnjj
J/(£2\(K)) spectrum based on the assumption of the constant collision density
Even if we
started with this assumption, we however met the difficulty of the calculmion of the integrals
over the probability distribution1'' of resonance parameters in unresolved resonance regions
Hence, the variuos approximation methods have been proposed for the estimation of the effective cross sections-"'1". These treatments at present however seem to be loss complete, and
we can not fxpect that high accuracy for the Doppler coefficient of reactivity in which calculation account must be taken the rapid energy dependence of the or-value of -MPu and the
sodium scattering resonance and/or heterogeneous effects into is obtained from no approximate
method started with the l/(E£i(E)) spectrum"*'.
In order to avoid the shortcomings of the semi-analytical methods based on the ].<KX,
(£)), the •xmstruction of a resonance sequence over the unresolved region of interest b*a
been considered to be useful meanings for the calculation of the effective cross Mxtion:<(° * "
: />
' . From the knowledge of ti?« statistica! distribution oi resonance parameters and levei spacing, it is possible to generate a sequence of resonances over some energy, interval by a
ramdom sampling method. These resonance parameters can then be used in the codes obovi?
described, which have been developed for the resolved resonance region.
Though the codes above introduced can be applied only for some limited cases, they have
merits, respectively For example, the RICM-I «>de'' can treat only isolated resonances but
it can treat fairly general geometries by an interpolation method of collision probability
prepared beforehand. The R!FF RAFF!e"«nd RABBLK1" codes adopt an excellent calculation
method of slowin-j-down source which serves lime and memory c«f computer to save, however
the geometries treated by them are limited to be cylindrical and moderators must have constant scattering cross sections. On the other hand, the SDR code'-"' cor. treat fairly general
problems, but the demerit is to consume long computer time The Monte Carlo procedure
c»n treat quite general problems with complex geometry but it comsumes considerable computer time and yields neither eflwtive cros* sections nor fJux distributions.
For general uses, we developed a code, PEACQ. for the FACOM 230/60 by taking the
merits of the above codes in This code consist* of three subcode* : The first code prepares
the resonance cross sectiont 8t ultra-hm groups, which wil! be presented in the section 4.
The second calculates the coHisjof. probability at the discrete interpolation coordinates given
beforehand, which will be shown m ihe section 5 The last one is the routine for the calculation of the flux distribution, the averaged cross sections, the activation, etc
MERI 1219
2.
Sch»m» ol PEACO Cod«
A* mentioned in the introduction, FEACO code consists of three mam calculating inns.
Computing time for resonance cross sections nerded for ilux distribution calculation usually
takes half of total computing time or moie Mrtowr, the cross sections for a fixed enrrjiv
range and temperature may repeatedly have io be calculated for different cases, if both the
calcuiation parts of the resonance cross sections and of the flux are included in 8 rode
Ifcncv. the first ;t»rt of the PEACO code (PEACO-MCROSS i calculates the resononre <ro«
sections from resonance parameters over the resonance energy ranges needed for usual ralciilotions and the results are written in library tapes. This separation of the cross jertion part
from the flu* part was uted also in the RICM nnd SDR codes.
The PEACO-COLLIS code prepares collision probabilities ir: the geometry ol rylindrwa!
rod or of slab at the specified values of the total cross section of resonance absorbing compositions, which are used as representative points for collision-probability interpolation later. As
for the interpolation coordinate, we adopted one used in the RICM code. The inclusion of
the collision-probability part into tha flux-calculation | irt will restrict the geometry that can
be treat by a code, ot it may make a code for solving the nrutron slowing-tlown unnecessarily
iarge. The separation of the collision-probability part will serve to save the computing time,
when many calculations must be done for a fixed geometry. Kor more complicated geoms'tnes
such as clustered fuel elements, collision probabilies should be calculated by other codes, as
INPUT
rnonanct paromatan,
INPUT
I tomparoturt, etc.
eras] ttctioru
of
tte.
PEACO-MCROSS
I
I
PEACO-COLLIS
Library Tapes
plan*, tq^nar*
Of h««»jo(wl
CfOtt MCtkWt
—T
—
I
•• ft.-
/ 1
PEACO-MA1N
sf
OUTPUT
tte ««lrikul!on ,
rtriou* cv«f«(«d or
l , ttC
t. Scheme of PF.ACO code
Another Cod*
for complicated
g*om«try
VKACO A
(imup
JAKHI \Ti1
tbf PEACO cod* {iocs not Stave computing device for such » case
The PEACO-MAIN solves the neutron Rowing-down and calruiatc<i the various avera^r'!
values such as effective rt-sonance cruss-swtions, usintf ihf library tape for resonamc iro.-s
sftcltoiis and the input tape or ami* prepared by ihe PEAC O-COLUS or In olher code-.The reistion tawwn thes-r three coties is srhrmatically shown in Flfl. !
JAKItr
3.
NunrMMTca! Mothod of Caiculating Neutron flux Distribution
We shall consider * heterogeneous system which may be homogeneous in special case
and may ferterally be infinite lattice of square or hexagonal cell, cylmdiicalized iattice or
cluttered type fuel element. Thit sy»tem may consist of several compositions of which the
comtpondmt numbers correspond generally io few different numbers of region, respectively.
The re*on*n:e-«b*c»-binfl materials are contained in some compositions, which are called as
rWl compositions and are restricted to the following two cases in present treatment.
1) Total eim* lection of each resonance-absorbing composition slightly differs fron others,
respectively," for examples, by non-uniform distribution of temperature or of resonance
ab*orb*r densities with fuel burnup.
2) All the fuel competitions have tiime temperature and relative rntio of resonance absorber densities at each compositiaw is tame as thmtt in others, respectively. For an
example of *"U being only a resonance absorber in the system under consideration, anydensity-distribution of S0U will be permitted. '
,^,
Special attention should be paid for other kinds ol fael compositions except for the above
two case*.
We further tMuat that neutron ballance in such a *y*tem can be de*cribed by using thp
irti flight collifion-probabiiitiei. In order to avoid the fl*t-flux assumption which shoultf
usually be *i*umed in the cailiiion-probiibility method, we divide each region of the system
into MM many mb-re^ion* a* nectt*»ty m po»»ible. Then, the equation of neutron slewingdown c*n be expresaed a*
r^^J
.
(3-2)
with
Here, the »ub#cripn (i) anf (j) Mand for the «a!>-refiets number* *ad the (k) correipondi to
the nuctsar specie*. Py h the probability that * ae«ir«i bora unifowniy and i#oiFOpkaUy in
region j make* it* finrt collitiors in region i, and other not»tion* have the cu»tom*ry meaning*,
By lettiflf V«f (»>«•-'.' equal to , Mat), we-Iwve
with.
. . .
He** it should fee noted that the cqu*4fani> 0-4) and (3-5>for #i(M>»« simpler thi« E«jt. (3~
t) *nd (3-2) ai»d the factor #• muhipii*^ t© #,(»> ««rve» to reduce round errors appeared Crt
the recurrence equation* introduced for thr'Ht«n«lic*J caletihtjon of slowinii-down source in
the next afctkm.
PEACO ; A
for CakuUtlon of Group Con»?«nt*
rajy Region in flvierogcne&uv Syviern*
JAKR1 IJI9
J- 1 OHro.fii>» Croup Cqwriion of Nowfren SfowinQ-Down
Ai the r««ofi«nce cross section* «re the rapidly varying functions of energy, the energy
spectrum of neutron flux will be afso. To describe this variation accurately, the energy
range of interest is divided into extremely narrow groups. FOT the calculation of. the slowingdown iource of Etj. (3-5) on this descrete energy meshes, • numerical method based on one
developed by KlERi0) wili be used. Hetic«, this ultra-fine groups are assumed to be extremely
narrow compered to the maximum Jeth* gy gain per collision with the heaviest nuchdes, but
they are not limited to be equal width.
In order to obtain the group representation of slowing-down equation, we integrate Eq.
(3-4) over the kthaty u of the(m-H) th mesh shown in fij, 2. In this case, we assume that
the resonance cross sections «re given «t the mid point of each ultra-fine group and the collision probability /*(,(«) is constant in each of the groups. Then,
U«0
U+
U-
m+t
Hf. 3.
Lethargy meshei for obtaining ultra-fine group e<|u«lron
where
(3-7)
with
The flowing-down source of Eq.
of integrations, i.«.,
(3-8)
Otherwise.
can readily be evaluated by interchanging the order
(3-9)
Therefore, equation (3*6) can be expressed by a mctrix form u follow*.
(3-10)
where
jAERI 1219
3. Numerics! Motho4 o( Calculating Neutron I-"!ux Distribution
"'\
Q
1
J«
B
,
1
5««/i)f5i
it
L
k 1 ) /
,
(311)
s
t
2(1—at.)
and
Fii(«)rf«=5lk+,
f.J.A^^.^)
(3 i.r.
1— ai
H
with
( i 1 , , ^ ) - •«•••== [""V.if.Orfi..
Ci l-i)
In deriving Eq. (3-12), it was assumed that Fjkm" u "'' was nearly equal lo />"' u'"
3- 2 Calculation of Stowing-Oowr> Sow* (5,k"i
In order to calculate the stowing-down source given by Eq. (ii-P), we must calculate
(i'lii^))1""1'* and hence store all the fluxes and scattering cross sections for exact evaluation
of this term if lighter materials are included in the system under, consideration. Because this
storing is actually impossible, we provide another kind of groups for representiition of i;V(u)
and St,k.(u) $ ; ( B ) , and these groups are called as fine group and usually have the Icthargy
width of ten or twenty times of ultra-fine groups. Here, we assume that the lethargy width
of this fine group is also narrow compared to the maximum lethargy gain per collision with
the heaviest nuctide. This representation of fa or 2,t$t by the fine group will lw reasonable
considering from its sufficient narrowness and rather siow variation of the<u> quantities lh.au
the resonance cross sections themself. Mainly important information in iho PEACO code is
stored in variable dimensions of which sises are beforehand determined by input data for each
problem. Hence, there i» no special limitation on group numbers of 4'\ or S^M's stored, unless.
total variable dimensions necessary for calculation do not' exceed 34,000. We will later discuss
the selecting method of the ultra-fine and fine groups and recommend a group structure there.
1)
Therefore, from above discusxion, we can evaluate Eq. (3-14) by
Using the averaged value of (X.ji^j) in the fine group to which the (m-Lii1*1) th ultrafine group belongs, (OPTS-1)
an interpolation using three successive averaged-vmJues of (X,,^ 1 ,), where the ultra-fine
group belongs to the second value of these three.- (OPTS-2)
For the second method, the lethargy width of the fine group is desirable to be narrower than
half of the maximum lethargy gain per collision with the heaviest nuclide.
2)
The source term of Eq. (3-13) can successively be evaluated by assuming an initial flux
above the starting energy Ei of calculation. We assume ihst ail the cross sections are energetically constant and the flux distribution is spatially constant at energies E>E\.
Then,
letting ^'j(«)equal to V,w(«) and summation over («) of Eq. (3-4) give
i=Smm~~~\—[
k
1— Ok.J*~n.
w(u')du'xSuMVi2,ik
(3-15)
j
If we assume »0«)=*'l'i we have
f}=Sum.l~^g~R,
(.3-16)
with
/
(3-17)
»
PEACO: A Code for C*!culalk>n of Group CoosunU of
8«OT»nc* Energy Region in JiewogenMius Syiicmi
The equation (3-16) for fi can readily be solved by an iteration method stalling with
Kenot, we c«o obtain the expressions for initial flux and source as follows.
frimy^Vf*
3. 3
for u>0,
p~l,
O 18)
Cotculetion of f k a Attribution
When live slowing-down source of Eq. (3-13) is evaluated by usin^ either option of
OPTS 1 or 2, the flux distribution can be obtained from Eqs. (3-10) «nd (3-11). Kii-K"""
negiected the non-diagonal term in Eq. (£-11), and the accuracy of ibis neglect will be
investigated in the later section. The PEACO code solves Eq. (3-10) lor the illux distribution
by any of the following three melhods.
I) Matrix inversion method (OPTFL-1)
The flux distribution can be given by
M-
J
Q; or tf» = (.4»)"'<?»*,
(3-30)
where (j4 m )* ! is the inverse matrix of A" and the matrix inversion is accoippiished by a subroutine in the PEACO code. This routine is prepared for accuracy check of the following
two approximate methods.
2) Matrix perturbation method (OPTFL-2)
Generally the diagonal elements of Eq. (3-11) is considered to be larger than any non-diagomil
elements. If the matrix, A, h rewritten as
A»=zD»-Bm
(3-21)
with
£,"9,, and (&")„= JfF\rS«m
f*^
•
QZt)
then we have
(>»»)-»= (Z>- i 3 ) - ' = X ) - I a - B f > - » ) - 1 S / > - i + £ > ' « / > l
Using this approximate expression for the matrix A, we obtain
(3-23)
Hence, the flux distribution is given by
4->K - -£ lQr
rQl<»)
(3-25)
where
3) Approximation of letting F,i""~fy->*— j n g q . (3-9) (OPTFL-3)
in this case the flux distribution is given by
4>lm~Qr/2im
with
Qr-AumSumP^)^,
it
This approxmation was mti in RIFF RAFF and RABBLE code, but the accuracy have not
been investigated. Ditcuwion will later be made for the accuracy of OPTFL-3.
JAERI 1213
4.
Rcsonanc* Crow Sections Ui»d in PEACOMCROSS Code
The single-level forms-'ia of Breit and Winner for the «'x;jres.sion of resonance cross
section has been adopted for analysis of resonance ubsorption in nuclear reactor As pointed
out by raany investigators, tor fissile elements the single-level formula fails lor reproducing
shapes of cross sections, because interference effects between levels* axe sigm/icant. AIM I i< atiti
AlH.I-.K"" derived a formalism for resonance cross sections which can be expressed by resonance
parameters independent of energy and the Doppler broadening can be treated by rather simple
expressions. For unresolved energy regions whetv are important !<jr ihe mh u)«tiun of Doppler
coefficient o! fast reactors, however the statistics for the resonance parameters and level
spacing is so complicated that its practical application is quite impossible''1. There is another
expression, which was obtained by Vex,!*' •'". This expression seems lo he very useful, because
the parameters for the R-matrix theory of which the statistics are well known can be used
and the Doppier-broadening cross-sections is also given by a simple formula, in the PEACOMCROSS code, an expression based on the approximation made by V'u, t will be used, and
detail* concerning this expression will be seen in Ref. (27).
Th<* predominant contribution to cross sections comes from s-wave neutrons in the resonance energy region of interest, but p-wave neutrons contribute very littie. Hence, the interference effect* between levels are taken into consideration only for the s-wave neutron cross
sections in our expressions. Thfse resonance cross-sections ore given by the following equations.
?
l
r
*J r *
ralr,
+r«I--
f
^r',rJ
(4-2)
«,(£)£-£
where
(4-6)
?3EACO ; A Code lor Tek-ulittKin of (iroup Consume of
Resonance Knrrjjy Region i:j Heterogeneous Sy&tems
-Ei-E, and jru^r,~r,.
1AFRI 1"MQ
""
(4-6)
Various notations used in Eq. (4-l)~(-l-<)) are she same as those used in Refs (26) or (27)
and the explanation concerning them will be omitted here.
The Doppier-broadened cross sections for Eqs (4-l)~(4-3) are also given in Ref. (27)
For unresolved resonance-regions, the resonance parameters needed for the calculation of the
resonance cross sections can be generated by using a random sampling method"". For this
purpose, we can use a code. BABEL*', which generates a sequence of the resonance parameters
for Eqs. (4-i) — (4-3) over energy range of interest and writes them in a (»[«• for later use
to the PEACO-MCROSS code The PEACO-MCROSS codi? was prepared for the prcsens
purpose by modifying MCROSS code37'
JAERI 1219
5. Calculation of Coliition Probability
A jireat number of methods have been proposed for the calculation of collision probabilities in various geometries. These methods have mainly used to calculate the group constants
of thermal energy region in complex geometries. Since the resonance energy ranges of interest are extremely wider than thermal ranges, the geometries that can be treated by a code
tire restricted, and various kinds of approximations had to be adopted for calculating method
of collision probability for a complex geometry. For an example, Winner's rational approximation and Nordheim-Sauer's model*3" for collision probability has frequently usetl to estimate
resonance absorption Even for a recent high-speed computer, however the inclusion of the
caicuiatioiMti routine of collision probabilities for complex geometries in a code w\il consume
considerable computer time.
A simple and convenient method was prepared for calculating collision probabilities in
RICM code- which estimated the resonance integral of isolated resonances m heterogeneous
systems. This is a calculating method using the interpolation of collision probabilities beforehand prepared and is quite useful for many calculations in a fixed geometry or for a calculation of wider energy ranges. We adopted a method ba$ed on this interpolation for the estimation of P,,m in Eqs. (3-1!) and (3-12), and the PEACG-COLLIS prepares the collision
probabilities at the specified values of total cros3 sections of fuel compositions in cylindrical
or plane geometries. The PEACO-COLLIS consists of several subcodes which w«s develo(*e<i
by K. TstxiilHASHl*". For a simple calculation of two region problem, she Nordbeim Sauer
model is also prepared in the PEACO-MA1N to calculate the collision probability
5. 1 Nordhaim-Scutr't Mod*! for Cylindrical GooiMtry with Two ttqlem
The approximate formula proposed for l'm(ii} by NOKJWIMM*' IS given by
1 — (1 —• Z ilira) (1 — G«)
where Pa is the etcape probability from au isolated system, 1\ and li(~4\'i/$)
are the total
cros* section and mean chord-length of the system, S is the surface are* and Gm is the Oancoff
coefficent. The combined use of this formula with S*uer's approximation231 for Gm is known
to give * sufficient accuracy for two-regions problem in thermal reactor. This approximate
formula by Sauer can be written for cylindrical geometries as
C » = l - e x p { ~ r W i ) / f l + (l-r)^sfe],
(5-2)
with
r(0.9069v'l+V'a/Vi"-lV(r J /V,)-0.aj (square)
r
1(0.8863 v'l+tVv'i -l)/(V 2 /V ,)-0. 12 (hexagons!)
(5-3)
(5-4)
where Si and h are the total cross section and meim chord-length (•-= <(IVZ/S'> of the moderator
region, re»pectively.
Using the reciprocity relation and the conservation of collision probabilities, we obtain
PEACO : A Cad* U'V Calculation of Croup Consvarils oi
Rewnanctt Energy Region in Mftei'cvfccm'oijsJnatems
).
IAFR! I'*i
jnr.Ki .,
fS-ff)
The escape probability Fo in Eq. (5-1) is tabulated in the PEACO-MA1N
5. 2 hmrpolotion Method of Cotlhlon Probability in WACO-MAIN code
At first we assume that nil the compositions except for fuel compositions hove constant
cross sections. For such <i system, we define a function of lot.il macroscopic cross section, i'<r
of a fuel composition. This function Z{Si) is defined by
7M
w
c , >
12-1/(27)
(ft 7)
UV>1)
where / ia a value having the dimension oi CM which may conveniently b<> selected as the
mean chord-length of fuel rod. The function Z ( i ' ) makes one-to-one correspondence with
27 and tabes a value between 0 and 2 for a value of 1\.
It is known that collision probabilities P.,(Z) for a fixed geomeSry vary quite smoothly
for the variation of Z. The fifteen values of Z was used as the interpolating coordinates ot
o quadratic interpolation method for collision probabilities in the R1CM-1 code. The coordinate Z is aJso used in PEACO cor!e. Fcr such a interpolation method, collision probabilities
should beforehand be fitven at the specified values cJ '£, n",i thU ui!>.uUt>u<t la dose in :h?
PEACO-COLLIS. In the PSACO-MAIR the collision probabiiitics in Eqs. (3-U) and (3-12)
are evaluated by a linear iturrpoUtion method u*;ng twenty points of Z *s interpolating
coordinates.
We shail consider at first a w.y*tfv\ rtii'iaisStstg of scverai (ar\-cafapo»h'tota which &*ti*fy
the condition 1 in the section 3 Then, the f"..,{Xi, X-,. Z.i,
) for the system n*n be approximated by
P,,{ZU Z3, Zy~>-)& P.JZ*. Z\ Z*
)4-$*m(Z1~Z*) Jl P,, ,« r.-i
(S-«)
wheie the subscript for 7. is the narafopr of fiw! conspftsition and 7? h tJv* ne«fe»t value of
tbt interpolating coordmaties to the averaged value of {Z%, Z?
). Here, we assumed that
(Z\, Z?, ) did not »o Kiuci» HSnt horn each ©iber, thai i», the condition 1 was assumed to
be satisfied. It should be oo'.ed thai the nottuniform distribution of temperature or of resonance-absorbing materials with SWs feyrnup can readily he treated by thi» linear interpolation
method because of its simplicity. TJw FEACO-COLUS prepTes also the d#riv«tives «»- P,, in
Eo, (5-8).
For the c««.e where the condition 2 for iue! compositions in the section 3 is satisfied, it
will be sufficient that one of the X, '« is detected as tin- interpolating coordinate, since the
coordinate seiocted can roake one-to-one corre«poridenc« with the total resonance cross sections
for all fuel compositions. Hence, in thi» ease, onSy one of fuel compositions may h» selected
ss a formal fuel composition if 'the values of ttte coilision probabilities at the interpolating
points is calculated by taking account of the tttnansaee crtmt Motions in other compositions
On the other hand, the eoSJisioi! prohabiliti*a must satisfy two important relations, that
is, the conservation iaw
SumPii^i
for all j
(5-9)
i
and the reciprocity relation
VJiPi,
for all i, j .
(5-10)
3.
CifciiUiitKi uf CotKjion PmbabiKtv
la the PEACO-MAIN code, the v%lue* of /% «ad -^
dimension* /or them.
Pis are input only for j&i
«o s»v<? the
Hence, »t first, the F',, for ji<i are calculated from the interpolated P,,
for j S * by using Eq. (5-10).
Then, the collision probabilities satisfying Eqs. ( 3 1 1 ) ar.d ('J
12) are toccexsively obtained by the fuliowtng equations starting from ,!-; 1
(5-U)
(.^
jl
where
N
i-l
(5-12)
)ji) a 5«m A j «n ^ r^^ Sum Pi,*
with
(5-13)
(or
c . J geometry
1
:
cylinder
^ion
•
»«•«<«' °f
1
0.2406
2
a am
i
o. sim
i
3
;
4
a :«sio
i
S
0,5684
1
:i
eylimier
I ;
2
0.5625
2
3
! plane
i (center
1 j
2 i
3
0.33
0.90
1.06
1.32
! »ymmeir ~j
!
* W?"*
4
i
-2C» -JOO-50
;
|
!
ATOMS. (V> cm)
remarks
1 sa»U(UW0K) : O-ffiiSM, C O.fKStl Tiie unit c t N
2 ' » U ( S3WK } : 0.025M. C : 0.035M • J00O-MW Wcuin3 N» : U. 01216
ghimst reactor is
4 ft : O.flKS2
1 «"U(290K) : 0.0473,
The example in
2 » U ( a > 0 K ) 0.0071, F* : 0.0UT, RABKLE code.
N« : 0. OKfll?
. 0.0*7.1
J
1
1
I
-10 - 8 - 6 -< -2 0
H : 0.079".'
2
4
6
8
The unit cell o(
the 11 i &:tH
core »f FCA.
iO " 50 tCX?
fl* J Error* ol colUnioti prvhthiikitf for iKc c » « no. 1 in TA*U I,
200
PEACO • A Cod* i<it C»k«btkm u\ Grcmp f oansamft yi
The collision probabilities given by Eq. (5-M) will readiry he known to *att»fy Eq». (S 9)
•nd (5-10).
For the purpose of the accuracy-check of the calculaung method o( co!!i»ion proi>Bi>tSily by
Eqs. (&-8) — (5-13), we shall show representative exsimpies c«lcsiiated m thvee different gecir'if.i's
for which the various imformalion is presented in TAKE I. Figurei 3 — 7 «.hovv the relative error
Ef = ( P,,* - Pij"*ct) X 100//1,,™"
{5- H )
where /"ij* is given by Eqs. (S-8)~(5-13) and A , " " is exactly calcu!ate<) by she PEACO-
-100-50 " -10 - 8 - 6 - 4 - 2
0
2
4
6
8
(0
50 100
200
x = 2 ( £ - E X )/rx
fig, 4
Errors of collision prohabilitie* for the cise no. 1 in TMtl I.
Ex * H 4 9 eV
0W»' 4 b.
50
c
o
M
IO
-200
~<00-5O
' -tO-S -6
- 4 - 2 0 2
6
8 10
50 K»
X*2(E-E x )/r x
fl*. 3
Errors of col&km probabilitSe* {or the c*»e no. 2 in TA*U I,
200
5. Cnkulalion of Cvflmon Probability
MEfl! m9
COLLIS. These t»lct
point* t i the v*r»b!«.
IS
n» were made for the typical resonances of i!*U and on iwenty
x-ZiE-EdlTi.
C5- 15;
In these figures, we illustrated only a few values of P^ which are c^nsuifreti to be more
important on neutron bsiance or have larger errors
It will be t « d itoxn thrse figures tKal the errors art* mostly no more ihun 2",j nnd,
esppcsaiiy for r^Sane jjuomelty, erroE* «rc stcarrely induced by ?K*1- pirsent »iL:!iw.i<i for caU'iiU-tinu
colhsion probwhitity For some v*!aM of J", few f',,'« have fairly Urn*1 e-m>r*. for example,
the errors of f'u M .r • 0 in fif. J and pf / ' i . at x - - 6 in F19. 5 arc abou; 2 .1 ami 7 % , respec-
-0.21
-200 -«00-50^ 1 -10 -9 - 6 -4 -2 0
Errors uf
2
4 6
8 .0
50 (00
200
ic^ for t*>£-
- 500
- 400
- 300 M
- 200
- !00
-0.3.
-200 -IOO-5O 7 ~K) - 8
50 100
fl#. 7 E n w i iif ^otiisicn prot»b?litk'» (or tilt c u t no. 3 in T*»U 1.
200
jK
PEACO : A Code for Cakufaitkm of Group Constant* of
Resonance Energy Region in Helerogwituut Systems
ti»tr!y. Howsver, it should be noted that the accuracy of such • f,, is not very important
for neutron balance due to smalln^ss of the value itself or of the volume where neutrons ure
bom. For an example of Pu at . r = - 6 in Fig- * /*» i« 0. 002933, while Pu iind /'ware 0 66S»
and 0. 2390, respectively. For one-fuel composition problem, we can raist1 the accuracy by sncressmg the points of Zi where /*„ is given, though the mumbcr of the points in the present
example is twenty. It will also be noted that the accuracy of the flux distribution does r.crt
general!}' so much depend on that of collision probability if Eqs (5 9) and (5 10) wen?
satisfied.
5. 3 Ccn»id»roHon of ft*ionon<» Scaftatar and Strong Abtorbel
When the resonant* absorption is studied in a thermal reactor, it will be mostly sufficient
that the cross sections of moderator and structural materials are assumed to be fartgiet>i-*Uy
constant. In fust reactors, however, the Doppler coptfieient of reactivity IOISIM from m'her
t.igher energies where ail the cross sections must generally b« considered to vary with energy
Especially an accurate value for the Doppler coefficient can not be obtained wit haul fjtkiug
account of she resonance scattering of sodium at £ « 2 . 85 keV. Moreover, the strong absorption by burnable poisons in thermal pow;r reactors may affect the resonance »b»orpn«»
and the neutron source into thermal energy region. Generally «pe*kiti|j. for *n *c-taf«ip
estimation of the effective cross sections in resonance energy region, the variation wrtb earrgv
should be considered (or all the cross sections in u system under consideration
The variation of the cross sections of these materials is usually (juiite slow compared! 6"
that of heavy resonance-absorbers. These slow variation will bring »Uo s slowly fTtsefgesk
variation into the collision probability in the previous section. Wr mume tluit thj» »luw
variation of the collision probability can be estimated by "a linear interpolation on the fsne
groups" and it can be neglected in a fine group. Under this as*amption, in ih* PEACOCOLL1S, the collision probabilities are calculated on the necessary number of energy point*
to express the slow variation of the total cross sections of the cutipositions except for the
a
fuel. In the PEACO-MAIN, these PJa and their derivatives - j - P»* are interpolated on each
fine group by a straight line and the resultant value* *r<- used to estimate the collision probability on untra-fine groups in the fine group, following Eqs. (5-8) — (5-13).
In the PEACO-MAi'N, the material are divided into the following five groups.
1) ResortaHee-abtOTDing materials (MAOPT-U;
Their cross section* tie read from the library tapes prepared by the PEACO-MCROSS.
2) Resonanc<?-*cattering ntaterisi (MAOPT-1)
This material is limited to one element and the cross section is calculated on each fine
group by
73^r
(5-17)
here
and 0**
3) Matericlt with pointwise cross sections (MAOPT=2)
For these materials, the crosr wclkms are gtven on a number of energy points that mrc
also used to calculate the collision probability in the PEACO-COLLIS code. The value of
JAER1
ttli
4. CsfcMlnSHHi ui CoSMoa Pn»b«iw!Uy
it
cross **ctkwj e«i each fine grasp are «hwm«i from a linear isierpointion.
4) Ab»orbing mtttrtiiki with poimwlse crass section (MAOPT—3)
Ths* m»t«ti*( is *l«o lijnited t« one and the variation of the cross section arc assumed lo
iaot affect the «s!!jjioi> probability when it exists in moderatoi." region.
5) M«tcri«l* «ehh i'Otistunt *.>«»#* s«tioirj (MAOPT=4)
For M A 0 P T » & 3 *SKS 4, !&e materials are permitted to hsve the i/v absorbing cross-section.
In thi» <:»»*, .«tt«ttti(Mt *ha«Ed be p«td for the consistency* of the cross sections usetS in tte
PEACO-MA1N with tto»* sn the »»EACOCOLL!S, . . .,«•.,
JFKACO A Ced* (at CtkubnUm of Croup Constanta of
8 « * O B » « « En«K» Rtgioo in Heterogeneous 8; slew*
6.
JAERI 12!9
Group S*fu*h*r» and! Accuracy of ColcuioHon
The various approximate method* were? introduced for th* calculation of the source and
flux distributions in the PEACO code. The accurncs' of these approximations w '! iargfiy
depend ofi the group structure of the ultra-fine and fine groups. An intorcomparison has
been made for the results obtained from these approximations using the simpte geometry of
the case 2 in TA*U1 »nd changing the group structure. The group junctures uvii in the
calculation ere shown in TAKE 2, and the 2S resonances oi ^ U samr as those in t! • example
of she RABBLE code were used between 999 and 1420 eV. The results obtained . re shown
in figure* from 8 to 10. in these figures is shown the deviation from the sta:v ;r4 value
which is calculated by using the options OPTS-2 »n<i OPTFL-1 and the finest m sh width
of the c«se 1 in TASU2.
In the present example, we cuult! not find any difference fetwcoi th« fesult obtained
from the mstrix inversion (GPTFL-!) and the matrix perturtjation (OPTFL-2) /or the emlca*
iition of flux distribution, though the neglect of the non-diagonal terms in E«j. (•'< U ) »ttlv<>duced * tittie error to the absorption probabiUty. This result* show th*t a very mxw-XLe re»uH
can be obtained for the resonance flux from the matrix perturbation method, It s?»uJd slsw
be noted that the increase of computer time la the matrix perturbation method is no more
than 10%, compared to the c»«c of OPTPL-3, while the matrix inversion method take* more
than three time* of the computer time in the case of OPTFL-3. Moreover, any difference
could not be seen between (he re«uH» obtained from OPTS-1 and OPTS-2, except for the
hut case in TA&U .2 where tfte calculation by OPTS-2 were imi>o»sib!f due to wider width of
2 Group structures for ifoc test cfalcultflkm in [lit tnrrvy
r.iiiRc from lf20 to 9WeV
numtjcr of (roup«
f'ne
i
3 j
s.|
750
378
150
SSO
as
ethargy width (xlO*>
ultra fine**
i
j
\
50
20
60
25
I
|
I
|
ise
i
fine
!
ultra fine
0.17
0.94
2,34
2.34
14.0?
j
I
0.017
O.ftJ?
j
0.W7
j ' . 0.094
j
0.094
»)- Thit »how» the nwmber in « fine group.
TMU3 A tecomnsencd jtrutture of fiae and ukr*-(in« gn>upti for the u*c of ilw PEACO code
Upper tfverjy
C«V)
23250
*
4650
No, of
Lttfwrgr"'
J. 1513
4660
4S6
4t.S
*
*
*
•6.5 .
4.GS
tt,<65
Urn group*
2303
JS3O3.
sm
a) Thtawkti*r«r wMth* for the fi»»t one *nt nkctti
it
?*t tettm•MI wMi **•*
b) ThwtWi
r
$u$* in a line <
Mt»fc wWris j Mesh width
of fraw
I of ultra-fi««s
10
20
0,0006
i
o.ooi
!
o-oooi
10
0.O03
j
0.0P03
&001
ft'OOS
aote
:
O.0J
;
O.O0OU6
t*4 the m«xim«m k tharjuy I«in
6. Oroup Strucsnrc and Accuracy of Ca!eu'»atioo
JAER11219
I'J
the fine group (2-4
It will be seen from Kgi. l~10 that in the energy range from 1 to 1 1 keV we can obtain
a (Sufficient accuracy from the calculation with the group structure of the case 2 in TA*I( 3,
On the other hand, a group structure such *& seen in \he example of RABBLE code can not
give a very accurate result for various averaged or integrated quantities in this energy range.
It will also be seen from these figures that the results from group structures with more
ultra-fine groups in a fine group are in general less accurate when the total number of the
ultra-fine group is kept constant in this energy range. Here, it should Sx- noted that «tw
infinitely dilute capture cross-section of Z*\J showed the »«me vnlut1 for till \he coses considered.
Now, from the above discussion and oar experience gained by the use of the PEACO
code, we shall recommend a group structure for the •ultra fine ami fine groups, which is
shown in TAILE 3.
2
Copturg
by " * U
fltxofption
Probobility
in Fuel Region
g
0
g
1
1o - 2
iOPTS-t, 0PTFL-3I
• « fOPTS-2, OPTFL-3)
{0PTS-1.0Pm.-2)
» IOPTS-1, OPTFt-n
* - (OPTS-2, OPTFL-2)
- (OPTS-2, OPTFL-1)
-1
-2
•-
i
'
OA
8
i
73Qf1t> 37V20 t5(VS0 150/26
n*, t
I
£Vl!liO
Deviation <rf cupture cross section of ""IT
i n r t n t m ! fuel region.
3 -
3
1
B
I
- o«(0PTS-1, OPTFL-3)
. • m (OPTS-2, OPTFL-3)
A-IOPTS-1, OPTFt-2)
-4
-(OPTS-1, OPTFL-I)
- 9 - A - (OPTS-2, OPTFL-2)
• I OPTS- 2, OPTFL-1>
-6 .
730/10
ft*.
•
373/20 ISO/SO J1XV25
Deviation of tolsl nfmorjHion prt>lwb*lity.
o - r O P T S - 1 . OPTFL-3)
• "(OPTS-2, OPTFL-3)
& - ( O P T S - 1 , OPTFL-21
- ( O P T S - 1 , OPTFt-1)
' * "(OPTS-2, 0PTFL-2J
- { O P T S - ! , OPTFL-1)
-
-2 -3
-4
j
25/150
i '.
..
t
mm
fly, 10 Devi*lion of arcr*j>*4 *tiK<r(NMi I K » » Mx-tbn in unit celt.
PEACO : A Code lot C«k»>latitin of Groop CooM-mut o*
Resonance iinef*ry Ke#>»«
U
^
7.
JAS8I 1219
PEACO-MAIN Cod*
This cods is written in FACOM-230/60 FORTRAN which m a smmdard type one commonly used. In this section, the input imformatinn uertMvry for operation «f tfw code wili be
given, then the quantities printed wili briefly be dencrihed, ami finally the structure of the
code will be shown.
7. I Input
C«rd i FORMAT
1 ; 18 A 4
2 | 1216
'.
!
Variables
: TiL(I)
KREG
I KMAT
j KR.ES
| KCOMP
j KCOMPF
|KTEMP
; NRMAT
f NOW
; NOPTFL
NOPTS
NREAO
3 j E12-5
LISTR
LISTS
: NOPTX
I EOIT
4 : 121 6
I KSOUCE
s
me
6 J6EI2.5
MORK
(NTEMPtl), J « i .
KCOMPF)
(TEMPCJ), f » ! ,
i KTEMf»>
IV.tcription
idtntifioation
Total no. erf muterisls ( S J0>
, Na of resonance absorbing nuetides
No. of compositions ( g 7 )
: No. eJ fuel compwsitkw* f S3)
No. of temperature* considered fur ittunanre cross seetwtfis
A number for special resortaftce-abaorfeiH^ nuclid* of which the
m»«s is commonly used for other reMjnance»biHorbin« nucSide*
when slowing down cource* ire e»fculat*it
•" — 2 : Hex*jgfta»l geometry erf twort'^WMi*i» intuled by Nerdheim model.
•» — l : Square geometry tresltd by Ntirdheim WodeJ.
««0 : Honwueneoas probfcm
<^! ; J'Une geometry
•»'? oe 3 : Cylindrical geometry
•;
JjS : Cither geometric* except fi«r at»»ve one*
« 1 :OITFL-I in sect 3,3
>l
; €»TFL~2
This option is operated below the energy, EOPT, sjjecified:":-]hy
the next c»rd, if NOPTFL «0. OFTFL-3 i> used.
*»0 ;OFfS-J in »ecL 3.2
> 0 : OPTS-2
Lcgka! unit number to re*d c«rds or tape for Z,, V",, P,t etc.
m »ul«««tine INPUT 2
# 0 ; Fin* group flux in e*ch region in printed
* 0 : Fine group f'u* >n wch sub-region is printed.
*Q ; The t*x 2 in sect. 3 i* tre*ted.
Iktuw this tflcrny (cV). OPTFI.-I or 2 in sect. 3.3 can he
>{| ; Tl*enn«l neutron source it c»k abud on KSOUCE point*. ( £ 4 0 )
If MORE«<^ wore problems t k n one are tre»ted in a s-uis.
uf tempemture number to ench fuel composition.
Tcmper«lur< {*K>
Card ! FORMAT
7
8
2/
7. PEACOMAiN Cod«
JAER1 121S
Variable*
216, E 12.5 CNCOMP(I),
NSUBR(l),
I RMAX(I).
i 1-1. KREG)
12J6
KBG
KKGP
KSO
KAG
N8B
NBH
NBR
6E1Z5
Description
I Cerapoaitkn number «! the I'th region
Subregion number on the outer boundary of the f'th region
Distance from center to ih* outer boundary of the I'th region
:
(This information is unnecessary for NOPT^;5.)
j No. of etarse groups for averaging various quantities (S40)
Tcta! no. of line groups med to aisign the iiu of variable
uimtfcsbn (£30000)
This may appropriately be selected as a number(i£ true one).
No. of energy point* where /*., or rrews sections of n>oder»tt>y
or structural material are given (the ca»e 2 or ,'i ir. sect !i.'A)
No. of energy point* where capture cmss section is given.
(the case * In sect. 5.3) (g20)
Mtximnm no. of the fine groups a neutron can be slowinpdown in • conlliswn with the lightest atom except for hydrogen.
Max. no. of the fine groups » neutroi< can be sfowir.g-down in
• coiliiiion with the he»vy re*o»*nt *tom.
Mix. no. of the fine group* » neutron can be »lowing-down
in • coUiiioa with the «»on»nt
Energy t«xind«ri«» for c<*rsf
i (EN(I). ! - ! .
KBG + !)
Not* : In the following cuit reUling to tiw in(«m«ttoo on ma^riais, the order of matcriaU must
follow to the magnitude of MAOPT in tccL S. 3. KMAT *cu erf the cards are needed.
10 A4.8X,
Nuclide ickntiik*t>on
NUCUD(l).
Atomic miM
•IE 12.5.16. AMU(l).
; Capture crc*» wctioi! !m K 5 starting
SIGAO).
Piuion crost *ect&n for K g itarting eikerjiy
SIGF(l).
Scattering crc*i «ection for K £ starting energy
SIGS(I).
D«£ned in Kct. 5. 3
22(K}-m/KC value of 1/v abtorption era* wriion
VCAPCl).
(DEN(1.J).
I
Atomic
number dentity (atonu/(b*rns-cm)) of the J'th «an11 6EIZ5
J - l , KCOMP)
, position
Note : The foilowing two cards are unnecessary unle»* KSG>1 and a nuclide with MAOP1'»1 h
included.
I ERES
| £» In Eq. (5-18)
12 6E12.5
PEAK
j a." in Eq. (5-16)
|GJ
i ,, In Eq. (5-18)
! GAMT
| P in Eq. (5-18.)
i GAMG
; /",. in Eq. (S-17)
POTEP
j • „ in Eq. (5-16)
13 6E1Z5
POTEM
[ <y. in Eq. (&-16)
Note : The following card S» *jnncces»ry for KAG£1.
II 6E12. S
{ (EAG(I),
Energy vberc capture cro» tcction is given to a nuclide with
I
; MAOPT-=3 (EN(l)>EAG(l)).
I SIGAA{I+1),
• The ctjxtire gjaa section (SIGAA(l)»SiGA(l))
. I « l , KAG-1)
i
Note : The following card i* unntctsury far KSO1JCE gO
1.5 j 6E12.S
(ESOUCEfl).
Energy boundary thermal neutron source is calculated
i 1-1, K3OUCE)
Note : All lit* above csnl* from 1 to 15 are read from subroutine INPUT 1, while the following
tnformMion U read from INPUT 2. It ihould especially be noted that a!) the information is
input from a read suit (pecified by the togka! unit number NREAD. When <he PEACOCXNLU3 code is uted for the calculation of f,,, all the following information it output from
it foHwing the medttl FORMAT.
16
7F10.6
! (Z«),l-t. KZ)
I Z in Eq. (5-7)
PEACQ ' A Otxle for Calculation of Grocp Constant* of
Ktsonsnce Energy Return in HeiemReneoua S> firms
C*rd ; FORMAT ;
Variables
Kescriptu.n
Nfefe : At present, in the PEACO-MAIN and COLLfS, ihc number K'l is fised m be 2<), but any
number can be u*et! by a ?roaH change of the ctxle*,
1?
7 F10.5
18
7FJ0.3
RF.
I in !•;<;. (5 ")
• (V(l).
l « l , KSRKG)
(EST,(I),
Subregwo volume
KSRF.C, •., N'SUim
(KKF.C)
F.n«r«y on V-SHCH /'„ and ihe <r<»s scutum- <.f Mrudural i.r
cJadding materials are jjUen.
I-^i, KSC.i — 1.)
If KSCi^l, th^ uifoTmntiun i> unnci j?*-vaf ^.
Nnle : The fuikiwing uir<l« arc mtt ncetlcd for tht case where K S G ^ l or the muicnaU \\ 11 h MA<^>T - - arc not t»i;3udc<l in sysscrn. Sisccia! iiilcniion '.should U? paul fur ihc con>i?.(cncy uf li-itfifder uf materials \*-->Eh ihft in of the card Syjn; 10. IVtails will br seen fri>rn ihc !*(>K'rRAN
list uf :be subroutine INPUT 2.
DO 1 - 3 , J ' A S T (uniil card 20) IKAST»no. of materials wiih MAOi'T J
19
7F1O.5
Macroscopic scattering cross srclion of the (I i ISSi'th material in Ihe (J + JCOMr')'tii composition
Here, <ht 1SS is the number of the last material wiih MAOPT
« 0 or 1 just before one wiih MAOPT - 2. JCOMP = 0 (or
(<:s2(r. j).
J - l . KCOMPKCOMPF)
7FIO.5
20
21 I 7F1O.5
(CA2(I, J), J - I ,
KCOMP-KCOMPF)
(W(I),
1 = 1, KIS8EG)
NOPT»>0, ~KCOMPF for NOPT*0.
Macroscopic capture cr<»« section
22 ; 7Fias
|
.(»'S/). KMSREG =»(KSREG*(KSREf; + !))*(K2-I )*KCO: 1 = 1, KMSREG)
MPF/2
Note : The informaliun from 19 to 22 must be prepared by KSG sets.
7. 2 Output
When the neutron flux distribution $,* for ultra-fine group is calculated in each subregion,
various resction rates or inte>jr*$<?<l qaantrtis?* can be obtained from thp use of mkroxopic cross
sections. In this bsstc quantities compaied arc
J4Mi =
5«merj')i^,"' and! #ii>a:.S*<«^,B'
(7-1)
with
<J,-==^i-«-««/^,
(7-2)
where M shows the fine group number-md x &t*nd* for capture, ftssion or scattering.
At first, we can print the flux distributions of the fine group, #, M /JU M , by using the
option LISTS=VQ, where JU M is the fine group width. Also, the flux for each region defined
by
at
can be printed by using the option LISTR^O, where I shows the region number.
The following quantities are computed and printed for each broad group :
1) Averaged flux in each subregion
with U lt =5«mJU K .
(7-4)
2) Averaged flux in each region
(7-5)
3) Effective cross sections in each tubregion
JAEKI BI9
7. PKACOM.MN I wit
•0 Eff«tiv« cross sections in each region
mi
al
*™
$
.
(7-7)
5) Effective removal cross section in each subregion
where («,.i—ek>^s» is assumed
6.) Effective removal cross section in each region
it!
For each broad f,*roup. the infitsilp dtlutioi! cross sectionsi of the ujjetiai eicnunts with MAO!'
—O—
' .T are also prjn.t«-d. The above quantities from 1) to (i; are always prmictj !'IT eavli
subrejjion, but for cfeh subnvgion ihc output of shern is omitted when LISTS • 0
The following qiliantUM'!! are obtained from ihe iaiegnlion over the whole energy liifigf
of calcuiation and elwaya primed as the ia*t stage of ousput.
7) Integrated flu*, in e*ch xegion
t
i 7 • 10 s
8) Capture and fission activations in each region
.4,it=. (.SusnSumA%t.V,)/Vi;
• i! M
(Ml)
«l
•S) Totai capture and fission probabilities
with
L
'.
1— at* I i
where ,3 is defined by Esj. (3-IS).
Th*r neulrou source slowed down iato thermal energy I'anjjc is calculated by the follow
equations;
*\"'
ffW-d*'
(7-14J
where «« i* th* lethargy corresponding to the cutoff energy Ec aoci u. >« — £.» is assumed.
Then, the integrated source over a narrow energy interval from E. to /•*, car. bt- given by
i — aw
where u. and v . correspond to E. and JS», re»p«ctJvely. Hence,
S,k~O, for e k < « - - u c .
(7-16)
For hydrogen atom,
du'~caa.stant.
On the other hand when «•—£
(7-17)
I'EAi.O : A Code for Cukulciiofl of ("truvp Cunaunt* oi
Romance Energy Kegiun in tttierugfwouj Sy«tmi
Ai
'
„. ..,.,,
*' ~iJ
For C I > B , — * ,
< £ • - £ - ) P*
7. 3
Sftvttur* of ih« KACO-MAEN eod*
This code consists of a main program and eight sbroutines. The biock JiKgram of lh<>
code is shown in Fij. 11, and the complete listing will be given in ih* Appersdis C T'tr
function of each program is given briefly below.
1. Main program MAIN-WAIN : Determine!! the size of variable dimensions HJKJ mils e-ath
subroutine.
2. INPUT 1 . Reads in inpat information.
'.t INPUT 2: Reads in Z;. I, V, and collision probabilities, and reads >» also cro*.* ttx-fon%
oi live materials with MAOPT = 2. and calculates Dancoff factor for the geometry wnth NOFT
<«.
4, DATA Computes group independent parameters.
5. OUTPUT 1 . Prints input data an,<i group independent perimeters
6 AVEKAG : interpolates collision probabilities on the fine groups, c*ica(*tes the r«gio»ul
;.fcvi-ift^-acv»-»i iiource by tfie use of Eq. (3-13) and sccumu'ietes reaction rales and mjfxrtfpd
ffu*r» over live fine groups and broad groups.
7, XSECT . Reads in resonan«% crus« cectiom from itbrary tapto *nd cajc«i*t«» coitiston
probn'mfities wwi the equations (3-12), (3-20) or (3-25).
K. OUTPUT 2: Lists the fine group fluxes, calculates effective cross tcctiom snd »ver*ged
of each broad group and accumulates various integral qctrustics over whole
MERl 1213
7
CEACO-MAIN ('ode
INPUT Z
• D»t»rminolk>n of
t i l * Of vartofel*
R»od
Calculation
o Group ind«p«nd«nt
ooteutqtlon,
Material* with
MAOPT'tor 2 or*
Calculation
of a.
jf NOPTS/Oj
Msnt
A8CD
ultra-tint
Mo«« (in* troupt 7
am*
Rttum to »hwrt>9 potnl
ki MAIN-MAIN
froupt ?
Calculation of
av*n>g*d ^
In a coane
I'fcACO
K
8.
A Cotle for Cckutttt'm of Group ronstants vi
i
R
M
Syslern-s
jA£RI 1210
PEACO-MCfiOSS Cod#
This cotfc ss &'%o written in FACOSi-2'M'l,J0 F O R T R A N , and it is prepared for the
present purpose by revising the MCROSS code that originates from the GKNl'-X. H*vre, only
tr*> input information and she structure of the library tape of resonance cross section? will br
dttctibed simply.
fi. 1 Input
Cord
1
FORMAT
6 15
JOUT
1
ion
9IS
Description
Vanabtes
%{), \'tfmiw\fiQ rt'Kirjance cn.n»s sevtiun^ aft priiitcd.
\itt
ptnnUvu<' resonance e n w t-cttiurji art' written in ii hi
IREAD
-<l, resoiienit! jwrameUrs ut iht s «rv<J ^-wavc are re«<i I
tapes, and * 0, ihey arc read by cards.
1GMO
™0. /\»'*=/ ( rt/ V £ .
•* I. /*,(, if- read in.
NONR
> 0 , am(-«>th pHrt tn>s.* sections arc added w the ecHtmant?
by a step Junction.
NSMP
MF
b read in,
No. of energy boundaries the smooib crt>a»
sue tea^i m.
No. of ftwion ch«nneU (
MK=^1, a sinjjc-ievcl lurmuta is »w;d lor lK-
MZ
No. of ineUisiic channels of the I>1. cxtitcti
NOKES
No. of bk>ck» the i-wave when rcs<niar?tc jsjar^iSisnift-!: n a** m.a>X
Nu. ui inc!iJ»tic chunncla of the 2 rv<i. t x t i i t
ievtiv
in.
NOS
NOLASS
NOREP
NCM'
NOMESH
N
MSI'
MSS
NMP
IMP
NT
. NEC
4
6E12.5
No. t^f the *-w»vc resonance level* m a l-kx«.
Ne- of Iiie 5-Hijvr ievcla m the la»i bluck.
*
No. of blocks when the jfr-wgve rc»onans.«? jjttfsjaeccrt: «f* rtad
ia.
No. ol ihc p'Wtxe resonance level* in a block.
No. of the />~w«vc levels in the las I h\ot\.
No. of energy groups where the finc-grwup width i» ddicrcnl.
No. of the s-Tfitnvt resonance levels of which (he resonance levels are summed up when the resonance cross ftecdun* are
calculated.
caO, the ^-wave resonances ere not taken into consideration,
^FO, the interference scattering eflect between (he *-wave re»is considered,
single-evei formula i* used for the rajculation ol" /eaonanvc
sections.
*0, multilevel formula is ui»cd.
No. of (he fi'Vfmvtt resonance levels tA winch the resonance
levels arc summed up when the rcstiDsrvcc cro»» wxiiuns are
I The lowest group number up u* which the /^wavc resonances
conskiersd.
• No, of temperatures
*=*0, ay**4xR* and U NEC^O,
Oi^'litR'fix).
1 he lowest energy boundary
JAEJH
Card
FORMAT
5
CE12.5
Variables
rVwription
AM
At«3i?c mass
R
F.FW 2
Atomic rediua on Fermi unii !
The lowest eirergy boundary above wb^h the contribution of
Dappler broadened function V < *\', can he »eg!ecfed.
Temperature CK)
Upper energy boundary
No. of fine groups in this energy ratine
(TT<!,, 1-1, NT)
CBOUND(I).
NOiG(J},
NFK!),
IMMF.i ! !.
! ; 1, NOMK.SM)
NSJ. of uhrti-Unc ^ritup in a fine Hrwup
No. »>(' th<r fine ^rvmps which ^ u^'d u> tetlucc i&w nwntS r*r*>r
apjjcarccl m the Ictlmrgy tali Julafcd uv ^iifrtiiiifi^ ut> tH.? u:w
Kr^up. A) every Ht«r> (ij ttit?- a u m k r ci ?!*u' iw. $<*;u';» SMlethargy i? renewed to W cxK^i^tfo* •ai'h s-iw^v vAirir i^ ^v
the tn,*rgy raftge *k'hf<? thi* frm' gruMu w'tlt-i ?*• * •:_"i??^*i'
No«« The Inflowing, cord iw>. <), ti r»o!
7FI0.5
(EBSiJ).
Low
ti
for NONK ™
cr.crgy
SFS(i),
SES(I).
SCP(tj.
SniiKXh Stjion
Smxjlh eiaslic
Smooth capture cross ac«"SK«n ?yr [*w ^ w-^s- !i*rwi^rfw
SFP(tX
Smootf
SEP(I).
Smooth elastic
ftssien
I™!. NSMP)
Note : The fmbwing no. 7 card* •* unacotssary
NOS + NOLASS)
7
i 6E32.5
ES
Kj for the f-w»vc ncutnxss
\
GS
*,
'
HNS
HCS
HFS
HIS
(CFfl). 1-5, MF)
: Note : The Wtcwing na 8 cardi ii
8
far IREAD•»». Ptte up(S * M S v - . ' V ) » K S
6E12.5
EP
GP
i *
cerd*.
/',„
/'„
A,
/".»
f.«-V/' x . (•'•^•)' " fof c » (.
unnMeysary unless KiEAD*O or MP*«. Pile up i(NORF.PEi lot She /*-wave neutruns
f,
: HCP
/'„
HFP
/«
Note : The above input, ntx 7 and 8, must be read from two separate iriput-iap^s in the binary
form when IREAD~O.
t
2 Structure of Th# Library Taps of t«ionon» Cron
The PEACO-MCROSS code can mafc» only a library tape for each resonant nucfide and
r«ch library tape corresponds So the special case of one nuclide on th« ntnurture shown below.
The content* of the final form of the library t«pe consist of a few informiticm specifying the
fine and ultra-fine groups and of resonance cross sections of several resonant riudnlr with a
temperature. Hence, a sorting of rapes must bs done for the rase of few nwlicir
PEAOO : A Cixk for Cakulation of O o u p Con ..tnn of
Resowanc* Energy Kepicm m Iiv[evozenton3 .S>»j-mu
Va ri» ble
JAr.a{!
Desc n pi ion
TEMPT(NN)
,
Temperature of the NN 'th library tape
KREST
(NUCLICI).
'
No. of resonant nuclide in this u p e f ;> 1 /
Ittcnlificaljon of nuciide
I - I , KREST)
NOXG
No. of energy renges with oifferent fine and uhra-fine witiths. nnJ 'hf^e enerKy
ranges »rc called '.he cross section groups (NOXG % 10)
Sturiing energy for th*. calculatton of resonance in»s &eclson«.
Lower energy boundaries oi tht i'th cro*H .^pi'tKin n r " u P
Kinc-sfroup width in the I 'eh cross tedton grnup
Tola! no. of fine groups in ihe T'th cross section frrnun
No. of L'lfra-fine groups in a fine group
EXf»(l)
(EXG(If I),
tri(jp(l),
Nf)Ki(l),
NFICIX
l » l , NOXG)
MAXNFI
Maximum no. of NFlrl)
Note . The following resonance cross sections are piled up by she nunl*r of (Su?n NOIG(i), ! — t,
N'OXG).
((SAACJt N N , K )
SFFU, NN, K),
&SS(J. NN, K),
!, KRESl
fit for the k ' t h uhra-fine g r o u p in a fine ^ r o u p a n d for the J'th nuclide
; »,
a.
~
12U
fc
PtACO-COlLlS Code
Thi* eede calculate* the collision probabilities Ptj *!«! their derivatives ,4, !',j for the n>ulti*
region lattices on twenty points of Z defined by Eq. (5-7}. Thcsp values of Z *nd the cartripondtOff value* of 12; used are shown in TASU 4. In this code,../' is fme<i to be the thickness
of fuel plate or the radius of fuel pin. When other Code*-'met used for the calculation of /*„•
the number of Z and the value of I may be (selected adequately. This code consists of several
.t«bj>rofr*m* developed by K, TisucHiHASlH^,
T/MKJ 4 The v»I«e of ^ »ntf /i"f used tot interpotatiiMi of
coil: >on
j
1
2 i
3 I
4
5
I
6
ii
7
8" i
.4
10
z
)
•
i
t
Hi
\
M
u;
•
0.0,'i
i'li
at
ai
^ 1 2 • !'
1.6
I
1.6667
2,5 ' .
0.2
0L4
0.2
! 13 J
; H ;
1.7
1.8
;
1
S.0
! • " "
at
:
0.S-
o.«
0.7
0.85
1.0
1.2
1.K75
0.6
0.7
17 |
1.0
!8 i
19 J
1.25
20 ;
0.JW5
;
1.9333 •
1.96
!
5.98
|
1.99
•';
1.99H
.'
3. XKS)
ao
15.
20,
SO,
\m.
500.
f. 1 !*»*<
Cart i FORMAT j
[18A4
2 1 HI5
3 i 541 5
Vcrwbles
'TU.
!NK£G
JNOOMP
;NCC»«PF
1 MATER
!
1WRITE
Description
of problem
No. of
No. of
f
No. of fuel £\xnpc*ititmt
No. of m«(*rs*}i t * « p « far ««*uiuint
No. of energy potnU »!s«ic ihe croew saeiion* of sjrwclaraf w
coolant mftttrsati *n «iveo.
LofictJ unit n a fiw sin output of the in'oi-RuSion need m the
PEACOMA1S.
^ 0 , the 4et»iS*d infwm«l»w5 is given for collision probainlii>s.
! MOTO
HIS
;NOPT2
i NOPT
iTBOUND
*<J, Ifx acruraef of the pre»*.nt inethnd of E<j. (5 8) in chctfcrt for the NOPTI r«*«M»ce ]rv«U rwkt Wo*.
#0, the cue 2 in »ee(. .1 o n be trr*(«l.
<0, pUne (•oaittrjr ir»n be Ireitetl.
•=•(), cylind«rictli«<l eel!
- 1 , w)u*re Uttice
-Z
"0,
»*(,
»2,
hcxmonul lattice
itertropic bouadnry condition at the outer
perfect reflection boundary condition
fixed reflection boundary condition
boundary
PKACO : A C*S* for CakttktkM eC tiro*.? Cwwunu of
Cwd (..FORMAT 1
V«rkb(*t
i LIMIT
JAER1 1219
Dew.riptksn
i M«JtUnurn ptfh length (em) beyond which th» neutron p*th
whkh «r* coosidtrcd in i r calcuktion ef
\
JNSYMM
S j 315, FHJ.S ; (NOOOMCt),
I
j NSREG(f).
i
6 [7FS0.4
j f a l , NREC.)
| (DENCM>,
! SlGMfM).
i
| M » l , NCOMPF)
f <DEN(M),
8 i 1413
9 I 7P10.4
10
7F10.4
No. ai ««t(».*:i!t(Kt in the I'th
Suhrcttibo nsmfctt *I she outer IxiunJtiry of the I'll)
) TIt«:kn«s« of i!» I'lJi region (cm)
] NCOMP)
!<MA0PT<t),
j 1-1. MATER)
j {ESG(I.}t
;
] l.«l,NSGAT~l.)
1
1 (SS1GS0, J),
j
k nvmlner ikfltiiy of rt«msnt nuclrilc in the M'lh fuc!
| Microscopic total cn»* section of admixed material in the M'
. th fuel composition per resonant atom
; This card it cot nwdtd for NOPT1 <*>0 or NOPT 2«0.
| Atomic dentil^ of resonant iwcBde in the M'th composition
j Thii c*r4 is also unne«iw««ry for NOPT2»>0.
MAOPT in Set. 5.3
\ Energy point wh«re the crow sections of suuctural or coolant
i matcrwl are given.
| Thi* car4 i« uniwceswy for NSCATSt.
| Sc*turin( ;ro« tection of the I'th mtttrtsl on she j'th encr-
*
J - l , KG5P)
KGP-S for MA0PT*,2. KGP»NSCAT for MAOFT«2.
AbxtrfHkxi crow tection
7P10.4
CSSICAOE. IX
f J ^ l , KGPJ
|
7 F10.*
! (OENSd, $')>
j Atomic nsmber denrity of the Iih m»tef«il in the N'th
i
iK-NCOMPF+l,
: powtioo
[
; NCOMP)
j
• Note: Pit up the MATER M U of tb« emrd no. 10, 11 ami J2.
j Note r The {oUowing card is unnecaMary for NSCATfil or for the case whet* the resonance
>
llerer it aiwent.
J3
i £* in Eq. (5-18)
;RE
; K5IG0
I RSIGPP
iRSIGPM
n
] RGN
j T.
: Plie up th* NOPT1 *ru of the card no. U and 15.
IF 10.5
CTEMPCM),
j Temperature Cm. "K)
M - 1 . »CC
i Eo.
7F10.5
ener(jr ievet for th« »ccur»c,v check
£R
G
i
|GT
1SGP
-I AMU
r.
r
U •
Atoaie a m
9. X Ortprf fr«M MACOXIOOK f*r KACOMAM
Tfct output for tit* P£ACO-MAIN *r* nudk by cw«l» or written on • t*pe
/
in*
t. s«A.cocou.rs c«fc
31
the Jogka! unit no. *p«rifi*d hf W R I T E . These output h a « the FORMAT specified befow.
Description far other output will be omitted b*c»u*« they *re printed foiiowi8# to self-evident
tfmbol*.
|
3 j 7F5O.i'
;
»!, SO)
I 2 tn B.;. (5-5Q
I (ESif-SCL),
I 1MAX<»NS«EG (NSEO)
I ITti* Wormjliwi » not ouiixs! when
11,-J, NSCAT-j>
I
j Note : A3) ihc folkwing inforavtliori it pikJ up by NSCAT [«K<.
.< J7FtO.fi
I CSSSfL, M, N),
' i', cf the, M'th auterisl in the N'lh
;
I N^NTEMP,
I
INCOMP)
S •' 7FJ0.6
!j
1 fAAA<t M, N),
\JNNTEMP
NCOMP)
I
}NTEMP«NC<:«4PF+I
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\ Note; The above miotmu'mo vf no. i «»<! 5 is pifeiJ up by MATEK pair.
6 S7F1O.6
'
|
.
I7KI0.6
j (W(rr J, K),
I Pi,
( I - 1 , IMAXJ,
| K ~ i , 20)
(W1(M, I, i,K),
j
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| M - l , NCOMPF),
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is
l*EACO : A Code for Cakuiafion oS tiroup Constants oi
KescuMmre'Energy Region in Heterogeneous Systran
IAKI?! M?i
JAfcKI 1.1
The authors are greatly indebted to Mr. K. TstiHIHASiii for hi* alienating them many
codes calculating collision probabilities before publication. It would be impossible to eonuttroct
this code without his comments and gu#gettions on programming. They are also grateful to
Miss Y. TAKAIIASIH for her excellent typing of this manuscript. Thanks are also due to Dr.
S. KATSUK.U.1 for fctis interest in the present work and his critical reading of this manuscript.
1) DKESNEK L. : "Kcn«»ance Absorption in Nuclear Reactor". Pcrgamon Press, New York
2) Nutt&iEitt L. \V.: "Th* Tcthjwfogy of Nuclear Kc«ctvr Saft)", Vot }, Ch*p. i. The M. I T.
1*1*8*. Cambridge (1!M»4)
3) GOLDSTEIN 8. and CoriKK !LR. : A W . Set. Bng., 13, 132 (JWia)
4} ISHIQUKO Y. r A'*rf. Sri. E»g., 32, 422 (JS68)
5) SEIICAt B, R. and GOUKTB??* K. : A W . Set. Eng^ 25. 174 (ISM*)
ti) ISHlGlfRO \ r . ai«I TASCANO H. . / . AW/. AW. Teitotoi. ('Mya), «, 3SW (J9S))
7) IIJIMA S. : A*«c/. 5c/. £»*-, S7, K (tJMg!)
8} MtCIEH J. : J. Nuclear Energy, Jl, 117 096*0
9) MKITTA H., AOVAMA K. aed KUKAI Y. : "RICM-An IBM-7(K« G»de of iJ>.sun*nre Integral
C*l«u!«tkm for Multi-region L*tlie«T, JAERI 1134(1967). and MlZUTA H-, KAMEI T. and PuKAl Y. :
"RICM 2 : A Code fur solving the Slewing Dumas of Neutrons over M«ny Resonance Levels in
Two-Region Lattices", JAER1 1170 096t«)
10) KlEK P. H. : "RIFF-RAFF-A Program for Comijutatto of Reison»nce Integral! in a Two-region
Cell", ANL 7033 (19€5).
11) KlEH P. K. »od ROBHA A. A. ; "RABBLE. A Prugr«m for Compuwtkm of Resonance Integrals in
Mulutttkm Reacior Cell*". ANL 7336 (1907}
12) BmssEMWN R. J. sod DUKSION C. : "A Us«r-» GENEX, SOU «iui Relaled Computer Codes",
A£EW-R62-i, Winfrith fjsm)
US) CANDELORE N. and CAST St : "RECAP H A iionte Carlo Pwgram for Estimating Epitherma!
Capture Rates in Rud Arrays", WAPD-TM-I2S, flttti* Atomic Pow«r Labtwatory (1964)
14) MtLUEk L. B. »nd MUMY C. H. . Nutdt, $ei. £sg.. 40, 438 (l?70)
15) Pt* example. PoRTCR C. E. : "SuliMtefti Thtotie* of Speetrit : F!Htt«»iJ«»*", Acadtmic Pres», New
York (1965)
16) HWANG R. N. •• AV/. &*'• *>?;. J*. S23 <!9tir>)
IT) Proc««iinK of a Confercr»:c «n Brrtding. Eccmomw* i"n<! Ssftf in 1 jiryc Fast Reactor. October
7-10, J9KS, Argonnc National Laboratory, ANL 6796!
18) TAKAHO H. « «/. -^, A W . 5 « . T<T*»^. cr»<ow), n 500 (1970)
19) BHISSENDeN R. J. and DtxRSTOS 0. : T h e CakulaiSon of Neutron Speetr* in the Doppkr Rtgkm".
ANL 7050, Argonne Nations? Laboratory &9S6)
30) DYOS M.W. and STEVBNS C.A,; Ar«rf. -.«.*««„ » . 294 (1966)
21) KEUKK C.N. and KlE* V. H. : Afcrf. Sft. £«^., 14, 383 (1966) and 35, 67 (1966)
22) ISrBCUBO Y. «* *T.; A'*f/. &<. £»#., « , 25 (1970)
33) AWJEB D . R and AMXR F . T . : TrttJi*. V»«, Nuei, Six., S, S3 (19S3)
24) HWANG R. N. : AWri Sf«. £««•„ 3*. 67 (I9ffi») and St. 32 (1970)
JAMS! 1213
9. f'KACO-COLLtS Code
25) VOGT E. : Pkys. Rrv., 113, 203 (J958) and HI, 29 (I960)
3B) UH& BUCKIJEK P.A.C. *ml PlJU. l.C. : "Doppter Broadening ui C U M Sections', AEEW-RXW
0962)
*
37) KATSUKAC1 S, <•<«/. : "jAERI Put Rtactoi Group O n u u n l * Syaitms Pan I I I " , ,'AERI 119S(197t»)
and TAKANO H. «I S(. : "A Program for CcSeubtion of Ooppfer Broadened Microscopic Cr<» Section
}>y Multilevel ForntuU", lo 1* pubiisbeti in JAERI-mcmo
28> ISMKiUSO Y. »nd KATSUKACI 5i : "BABEL, A program of Construction at StiUstical Resonance
R»r«meters in UnresolvesS Rcsonaftce Region", to be publisKed in JAKRl-memo.
JS) S*«KR A. : xV«i/. AV». &ifr, 14, 329 (I9E4)
.JOj TSlWHHASHi K, : io be puUiihccl ia JAERI report
i
PEACO : A Code lox Calculation of Group ConitinU of
r,
Energy Region ta Helttogentoas S r "
Appendix A
IAII-DI I?IO
Sampta probttm lUfino of PEACO-MAIN coda
To illustrate the u»e of PEACO-MAIN, « sample problem is presented fov the ca&f no.
I in TA»U $. In this problem, the fife-regions problem is reduced to two-regions one where
fuel rr>d has an averaged temperature, 1500 °K, and the mixed region of sodium bond, cladding
and coolant is taken in outer region. For the simpficity, the resonance cross sections of ~"Pu
are given by effective <?r«»s sections and MAOPT lor fflJPu i» assumed to be 3. The energy
range treated emends from 7. 73 to 0. 1 keV.
I vwui '**** t u t ••(!'•'«. M i t
O i l C1S.1
i
1
iw
is n
u f•
jTAER! 121$
*-p»cn&x A S*rapie pwhtcm K.&tty o! PEAC0-M.MN rod*
Vi*. i#'>
I
»
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1
f^'
W* I f l
»""»• t
35
41 ft
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V«>
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M
MS
JJI
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*Jjft
PEACO ; A Cede UIT CaicuU'kw* of
Resonance Energy Re^ton m Hcirr
*,.}<-''•• >-•(' J W / . X . !
71
Tt»«.«J
!-T*«^ 1 . : . * * ^ J-
c.-S
JAER1 1219
Appendix A Sampk problem Ratiitje of PEArO MAIN
'•••"-•
; x.«
,-^ 1
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O.i'
f' J
V'tr
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0-j
«,
PEACO : A Code {or Calculation of Group Constants o<
Resonance Energy Region in Heterogeneous Systems
JAf'RI 1218
Appendix B
Sompi* input listing of PEACO-COLLIS and PEACO-MCROSS cod«*
I
*
00.(1? I *
?*•*.
il
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j . •>•••!*
.,
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t
MERI 1219
Appends* C L»(in* of !h« PKA CO-MA IN «>urcc deck
Appendix C
G • •
Lilting of »h« PEACO-MAIN tourc* dmck
1***1
•
) 1-4» * * f j ^
If (<MUPT.tft.01
J'RiMXJ
A O
JJKMI in
)
3
,
»
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JAKJ8E 122f
Amet&x
C Urintf til
J*
--
FfttOO; A Cad* /or C*lcu[jfiors of Group Constant* af
» m M w e Energy fttg&n fo (kicrofcncmi* System*
lt^/.j)**!
vp
SXBWL 121*
Apixnvi*. C Usiifitt af th* PEAOOMA1N *ow
to
MNnt.
f
wr tit.-
v%Z"*«, «<. ^
PEA CO ' A Ccsdc far Calcutatsem *if Group Constants ot
ReswiMK'e Etifiifl:>' Sexton m Heies-oger-eiMi* Systems
7* IM*>MWQ,
^^W*r
<[ M < i i ' i f P l
%ts'
J« HI(.(41,
(\F£!
'
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Appeodk*. C U«ing of the PEACO-MAIN source deck
J * * * ! * * ^ f • MLfip.!itfi3
IJtf )#<,,J*** f.'W, ( i f » * ! *
i 5 t > ' ' U '-^' f lit.)
»
PEACO : A Code tor CsicuiaSton of Group Cvns-tsnt* of
Rc*OR»r,>cc Enefgy Region in Heierogfrtcuus Systems
i f f l!i
«*l Tw I I
&1 ( f f ! * S . | f * 4 t « f l M Ml TO
?
^
. %5
CHf.l,!
S: • c (
re 5i"H'•3.
T
tt'
IAFR!
"
JAERl 12J9
Appendix C
?0
Listing ot ib* PEACO-MAIN w t » « det:k
1
ift=.
1(1 •MMfMBf
w> Tw w-'ii
f i. y it O i *?J»«C I n * < t»WJ J " " H ! 1 i I
jawtij^n'^'Sii"
PfcACO • A Ortfc for CakuUtwft <jf li-ru-up t\i«itiants at
>?
OH-.'**
dti ^Ii,rt^Tm^i-tlt
,
JAERI 1219
Apjwndi* C Usring »( ih* 1'EACOMAIN sou«<- deck
BO A -J»J «t
jit»J"
!« O .t«,tl»<<.l
Lf iJW f Ii-"r5 l i
PEACO : A Coda for CfckwUlJon of Group Cuiuttnta of
R«nwa»nce Ka*ryy Region in Heterojgeneovi? System*
"Vvi row*** ((•*s»,fo-*.i*,ii
u ; ft ,-x -iH*r
JAKK! 1219
jAKRI 1219
Append)* C Utting of tht PEA CO-MA !N tource deck
*
*•* s *» r i ? *.<.
l,W*1UHi;O.fi)l^*H-
t«TF
<x> * (>i ...aft.
n M if i i.
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i, Hi!
j^ii.
f ,-.3*
l^^
3* ^ • : .
tf/f^.t
L Pi-'O*
Ut 1
oTMM^ vr*l*<| -
I
PEACO: A Coat for Calruteffon nf r..-.-up Constant* of
JAERf !?I
HtftW/HK^ce Kncf$t?' R^gwun in Ku-ieroitenirauu Systems
<
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