Смелов В.В. Лекции по теории переноса нейтронов

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ypaBHeHHe

rrepBoro

nOpH,llKa

nepe~

MeHHOH

0,+

-dr

Ecp(r,

-I)=/(r)

Haq3JIbHbIM

yCJIOBHeM

(i7.10)

cp(R

-1)=0.

(17.11)

ECJIH

renepr,

sanauy

Koum

(17.10),

(17.11)

pemars

KaKHM-HH6y,nb

KOHeQHO-pa3HocTHbIM

MeTO,llOM,

MbI

npanex

KOHeQHO-pa3HOCTHOH

125

tj>opMyJIe

OTHOCHTeJIbHO •

BeJIHtIHH

q>kO,

npHQeM

(J'NO=O*.

npe,llDOJI01KllM,

lITO.

3TOAOnOJIHHTeJIbHOe

ypasaeaae

peureao, T. e.

H3BeCTHhI

saaxe-

HHR

(j>

Y3JIaX

JIHHHH

(CM.

pHC.

25).

3HatIeHHSI

<PA;

yaJIOBIiX

TOtIKaX

(k,

np~

l~i~n/2

MorYT

6biTb

onpeneaeau

epopMyJIe

epkj=G;fk+l.

i +

R~,

(17.12)

r.a.e

a;;;

==- D

•

-lICk,

i-I;

Caenyer

ocoticrronsepxnyrs,

sro

CtIeT

ecaepxy -

BHH3»

$opMyJIe

(17.12)

npa

1~j~n/2

cosnanaer

HanpaBJIeHHeM

yCTOHtIHBOCTH

CQeT8.

Ilpa

nt2<j~n

cser,

aanporas,

YCTOHqHB

«cH'Hay:..-

saepx».

3TOM

cnysae

pacxeraaa

$opMyJIa,

noaynaexas

H3 (17

7),

H~eeT

BH,l{

(17.13)

r,11e

= - C

•

i-I;

Rt=

Dk.1f_1

(Ekfk+Fkfk+J

Ak.i-llPk.i-1

-8"k.l-llk+a.I-

) .

,[lJIR

peaJIH3aUHH

cxera

<popMyJle

(17.13) HY>KHO

3HaTh

saane-

HHB

epyHKUHH

JIIHHHHOd

('OM.

pac, 25).,

HaH60Jlee

eCTeCTBeHHO

-3TH

3Haq~HHSI

nOJIytIHTb

CJIe,nYIOIUero

YCJlOBHSI, otiecnenaaajomero

HeHTpOHHhIH

OaJIaHC

B ueurpe

c<j}epbI:

q>Oj=<Po,

n-j,j>n/2

nOCJIe

roro

KaK

3HaQeHHSI

<pyHKUHH

q>kj

nOJIyqeHhI

scex

Y3J1ax

paCtIeTHOH

ceTKH,

nerxo

no,nCQHT3Tb

,nJlSlCJIe,llYIOIUeii

HTepaIlHH

SHa-

qeHHSI

<pyHKUHH

<po

(Ir) (17.2). B COOTBeTCTBHH c

annpoxcauanaea

(17.5)

HMeeM

lP.

(rl)

= A;

~lPk'

+2lPkl

2lPki

+ .••+

2rpk.

"-I+lPkn)·

Zlaaee

HTepaUHOHHhIH

npouecc

MO)KeT

ObITb

nosropea

aeotixonaxoe

QHCJIO

pas

(B03MO>KHO,

npHMeHeHHeM

yCKopeHHSI

CXOJ(HMOCTH).

06ocHoBaHHe

(aecrporoel) CXO,llHMOCTH

HTepaUHOHHoro

nponecca

Sn-MeTo,na

CJIe,nyeT·H3

'§ 15

(CM.

TaK2Ke

3aMetIaHHe

2 K ,§ 16).

3 a Me.q a H

He-

BWlIe6hIJI

yxasaa TOT

lIOPSIJI:OK

'IlOCne,Zl.OOaTeJIbHOrO

BbIlIH~Jle

HHH 3

eHiHH

«P1&l,

KOTOPblH Heo6xo.ll.HM )J.JUI

YCTOHQ'HBOCTB

S,,-aJIrOpHTMa

[12].

·B

JJ.yxe

asaaraeaoro

MeTOAa

!KOHetIHo-'pasHocTHYIO

a1I1IpOKCBMaUHIO

ypasne-

HHH (17.10) ecrecraeaaee acero

peaaasoaars

'B upeA1JIOJlO1KeKBB

JIH:HeHHOCTiH

4JYHK-

~HH

<p(r,

-1)

Me1KAY

'Ka1K,ZI.OH

napoa pacseraux YSJIOB aa

JI'B1I'HH

ac '(!CM.p'HC. 25).

126

O.!t-H8KO

npeJlJlaraeMhlH

nOpH.D.OK Bbl'IIOJlHeHHH BhIqHCJleHHH

-~BJISIeTCH

.Il0CT3TOllHhIJE

YCJlOBHell

ycroiiqlmOCT.H.

ReKOTOpOM

CMhlCJle

POJIb

.ztOCTaTOllHoro

YCJIOBHB

-lIDpaeT

Tpe6oB8HHe

nOJIO>K'HTeJIbHOCTH K03$«I>HUHeHToB

Gkj

H·

ai!i

Stj)opMyJlax

f17.12) B

(17.13) (12). 3anHUleM

ynoMSIHYThie

Ko~clJHU'HeHThi

B q>op-Me ·paSJlO>KeHHB no

nap8--

MeTpy

8k=4I-",/r/&:

rae

akj

= (2

Ip.

/ I- Ek

/(2

IIJ'

11+Ek

4rk).

llJUI

nOIJ.3

&TISlIO~erO

60JIbmH~eTBa

TO-

qeK

ceTKH

ecr

s M3JIh1H napasserp, 3

nOToMY

SHaK

G:J

npeHMYnt.eCTBeHHO

onpene-

JISleTCB

3BaKOM

akj.

CJle.n.OB3TeJlbHO,

Tpe60Ba

HHe ali>0 CBOJI.HTCB B

arore

He-

p3seHcTBy

3 3

e q a H

e 2.

KaK

caenyer H3 y-p3BHeBHSI .( 17.1)

-T3K1Ke H3

co06pa>KeHHH

CHMMeTPH~H,

B IleHTpe

-H3yqaeMoH

ctj)eplll lJIOTOK He.

.llOJDKeH

aaancers OT -

yrJIO'BOH

.KO-

opllHHaThI,

T. e. BO

scex

TOtIKaX

JIH'RHH

'(eM.

'PHC. '25)

SHalleRHH

epyHKUHH

,l1.0JDK-

HhI

6hITb

pa'BHhI

,llpyr .u.pyry:

<J'OO=tq>Ol=(f>OF=

•••

=q>o.n-l=q>On.

3To

06cTOBTeJlb-

CTBO B

paCqeTHOM

aMopHTMe He

OTpa>KeHO,

a 'lIOTOMy

'IlpHBe.aeHU8S1

uenoxxa

paaeacra

MO)f(eT CJIY>KHTb KOBTPOJlbHhIM K-pHTepHeM TOlJHOCT1i· pacsera.

§ 18. &AnAHCHWM

METOA

YCKOPEHHSI CXOAMMOCTM

MTEPAL\MOHHOrO

nPO~ECCA

cnynae

Me,l(JleHHOH

CXO,l(HMOCTH

MeTO,na

BJIa,l(HMHpOBa,

Sn-Me--

rona

HJIH

JII06oro

npyroro,

OCHOBaHHoro

HTepaUHOHHOM

npanuane

(

15.1

)

npHxo,nHTCSI

npaoerari, K

npaexaa

YCK.OpeHHH CXO,nHMOC-TH.

Onmnex

O,nHH H3

TaKHX

npHeMOB,

KOTOphIH

MO}KHO

Ha3BaTb

6aAaHC;..

HbtM

MeToiJoM

YCK,OpeflUfI,

CXOOUMOCTU.

3TOT

MeTO,n

H3BeCTeH

yJKe

cpaBHHTeJIbHO

,naBHO

[~2]

C ycnexox

npHMeHHeTCSI

KaK

npa

peme-

HMH O,nHOCKOpOCTHhIX sanas,

TaK

H B

np06J1eMaX

MHorOCKopOCTHOii-

reopaa

(ca. § 30). Hs

MHorHX

npHeMOB

yCKopeHHH

CXO,nHMOCTH

OCTa-

HOBHMCSI na

6aJIaHCHOM

BBH,ny

ero

npocroru

rJIy6oKoro

<pH3HQeCKo-,

conepacaaas.

MTaK,

paccaorpax

cxO,nHIUHHC~

HTepau.HoHHbIH

npouecc

(15.1),.

KOTOphIH

KOpOTKO

MOJKeT

6bITh

sanacaa

<}lopMe

(18.1

Oneparopsr A, B H $Y'HKUHSI Q onpeneneasr na

c.-

li13.

·npe,llnOJIOLKHM,

qTO

··B

ypaaaenae

(15.5)

no,nCTaBJIeHO

pemeaae,

peayasrare

qero

nOJIyqeHO

TO)K,neCTBO~

YMHOLKHM

3TO

TO}K,neCTB()

CKaJIHpHO

'[eM.

(15.3)

]Ha

aexoropyro <PYHKII.HIO p

(r,

>0:

([A-B]<p,

p) = (Q,

p).

(18.2}

• H3 CCYUlOK

1Ia

OTJleJIbHble

'l»opMyJIhi

§ 15 He

cnenyer,

xro

npensapareasaoe

HsyqeHHe

'§ 15

o6BS8TeJlbHo.

127'

ECJIH

~TepaUHOHHbIH

nponecc (18.1)

npa

,naHHOM

erne

H~JIh3H

ot)o-

paart,

(paanocrs

cp(n)--Icp(n-l)

3aMeTHO

OTJIHqaeTCH

HyJISI)

, TO

paCCMaTpHBaeMOM

HTepaUHOHHOM

arane

<pyHKUHH

(()(n), nootime

rOBOpSI,

y,nOBJIeTBOpSIeT

paseacrsy

(18.2),

TaK

KaK

OHa

3TOM

CJIyIIae

nO,ll.IIHHSIeTC5I

,npyroMy

COOTHOllIeHHIO:

([A-B]ep(n>, p)===(Q,

p)+(B[cp(n-l)_ep(n)],

p).

113

nOCJIe,nHeH <}>OpMyJIhI BH,nHO,

lITO

rpedyexoe

paseacrao

(18.2)

3a-

Be,ll.OMO

y,noBJIeTBOpSIeTcH

TOJIbKO

npH

n-+oo.

A eCJIH

MO,nHq>HUHpoBaTb

HTepaUHoHHhIH

npouecc (18.1)

TaKHM

o6pa30M,

IIT06bI

Ka)K,nOM

HTepaUHoHHoM

nrare

BbII10JIHHJIOCb

pa-

BeHCTBO (18.2)?

XOIIeTC5I

,nyMaTb,

lITO

3TOT

npaesr

enoncrer-aer»

HTe-

paUHOHHbIH

npouecc

yCKOpHT

ero

CXO,nHMOCTb.

nOCJIe

Toro

KaK

BbICKa3aHa

06ruaSI

H,ll.eH, MO,nH<pHKaUHIO.

HTepaUH-

ormoro

npouecca

yxce

JIerKO

npnnyxars.

MOJKHO,

aanpaxep,

nOJIy-

IIeHHyIO

<PYHKUHIO

!(j)(n)(r,

YMHO)KHTb

aexoropyro

KOHCTaHTY

C«,

nO,ll.06pa

IHHYIO

T.aKHM

pacaerox,

qTo6bI

<py'HKUHH

Cnq>(n)(r,

y,ll.OBJIeTBOpSIJIa TOLK,neCTBy (18.2). B

arore

npHXO,ll.HM

cnenyiotneay

MOJJ:H<pHUHpoBaHHoMy

npoueccy:

(18.3)

rne

~(n)

CJIeJI.yeT

paccxarpaaars,

KaK

aexoropsra npOMeJKyTOqHhIii

pesynsr

ar,

(18.4)

(18.5)

nOJIYIIeHHoH

tpopxyne

(Q,

p)=F.O,

TaK

KaK

p>O, a <PYHKIlHH

HCTOq-

HHKOB

ueorpnuarensna

panna

TOLK,neCTBeHHO

HyJIIO.

CJIe,nOBa-

TeJIbHO,

Cn=FO.

XOTSI

,nOKa3aTeJIbCTBO

CXO,nHMOCTH

npe,nJIOJKeHHOrO

MO,nH<pHIlHPO-

BaHHOrO

npouecca

06meM

cnyxae

nOJIyqeHO

[;15],

npaxranecxoe

npnxeaenne

aJIrOpHTMa

(18.3) B

pane

cnyxaea

secsaa 3<p<peKTHBHo.

3 a M e q a H H e

MSI

BblQHCoI1eHHSI

K03<fl<flHUlieHTa

HCnOJlb3yeTcSI,

npelKAe,

acero,

Y}I{e

HaiiAeHHaSl

4lYHKIlHSI

A~(n),

pasaas

B,(n-l)

[CM.

(18.3)].

BhlQHCJle-

HBe

BTOPOH

~YH~HH

~(n)

«He

nponaAaeT

AapOM»"

TaK

KaK

BenHqHHa

B,(n)

Heoo-

XO,!{BMaSl AJISI BhlnOJIHeHHH

CJIeAYIOm.ero

HTepaUHOHHoro

mara,

onpeneaaerca

epop-

qMyJTe

B<p<n)

CnB'<n).

ella

H H e 2.

HcnoJ1b3YSl

nepsoe U3

paeeacrs

(18.3), HeTpYJJ.Ho

nOJIy-

8Th

AJISI

CJIeAYIOmyK> epOPMY

JlY:

(B (;<n) _

ep(n-l)],

= 1 +

---..;...,

--.;.----~---.;---

(Q, p) - (B

(f{n)

- <p<n-l)] , p)

OueHHM

B 3TOM Bblpa>KeHHH CKaJISlpHOe

npoasseaeaae

(B [f(n) - <p(n

-1)]

p).

nOJIa-

ras,

liTO

ecrr, TOqHOe peuieaae 3a,u.allH (15.5), 0OO3HaQHM

~<n)

-fen); e(n) =

,en).

BbIqHTaSl

nepsoe

paseacrao

(18.3) H3 (15.5),

nonysaea

A7<n)

=B€(n-l)

HJlH

7(n)

-lB€(n-l).

128

c·

yqeTOl\1

oiJ.eHKH

(15.16)

HMe6i·

OTCIO,lla

1I,<n)

- ,<n-I)II = IIs<n-l)-

;(11)11

lle(n-l)1l

+1I;(n)1I

ns(n-l)lI.

Haxoaeu, c ,HCnOJIb30BaHHeM

lIepaBeHcTBKoWH

BY'HSfKoBCKoro

IJI

IIBII~I

i(CM. § 15)

HaXO.llHM

ouenxy

CKaJISIpHOrO npOH3Be.ll.eHHH

I(B(fin) -

,en-I)],

p) I

liB(;in) - ,<n·-I»II·lIp(l<

Bll·llpll·lf'<n)

,<n-l)1I

IIpU·ll

(n- l}

1/.

B cnysae CXO,ZJ.SlmerocSI

arepanaoaaoro

nponecca (18.3) Jl8(n)II--+0

'D-pH

n--+oo~

BcooTBeTcTBHH

lJIOJIyqeHHoH'

uenOtIKOH

HepaBeHCTB

3TO osaaxaer,

liTO

CKaJISlpHOe

.npOIf3'BeJJ.eHHe

(IB[q;(n)~cp(n-1)],

p)---+o.

nOCKOJIblKY

(Q, .p)

=#=0,

TO ;H3

q»OpMyJlhl

(18.5) CJIeJly.eT, nro

Cn--+I

'IIpH

.'n~oo~

,nOCneJliHHH

pe3YJIbTa1'

n03BOJISleTB

H3BecT·

HOH

CTeneHH·

CJIe,Z(HTb sa

'HTepaIlHOHH~M

'nPOIlecCOM~

JII06.0M

cnyxae

ero

cnenyer

06pbfBaTb,

noxa Cn

«3aMeTHO

OTJIHlIaeTCSI 01'

eAHHUUbl».

Haxonen, iJIOJIe3HO

06paT,HTb

'BHHMaHHe

erne na O,Z(Hy neraas, A

UMeHHO:

'lIO

Mepe

upH6JIHiKeHHSI

napaxerpa

Cn K enanane Bee cnaeee -ecpatiarsraaer»

aropoe

ypasae-:

'HHe (18.3).

3TO

osaaxaer,

liTO

C'pOCTOM

n MO.lI.H<pHllHPOB8HHhiH nponecc UOCTeneHHO

cr3HOBHTCSI

nosra

3KBHBaJIeHTHhlM

arepanaoanoay

anroparay

,(18.1).'

,llPYfHMJI

CJIO·-

Ba'MH,

MO,Il.H<pHIlHlpoBaHHbIH nponecc

(t8~3)

HaH6oJIee3qx}leKTHBea

nepssex arepa-

UHSlX,

lITO

n03BOJISIeT-

6hICTpO

'IIOJIyqUTb

:«13

06mHX

'lIepTax»

Heil'rpOHHOe

-ooJle.

no-

CJIe.LI.YlOlIJ:HX

HTepaUHSlX

(npa C

n~l)

HallHHaeTCSI

y:>Ke

60JIee

Me,ZlJIeUH3S1

«mJr~OBKa»

sroro

nOJISI. ·KCTaT,H,

TaKOH

xapaxrep CXO.llHMOCTH

npHCYIIl

·K !JIeKOTOlpbIM

p-pyrBMHTe-

pauaonaua

.MeTOj{aM,

aanpaaep

MeTO.llY

MJlHHMaJIbHhlX

;HeBSl30'K [33, 22J.

aCCMoTpHM

Tenepb

BaLKHbIM

qaCTHbIH·

"CJIyqaH,Kor.na

·1.

BbI-

nHllIeM

Heo6xo,l(HMbIMH

;~pe06pa30)3aHHHMH

lKaJK,l(bIH H3

3JIeMeHTOB

paBeHCTBa

(18.2): .

(Aep,p)

ldVJ

dQ(Q\7ep+~ep)=

~dV

SepdQ+ J

dQf

'\](Oep)dV.

v V

Ilpeotipasya nropoe

cnaraexoe

$opMyJIe I'aycca-i-Ocrporpancsoro,

nOJIyqaeM

(Aep,

p) =

~dV

SepdQ

+SdQ

QnepdS.

v "--8

yserou

rpaaaxuoro

yCJIOBHH

({)(rs,

==0

npa

gn~O

HaXO,l{HM

OKOHtIaTeJIbHO

(Aep,

p) = SE

(r)dV

(r,

O)dQ+

dS S Qnep(r, a) dQ.

v . s

00::::;0

BTOpOH

3'JIeMeHT

paseacraa

(18.2)

npHHHMaeT

COOTBeTCTBeHHO BH,n

(Sep,

p).=

IdV S

~s<r)dQ

Sg (r,

00')

(r,

a')

dQ' . .

v ~

(r) dV S

ep.(r,

a')

dQ' Sg (r,

00')

dO,

9-301