Table Of ContentNICOPLAAC ILIO
ReactPoh�y siDcsi vision
ArgonnNea tionLaalb oratory
and
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Rome,I taly
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jb
ATOMICE NERGYC OMM ISSION lvlsolfoT ne chniIcnaflo rmation
U.S.
OR�
1!1SJRJ8Ull0N D'D OICHUIMSE IISU
AvailaabsTl leD -24512
fromC learinghfoourFse e derSacli entainfdTi ecc hnical Information
NationBaulr eaoufS tandarUd.Ss .,D epartmoefnC to mn1erce
SpringfViierlgdi,2n 2i1a5 1 $3.00
LibrarCyo nogfr eCsast alCoagr dNu mbe7r9 ·600321
Printiedn t hUen itedS tatoefsA merica
USAECD ivisioofTn ec hnicaIln formatEixotne nsion
Oak Ridge. Tennessee
April1969
FOREWORD
Thibso oiksp arotft hceo ntiCnruiitRniegcv aiSlee wrp iuebsl isthheed
by
DiviosfiT oecnh nIincfaolr moaftt hiAeoE nCT .h isse rpireess deinstcsu ssions
ont he sttahtaeer ts opfe cainfldii cm istceide nfiteilofdfisi c n terest,
in
identsiifgnyiifncigoc natnrti bbouttphiu obnlsia,sn uhdne pdu blWihsehneedv.e r
posscirbilteeiv,ca allua antadil otnesra nvaetnwiuivbeleesgil v eHno.p eftuhlel y,
authortsa kaed vanotfta hogepe p orttuopn rietsayme unltt iotffu adcet s
will in
ano rdemralnyna enrtd o s ynthneescwio znec oeupottft s h ceo ntributions of
many.
woublede xtreimnetleyr eysotucerod m mecnotnsc etrnhibinosgo k
We in
oro thienr ss eriehso.pt eh iantt hsep iorfti htpe r ogyroawumi bleal s
the We
constrcurcitatiysivo ceculaay ln .
JohMn.S ullCihviaenf,
SciePnutbilfiicc Bartainocnhs
DiviosfTi eocnh nIincfaolr mation
iii
PREFACE
Thibso oki sd edicatteodP rofesUsgooFr a rinaenldl iD Crh.a rlEe.Cs o h!l,
twoi mportmaennti ni tdse velopment.
UgoF arinienltlrio dumceet do nuclear cnoorirseel amatenitdoh now dhsi le
l wasw dtinmgy doctortahle sis uhinsdd eirr ectLiaotne.hr e s uggested,
encouragaendds, u pporat efdu lplr ogram of thaenoder xepteirciamle ntal
p
studiienls c alayn,dh e ropostehda1 tm akeal ong-tveirstmio to neo ft hmea in
laboratoofrt iheeUs n iteSdt atteosr efl.nteh ef undamenotfar lesa ctor-noise
analyMsiys g.r atittuodC eh arlEe.s Cothrna nsceanndyfs o rmaelx pression.
Withhi cso ntinuaosussi stcarnictei,c al coopeexrtaetnisokinnv,oe w alneodd fg e
thes ubjehceth ,e lped not onwlryi tiiannng de tdhiet ionftg h weh olaen alysis
buta lsion s �pervitshiecn ogl lecatnidoo rng anizoatfti hoemn a teriIta hla.n k
botohf t hesgee ntlefmoetrnh ee ntiprreo duction.
Certaiandlvya,n cemaennddt esv elopmienrn etasc tor-tniomaiens ael ywsiilsl
occuarf tetrh iwso rkg oeisn tpou blicaatnidot nh;e secan n ot,of c ourseb,e
includientd h issu rveDya.t ac ollecwtiiloclno ntinue wiinttheorurtu ption. I
shablelg ratetfoau nly r eadoefrt hicsr iticalw hroew viilielnw f omrme o fh is
opinioonn a nys ubjebceti,nt. gsc austaishc e l ikePsr.i vate communications,
memorandutmesc,h nirceaplo ratnsda, r ticolnre esa ctor-tniomaiens ael ysis and
related tercehgnairqdubieonstgp h a setx perimneontqt uso teidnt hibso ok and
futuarcet ivwililt bye g reaatplyp reciFaotrte hdes. a koef g atheriinnfgo rmation
on thipsa rticfuilealrad n, a ccumulationm aypb oeio nftg enerhaell tpo
interessctieedn tPilsetassse.en adt l eaosntec opoyf a lylo uwro rktsom ea t
LaboratFoirsiioec C aa lcoRleoa ttori
ComitaNtaoz ionpaelrle ' EneNrugical eare
C.S.dNe.l ClASaA CCIA
RomeI,t aly
Nicola Pacilio
v
CONTENTS
Foreword
i1i
Preface
v
Introduction 1
LisotfS ymbols
2
GeneCroanls ideroant iRoenasc toArn-aNloyMiseSe!it sh ods
4
Time-CorrAenlaaltyisoins t hANeme ountgr on-DPeutlescatsi on
1: 8
Rossi-AMiepthhao d
9
Endogenous PSuolusrecMdee thod 17
AutocorrelaantdCi roonss -CorreFluantcitoinoo fnt sh e
NeutrCoon untiRnagt e
21
IIA:n alystihsMe o omfe notsfN eutrDoins tribution
28
FeynmaVna rianMceet hod 28
ExpectNeudm beorf C orrelated-NeutMreotnh-odP airs 39
BenneVtatr ianMceet hod 44
Third-OrAndaelry asnidst heEx pectNeudm beorf
Correlated-NeutrMoent-hTordi plets 48
CovariaMnecteh od
63
IllA:n alystihsPe r oofb abiNleiuttyr Doeonft ection
60
Zero-CoPurnotb abiMleitthyo d
61
Polya-MoMdeetlh od 66
OtherR elatPerdo babiMleitthyo ds 70
TimeI nterfvraolma RandoOmr igMient hod 74
PulsteoP ulsTei meI nterMveatlh od 79
IV:C orrelAantailoyAnsm iosnt gh Sei gonfst hFel uctuations
86
PolarAiuttyo correlaantdCi roons s-CorrMeeltahtoido n 86
CovariaMnecteh ovdi aP olarDiettye ction 89
A Synoposfi sN euStrttoains Mteithcosd s
96
Conclusions
97
Index
99
vii
�I
INTRODUCTION
Theq ualiatnyd q uantiotfthey growinnugm beorf t heoretsitcudailae nsd
practicaaplp licabtaiseodno sn t heo bservatoifon ne utrfolnu ctuaitnia o ns
nuclearre acteomrp hastihzeie m portaonfthisc e subjeicnrt e actkoirn etaincds
dynamics.
In manyl aborataorroiuenstd h ew orlgrdo,u pso fs cientwiosrtkos n
reactor-annoailsyoesna i fsu ll-sticmhee dTuhleei.cor n tribuitsic olne avirsiblyle
int hec ontinuiamlplryo ving odfe sgroepeh isticoafthet iotnh eoretical
treatmenitnsthe , highaluyt omatviezresdi oofne sx perimetnetcahln iqaunde s
datpar ocescsoidnegas n,d i n tnhuem beorf s pecial-piunrsptorsuemn eonwt s
conunerciaavalillya ble.
·
Oneo ft hem osti mpresfsaicvcotens ce rninrge actor-mneotihsoedi ssthe
multidpilfef erentiaantdbi roann choifnt gh eo neo rt wop ioneering procedures
intseov eraslp ectieaclh nifqouthere sm ethodologyd,a tctaoh lel ecatnidto nh,e
interpreotfta thireoe snu lts.
Todaya bibliograopnrh eya ctor-annoailseyi ssni os l ongaenra cceptable
guidteo t hes tatoef tahrets ,i ncthee q uantiotfp yu blislhietde rawtouurled
makea neophyatpep roatcohthe subjeecxtt remcoenlfyu sinIng .o rdetor
partiaalvloyit dh icosn fusiowne, have restricotuerdp resentattoi otnh e
techniques anrdea lpaptJeJtode s dp ontanfleuocutsu atiinot nhset imed omain
andh avdei sregather dFeodu rier-conjmuegtahtoetddhs a t itmhpoel byae rvation
anda nalyosfithe s samep henomeinnat he frequencdyo maiTnh.i rse view
describes theev olutoiftheo na rttr eateidnC hap6.o fJ .A .T hiReea,c tor Noise,
ANS-AEC Monograph, RoawnmdLa int tleInfic.e,lN de,w Y ork1,9 63.
Althouag h bibliogorna rpehayc tnoori sei s adeqfuoartt hee oretical
referesnocuer cietsi, s o ftens horotn pracitnidciacla tfioorpn esr forming
experimoern tcsh ootshiebn egsf to rmufloair n terprtehteimFn.og r t hirse ason,
wed edictahtibeso okto t heex perimenters.
1
LISTO F SYMBOLS
2
Thirse vieenwd eatvoso yrnst haensddie zsec trhisebt eao tfte h aer att
presbeydn itv itdhimeno gs t-tuescehdni inqftuooecu sar tegeoarcoihfwe hsi,c h
featsuurbesst dainftfieacrloe nncceeernsii tnthghem ere asuarpipnagro artt hues
elaboradtaiatonatnd h eo fe xotfbr faocntniaootnnti hroeen a cptaorra meters.
Thes ubjiesac ptp roaicnmh eeddi raessn :og eneral doerf mitions
mathemaitnitcraodlu tcont uicolnresea arc ldtnoert incosit,sh ee oarnyd,
stochparsotcieacsr giseve estn h;e csaeben fouinndm ornee acormlpyl ete
bookIstw. o ubledw orthtwohr ielatedh e f..�u nntPdahaylrs iroc fNs u clear
KinetibcysG .R .K eepRienac,t oNro iseb y A.T hiaen,dt he vast
J.
comprehternesaitvmReea nndto amn idPn s eudoranNdooimsT ee chniques in
J:lttcleSayrs tebmsyR .E . Ubherfieognr ceo unttheserp iencdgie ftioacfit lhse
presseunrvteI yna. d ditai foanm,i lwiiatrrhie tayco tpoerr atiboans iacn d
nuclienasrt rumiensne tcaetsisoanr y.
Thef infaolr muolfea ascp hr ocedaurrneeo dt eribvueatdr sei mply
descrainbiden dt erapcrceotretddoai nen xgp eripmeernstpaeslci tntivcehe e,
maienm phagisviest� o mtehteh odtohleeof gyf,i cainetdnh mceoy r,oe rl ess
is
diffriecaulliotzf ta htteie ocnh niques.
extemc:hivreo norleofgeirceanlrce e pofroterea dcc ha tetgoo ry
An list is
provais duef ficcioemnptdlloeyct eu menotfthea tthieoonr beatcikcagrlo und of
thmee thaondodsft hmeo swti dkenloyrw ens oublttasini ldnneedtp iacr ameter
measurements.
LISOTF S YMBOLS
A Counrtai(tncego unts/sec)
b(t) Binaroyr1 )v ariable
(0
B=e=:FD K'2 Probaobfdi eltietcycot rirnegl aat epda pinerer u turnoint time
2oo.2 intervaattl i tme 0 =
Cjj(T) Autocorfurnecltafitooairtn o i nml ea g
T
Cross-corrfuenlcatftioi�aron n t ime lag
c1j(T) T
Cov(.ot) Covaroifta hnneceu mboefr etvheoancttc su irtnrh etedi me
interval.ot
Mz-M
D Divfeins psairoanm eter
= 2
M
F Fissria(otfnei ssions/sec)
fk(.ot) FrequoefsP arm.pycl oenst akie nvieinnangtt si minet erval.ot
G(s) Probtaydb iis trgiebnuetrifuaontnci tnigo n
I Amplita�s Jpeo notfab nueorousfnst e utrons
K Effemcutlitviep flaiccattoiro n
Kp Prommputl tiplication factor
N Totnaulm boefsr a mples
LISTO F SYMBOLS 3
Numboefrs ampwlietsch o nteeqnuta kle vteoin ntat s i me
inteArtv al
Numbeorfs ampcloenst aainne ivne(gno ord dn)u mboefr
NevenN,o dd
events
n(�nt ), Numboefer v enatt si miera te.c:.rtv al
in
n(+t .c:. j� ), Numboefre vents thatth teoi cmceu rirtn+e Gt d- e Iriv)na l
nj(.c:.t) .c:.tt otj.c:.+ t
no Ampliotfua sd peo ntabnueraosbutos tve h mee apno wleer't cl
p(dtt) Probabtihlaaitft,tya ep ru lasett imte =t hnee xptu lse
0,
arrbeitvweeste a nn td d+t
Po(t) Probaboifln iotco yu ntisn a timei ntervoarlt he
Po(.c:.t), .c:.t
probabtihlaaitft,tya et ri mte= 0 choseantr andom,
the
ftrpsutla srer ainvtyei sma ef tetr
Pk(.c:.t) Probabtihlaasit at mypc loen tka eivnesn ttsi miienn t ea�r tv al
Probabtihlaaist ta ym pcolnet aainne vse (noo rd dn)u mboefr
PevePno•d d
events
P(dtt) Probabtihlaitt,ya p aufltasettimeer t 0=, a p ulaserri ves
betwtea enntd dt
+
Probaboifld iettye cat +i+n,g- + , -,-a nd pair,
+-
respectively
Pol(-r) Polacroirtrye fluantciftooinarot ni mel ag
r
C ()
R('=r )h kr Normalciozrerde fluantci(toh=ino o rn k )
=I=
chk(O)
Repetriatotifebo un rosfit nst eIn( sbiutyr sts/sec)
RR
Sourscter ength
s
T Totmaela surteimmeen t
U(�t) Ratoifto h eex pecntuemdbo efcor r relnaetuetdr ont o
triplets
theex pecntuemdbo efrr a ndnoemu ttrroinpi lnae ts
time
inte�rtv al
V(�t) Relavtairvieoa ftn chneeu mbere voefin nt st imien terval
the
At
Correlraetleadvt airvieoaf tn hcdeei ffebreetnwcseeue cnc es
sivsea mpolfed su rataitao nn di tass ymptportoimcp t
val(u=ie 1 sa) n dde lavaluesy e(id =2 ,3 ,., 7 )
..
Third-coorrdreC?rlp aatriaomanen tidet arss ymptporotmipct
val(u=ie1 )a nd deia(y=ie 2 d,3 ,v., a7 l)u es
..
Correrlealtaevariantdci ev eo f numbeorfc ounitnas
the time
intervali ta.6.sst y mapntpdor toimvcpa tl ue1 )a( nid =
delavyaeldu= e 2s,3 ,(, 7i )
..•
Ratoifo expecntuemdbo efcr o rrenleauttrepodan i tros
the
thexep ectneudm boefrr andnoemu tpraoininr ast ime
inteartva anildt psr om':'a!Uep t (=i 1 )a ndde lavyaeldu es
(=i2 , .3,.7, ).f o art imien terva0l At=
GENERALC ONSIDERATIONOSN TIMEA NALYPIS
4
$ R=/ (3 Reacti(vdiotlyul nairt s)
� Summatisoynm bol
n Produscytm bol
Expectaotpeiroant or
0
Prompt-neutreoing-ednevca(alRyuo esal,sp ih a)
a
Reactor-transfpeori-e(fsfu onir=c t1 ii otcn o inciwdietash )
ai
Effecttoitvfaerl a ctoifdo enl ayneedu trons
(3
Pal-Miolgner-Zoloctourk.'tinr elaptairoanm et(e=r sf 1or
purerlayn dodmi stribution)
Samplitnigmi.en terval
Detector effi(ccioeunnctys /fission)
Prompt-neluitfreotni me
Prompt-neguetnreorna ttiimoen
v Numbeorfn eutreomnist tedf ipsesri on
1l(td)t Probabitlhiatty, aa tfitmeter= 0 c hosater ann dotmh,efl rst
pulasrer ibveetsw ete na n+dd tt
K-1 Reactivity
p=
K
a Nondimensosuirsoctner aeln gth
s Generating-dfuumnmcytv iaorni able
( + 1
signx (t) Polaritoyr - 1)v ariable
Timel ag
T
Timceo nstants
Tr,Ts
t Time
(top> Averatgiem ei nterbveatlw eean r andoomr igiann dt he
first-arriving pulse
Averatgiem ien terval cboentsweeceunpt uilvsee s
GENERACLO NSIDAETRIONS
ONR EACTOR-NOISE
TIME-ANVASLiM SE THODS
Thef irfsutn damenatsaslu mptoifeo anc mht\ thoisdt hatth er eactoirna is
steady-ospteartaetco inndgi tiio.ne,. , crsiutbiccrailit neio qcrua ill ibwriituahm
,unifosromu rcTeh.ia sss umptimoanyb ed ispensfaobsrlo em ep rocedubruetis t,
represeanu tsse fsuclh ematization.
Then eutrpoonp ulatiisoo bns ervtehdr ouag hsu itable dewtheocset or
output isi netiegtrhaoetrres di mpcloyu ntiendt het ime domtahienn e;u tron
enerigysu sually disTrheeig nasrtdredu.m enftoartt hieosemn e asuremcane nts
bet heco rrurn.eount ron-monaiptpoarriaontfgtu hser eact(opru lcseo untae.r,sd
GENERALC ONS IDE RATI ONS ON TIMEA NALYSIS
6
currcehnatm bertsh)ee x cwietpohtft i hoocnsa seine sw hi:cuhghd etection
efficairneeen ecaidneesdsdo mien -core instbreau dmdeemda.t ion must
A commfoena taumrotenh gpe r oceidtsuh raaelttslh t ee chnairbqeau seesd
ono bservtihpnerg o mpt-neutthra.doten c baceyhca auoiseftn hse prompt
subcrciotnidciTathlie odinersca c.yo nsdteatnetrs.tm hoiern deoe fmr a gnitude
oft htei miensv oilsnvu eacdnh a ltyhs!.teiryase p; p roximately
msefco Dr2 angdr aprheiatcet ors
100 0
1m0s efco Hr20 anodr ganic reactors
1m secf otrh ermal repalcu·tt oornsium
10J.L Sefco rre flfeacrstetea dc tors
1J.L f!efco brar fe arseta ctors
Sincet htee chnhiaqvause et sa tistictahlwe h odaelnreai livyasst iiso n,
all
basneodot,n s inegvleeni t.tesi.,ms, ee parbaettiwtoewneno ne utbruootnns ,
obsearvnisdnt go arl ianrgg e oafdm aotuan.t
Twoa pproaarcpeho esss Oinbeile ss. t riatc itmlaeyn alysoinst heb ased
localoiftz haaetr iroinnve iuntgor noa tn isma xei s;o thieasrn inttyepgrea l
the
thcaotn siincs otusnt thinenu gm boefnr e utdreotnesic fntJX eetddi imnet ervals
ancdo lltehcdeta iatnsaag t imsee qunejn(cjAte t ,o)+ r,a ss uitraebdluec tions
ofi tT.h es ymbnojdl e notthenesu mboefpr u ltsheast occurred (or the
integroauttepidunt t i)m e itn t+-e lGr) vtAaottl +j A tw,h etrsiae n i nstant
inr eacltioAfrtei ,st h et imie nteardvoaplat nejddi ,sa n i nteG=g 1e,2r ,
, Nc)h aractetrhiseze iqnuge nce.
3,...
Foudri ffteyrpeoenfstp rocedaucrceo mmo1d5a mteitnhgco adbnse
distinagcuciosrthdohe iodtnw hg e osuet paurmteas n ipulated:
1.M ethboadsesod nt ime-corarneallaytsitiohspne u la semiso,wn n hgithe c h
varicoorurse lfautnicotni boenca sl culoafft-elodira n pep roxbiyma a ted
can
suitaarbrlaen goefm etnhte meas(udreilnagy ead-copipsnaycrsiatdteeumns,ce
time aonrma ullytziewsric,tra hel peerct yictlTiehvseei)se n. c lude
thReo ssim-eatlhpohda
•
theen dogpeunlosuesd m estohuordc e
•
·t·haeu tocorarncedrl oastsi-ocnmo ertrheolda tion
thtiem ei ntefrrvoaram al n domm eotrhiogdi n
•
the tpopu ullstseei mien temrvaelt hod
•
Althotuhgebhye lotnotg ht eim e-analysithesse lcaatstwtteo e gcohrnyi,q ues
will iben cliungd erodu pf otrh sakee ofa morceo ntiannuldoo ugsi cal
3
sequeanncfdeo a rb etutnedre rsotfa ntdhien gip nrfeosremnatteido n.
2M.e thboadsesod n a naloyfts himeso menoft sd istr{inbjui}tn,i on
the
whiacc ha lculliamtiitototneh sdee coortn hdi orrdd egre nesruaflfltiyoc ient
is
givteh dee siinrfeodr mTahteiisnoecn l.u de
thFee yrvuanrainma entcheo d
•
theex pected ncuomrbreerl aotfe dme-tnheoudt ron-pairs
•
