Смелов В.В. Лекции по теории переноса нейтронов

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(CM.

§ 20)

HeBH3KH

F<n+l)(z,

J.t)

[22].

134

,UJIH

QHCJIeHHOH

peaJIH3aUHH

MeTO,ll.a

paC~eIIJIeHHH

KaJK,llhlH

npOMeJKyTKOB

-1

~~~l,

O~~HHaHocHT

cerxy Y3JIOB_

HH'TerpaJIhI

(19.14)

armpoxcaaapyior

IIO,nXO,ll.Hme

Bhl6paHHoH

xsanparypuoa

epop-

MyJIOH,

peurenae

,[(I-HlupepeHUH3JIbHoro

ypaBHeHHH

IIepBOrO

IIOpH,nK3"

(19.15) (aenpepsraaoe

rp3HHU3X

p33,[(eJIa

cpen C

p33HbIMH

CBOHCT-

BaMH)

HaXO,llHT

IIOMOmblO

xoporno

H3BeCTHbIX

KOHeQHO-p33HOCTHhIX

cxex. B COOTlBeTCTBHH C

rp3HHQHbIMH

Y'CJIO"B'HHMH (19.'16)

CtIeT

IIO

JIIO"-

60H

KOHetIHO-pa3HocTHOH

cxeMe

HyJRHO

BeCTH

CTOpOHy

B03paCT3HHH

KOOP,llHH3TbI

IIpH

1J.,t>O

H B

IIPOTHBOIIOJIOLKHOM

Hanp3BJIeHHH

IIpH

~<O.

CeH'tIa'C

MbI

O'IIHIIIeM

OJlHY

TaKHX

cxex,

o6eClIetIHBalOI:UYIO

yCTOHtIHBOCTb

BbltIHCJIeHHH

coxpaHHIOIllYIO

BaLKHeHllIHe

KaQeCTBa

oneparopos HCXO,llHOH

aanana.

Pa306beM

IIpoMeLKyToK

-1~1J.t~

tIeTHOe

tIHCJIO

p3BHhIX

HH-

TepB3JIOB_

Cepennay

K32KJlOrO

TaKHX

HHTepB3JIOB

npHMeM

Y3eJl

ceTKH

yrJIOBOH

IIepeMeHHoH.

YCJIOBHMCH

OTMetI3Tb

OTpHU3TeJIbHH-

HH,[(eKC3MH

Y3JIbI,

OTBetI3101I1He

oTpHuaTeJIbHbIM

3H3tIeHHHM

Jl, H

nOJIOLKHTeJIbHbIMH

HH,lleKCaMH

nOJI02KHTeJIbHbIM

3HatIeHHHM

Ta-

KHM

06p330M,

HyMepaUHH

Y3JIOB

npoxeacyrxe

-1~!J1<

6YJleT

cJIeJlYIOIlleH:

-jo,

-jo+

1, -

-2,-1,

...

io-l,

ifJ'

CJIe,llOBaTeJIbHO,

....

;=--1,..,-;.

npoMeLKyTOK

O~z~H

H3HeceM,

sooome

rO·BopSI,

npO'H3BOJIbHYIO

cerxy

Y3JIOB

(19.17)

CeTKy

(19.17)

scerna

MO:>KHO

nocrpoars

TaK,

tITo6hI

KOHT3KTHpYlOIllHe

rp3HHIlbI

cpen C

pa3HhIMH

<pH3HtIeCKHMH

CBoHcTBaMH

npHxOJlHJIHCb

Y3JIbI.

3TO

Tpe6oB3HHe

snpens

CtIHT3eM

BhIIIOJIHeHHbIM.

Haxoaeu

(oaeas

B3LKH3H

,lI.eT3JIb!),

napa

HH,neKCOB

k, i B

CHMBOJIe

JII060H

nsy-

MepHO-CeTOtIHOH

<pyHKUHH

«hj

nnpens

03H3tIaeT

cnenyrouiee

COOTBeT-

CTBHe

TOtIKaM

ceTKH:

./I'(Zk'

POj),

eCJIH

i>O;

(19.18)

~(Zk-l'

J1'j),

eCJIU

1<0.

P33HocTHhle

oneparopu

o603Ha9HM

npeLKHHMH

CHMBOJIaMH

Al H

•

nOJIOJKHM

'k;-'k-l,j

1->0;

P-j

hk '

fk+l,j-'kj

-<0

fi'j hk ' 1 ;

(A.Cfl)kj

EkCflkj

l.:

(Es)k

Cfl

k.,

v:::-jo

(19.19)

(19.20)

135

rne k===l,

.2~

.... ,

i-±

1'-~.~,.

±j.J.

hk=Zk

-Zk-l;

J1".----..:.lfj.;'f.k~

E(Zk_lj;);

(ES)k:'-Es(Zk_l/2);

Zk-l/2=(Zk_l

+z~)/2.

<popMyJlax

(19.19)

HYJKHO

CtIHTaTh

ep,/=

'=epN+l,j=O -

3TO

cnenyer H3

rpaHHlIHbIX

YCJloBHH'(19.1)

HHAeKcHoH

cxexsr (19.18).

pa3HOCTHOM

aapaaare

neransaas

saIIHCh

aaroparxa

(19.11)

IIMeeT

CJIe,llYIOIIlHH

BH,lI.:

F~;)

= --(A

1ep(ll)h,-(A

(Il» k/ +qk/;

F~n)

I1po

F~;);

i=-Jo

~(n+I)'=

2't

[ (1:

F(n)

+F<n)].

. k

1+'tnEk 2 (I +

(EcJk't

)

kj t

a(n+1)

('~;I

'en

a(n+l)

A(n+l»).·

kj 1+

I~itcn/hk

k-lil/j,

t .

rkj

. '

m(n+l)

= men)

+a(ft+l)

Tkj

kj'

rJle

k=l,

...

, N;

j=±

1, ... ,

±io;

Qk/=q(Zk-I/2'

po,);

(Ea)k='f.k-('f.s)k;

tX~;)=

tX~+I.i=O

•.

MbI

pacCMOTpeJIH

MeTO,lI.

pacmenneaaa

,lI.JISI

pemeaaa

ypaBHeHHSI.

nepeaoca npocreaurero BH,lI.a .( 10.23).

npHMeH;eHB;e3TOro

~MeTOJla

,,lI.JIH

6.oJIee

~JIO)f{HbIX

<!>QPM'

ypaBHe,HHH

nepeaoca

MO*HO

HaHTH

paoorax

[34, 36,

37]~

<p0pMaJIbHbIM

,oIIH~aHHeM

npanuana

pacmenaeaas

MbI

3aBepIIIHJIR

3H8KOMCTBO

HeKoTopbIMH

npHOJIH)KeHHbIMH

MeTO,lI.aMH,

xoropue,

'pa3YM~eTcH,

,naJIeKO

acxepnsraaior scero

apceaana

M'eTO,l(OB

pewe-

HIUI·

O,nHOCKOpOCTHoro

ypaBHeHHH

nepeaoca.

113

onytueaasrx

MeTO.AOB'

B nepsyio osepens

CJIeAyeT

Ha3BaTh

BapHaUHOHHbIe

MeTO.AbI

,M~TO,ll.hI

MOHTe-KapJIo,

KOTOpbIMH

qHTaTeJIb

MO)KeT

n03HaKOMHThCH,

B patio-

Tax

[27,

38-40].

§ 20. MATEMATH'fECKOE ·O&OCHOBAHHE

METOAA

I!ACLqEnnEHMSI

Hexoropsre

<pOpMyJIHpOBKH

AOKa3aTeJIbCTBa

aacroauiero

napa-

rparpa

(uanpaxep,

JIeMMa

3 H

,noKa3aTeJIbCTBO)

xaxoa-rocrene-

HH,

,ny6JIHpy1OT

MaTepHaJI

.§ 15.

MhI

H.AeM

C03HaTeJIbHO

.TaKHe.

nOB-TopeHHH,

lITOOhI

lIHTaTeJIb

Mor

3HaKOMHTbCSI

JIIOObIM

3TH

naparparpos

He3aBHCHMO.

,nOK31KeM,

xro

HTepallHOHHbIH

rrpouecc

(19.7),

'KOTOPbIH

MOLKHO

nepenacari,

eme

H B

TaKOM

BH,lI.e:

(I+

't:

(I+

't:nA

)

f(n+~)

= (I -

'1:~A2)

(I -

~nAl)

ep(n)'+

2't

q, (20.1)

CXO,l(HTCH K

peurenaro

aanaxa

(10.23), (19.1)

xnacce

<pyHKUHH

!!f)

[cu. ' (19.8) ];,

eCJIH

COOJIIO.AeHPI

,nOCTaTOqHbIe

yCJIOBHSI (19.9) H

(19.-10).

OIIHcaHHaSi

B -§ 19

HTepaUHOHHaSi

npouezrypa

CBH3aHa

npo-

neCCOM

paspeurenna

ypaBHeHHH

(19.7)

:[HJIH

3KBHBaJIeBTHOro

eMY

i36

ypaBHeHHB

(20.1)

]OTHoCHTeJIbHo.

<pyJIKUHH

(J)(n+~)

•.

ep~KTHqecKll

·~TO·.

CBO,llHTCSI K oopameamo

·onepaTopoB

I +'t

I:+~nAl.

HaIIIa

g·a,naqa··,

ce:ntIac"

COCTOHT B

T~M)

qTo6hI

,ltOKa3aTb

cytnecrsoaanne oneparopos

(I +

"tnA2)

-I

H (I +TnAI)

-1

-BbIBBHTb

HeKOTOpbIeHX

CBOHCTBa.

YCTaHOBHM

C.Haq·aJI3 OCHOBHbIe CBO:nCTBa orreparopos Al H A2. ,UJlSl .

npoaenenas

,llOKa3aTeJIbCTB

npHJleTCSI

HMeTb

zteaoco

CKaJISlpH'bIM npo-

H3Be,lleHHeM

B npocrpaacrne L

(G») a

TaKLKe

c .

HopMaMH

3JIeMeHTa

H oneparopa B

3TOM"

npocrpancrae.

BOT

3TH

onpeneneaaa:

i H

(<p,

0/)=

dlJ.

dz<po/;

Wp!l2=(<p,

<p);

W= sup

(A<p,

A<p)/1I<pU

2•

0 .,

Oneparop A

6y}.(eM

Ha3bIBaTb

nOIlO~UTellbHblM

aa MHOJKeCTBe M,

eCJIH ,llJISI'

JII06oro

OTJIHQHOrO OT

HY~SI

.3J1eMeHTa

fPEM

HMeeltl

(Aq>,

cp)

>0.

OneparopA

oy}.(eM

Ha3bIBaTh

npJlO3ICUTeJlbHO

onpedenes-

HblM

eCJIH(Acp,

q»,;a::cll¢tl

(c>O)

,llJIH scex

q>eM.

Mal.

Oneparop Al Jl,BIlReTCJI;

noJt03ICUTellbHblM

«nacce

l/JyflICijUU

0 K a

T e JI s c r B O.

n~CTbq>E.!?l>.

Torna ,

1 H 1 H

' <p)= S.wfLS

dz= +J

pdlJ.

S:Z

<p2dz=

-1

-I

Sp.f2(H, fL)dlJ.- 5

1J.<p2(0,

fL)dl']=

-I

fL<p2

(H,

fL)dlJ.

1J.<p2

(0,

lJ.)dlJ.

]~O,

xro

Tpe6oB3JIOCb

,llOKaaaTb.

M"a 2. Oneparop A2

JlBIlJleTCJl

nOIlO{)ICUTellbHo

onpeoeAeHHblM

npOCTpa'H,CTBe

( G) . '

Jl0 K a 3 a r e JI s c r B O.

H 1 [ 1 ] ,

<p)

Jdz J

~<p

-:-

(a,

fL')dlJ.'

(z,

fL)

dp.=

•

-I

-1

. .

Idz

J<P

dlJ.

- J

<pd1J.)2

dz,

-1

0-1

Ha .

ocaose

HepaBeHCTBa

.I(oIIIH

OyHHKOBCKOro

MOJKHO

sanacars

CJIe-

JlYIOIllYIO

uenosxy

HepaBeH~TB:

[_i<p(Z,

fL)dlJ.

I<;;[_/I<P(Z,fL)ldfLI<;;

1 1 1 .

<;;

2dIJ.

<p2

(z,

IJ.)

dfL

2 S

<p2

(z,

fL)

dfL.

-1

-I

'187

OTCIO,na

HCrrOJIb30BaHHeM

OrpaHHqeHHH

(19.9)]

noaysaea

rpeoye-

MbIH

pe3YJIbTaT:

H I H I

(AICP,

cp)

Edz

- S

~sdz

-I

H I

= SEadz S

cpldp.~a.

(cp,

cp)

=a.1I

ill.

-I

e x M a 3. Oneparop

(I+'rA1)-1

(,;>0)

cuuiecreue» U oepanu-

«en

npOCTpaHCTBa

(G)

!!lJ

0 K a 3 a r e JI h C T B O.

IlYCTh

fEm.

Torna

B cuny onpeneneaas

saacca flJ (19.8)

3J1eMeHT

(I+

-cAl)

npHHa.u.JIe>KHTnpOCTpaHcTBy

L.(G).

IlYCTb

onepanas eJI==(I+-cA

)

nepesonar

m B aexoropoe

MHO)KeCTBO

MeL2

(G).

nOCKOJlhKY

II (I +

-cAl)

lI~ff2+

2'C

(Atf,

+'C

A1f

1l2~

fill,

(20.2)

TO na

OCHOBaHHH

reopexsr

cymecrsoaanaa

JIHHeHHOro

oriparaoro

oneparopa

[26]

aaKJIlOqaeM,qTO

na M B

!!lJ

cyurecrayer

JIHHeiiHbIH

o6paTHbIH

oneparop

(I+'tA1)-1,

npaneu

(I+'t'Al)-lll~

Oneparop

(I+'tA1)-1

naurex

qaCTHOM

cay-rae

JIerKO

BbITIHCbIBaeTCH

aHaJIHTH-

qeCKH,

orxyzra

cpaay

BH,nHO,

qTO

M=L

2(G),

(I+'t'Al)-l

nepe-

BO,nHT

2(G)

!!lJ.

3TOT

iKe

peayJIbTaT

CJIe.nyeT

Henocpe.ncTBeHHo

paooTbI

[27]~

AHaJIOfHqHO

,nOKaabIBaeTCSI

JIe

4. Oneparop

(I+'tA

)- 1

(,;>0)

cqtuecreqer U oepanu-

«e« B npocrpancree L

(G).

B CHJI'Y

JIeMM

3 H 4

ypasaeaae

(20.1)

MO'iKHO

nepenacars

B'H,C(e

f(n+l) ==

Knf(n)

gn'

(20.3)

==(I+

)

- I (I+

'CnA

)

-I

(I -

'CnA

)

(I -

":nAt);

2'C

(I+

'CnA1)-1

(I+1:

)

-I

Herpynao

BHl\eTb,

xro

gnE£J.

caMOM

nene,

yCJIOBHIO

qEL

(G).

Oneparop

(I+

'CnA

)

-I

JleMMe

nepeBOAHT

(G): (I+

+'CnA2)-tq==~EL2(G).

,naJIee,

3J1eMeHT

oneparopox

(I+'CA)-I

coraacno

neMMe

3 nepesonarca B xnacc £J: (I+

'CnA

)

-I

eJI

fEme

Cnenosarensao,

2'C

E m.

TOqHO

TaKHMH

JKe

paccyscneaaaua

yCTaHaBJIHBaeM,

xro

oneparop

nepeBOAHT

JII060ii

3J1eMeHT

KJIaeCa

CHOBa

xnacc

·m:

Kn(~)C~.

KpOMe

roro,

OlleBHJJ.HO,

liTO

-JlHHeHHbIH

oneparop.

Teneps

paaencraa

(20.3) aenocpencraeaao cnenyer,

qTO

f(n+l)EaJ

npa

f(n)Em

qEL

(G).

Ilepefinea,

HaKOHeu,

3aKJIIOqHTeJlbHoMY

arany

HaIIlUX

paccysc-

.IleHHH

- K

,lloKa3aTeJIbCTBY

CXO,llHMOCTH nrepauaoaaoro npouecca

138

(20.3).

nYCTb

q>E.!'D - peurenae

ypaBHeHHH

(10.23)

(cymecrsoaaaae

xoroporo

MbI

6y~eM'

npeztnonarara) :

Aq>=q,

A-A

1+A

•

Baenex

paCCMOTpeHHe

norpeIIIHOCTb

B(n):

B(n)

<p_q>(n).

(20.4)

113

(20.4) H (20.1)

JIerKO

nOJIyqHTh

ypasneaae

~JIH

norpenmocra:

(I+'l:

) (I+

1:~AI)

s(n+t)

(I -

)

(I -

'CnA

scn)

s(n+

===

Kns(n).

ECJIH

6bI

ynanocs

oueaari,

HOpMy

oneparopa

H eCJIH

6bI

"PH

aTOM

OKa3aJIOCb,

xrc

IIKnll<H<1

(n=O,

1, 2,

...

TO OTCIO,ll3

CJIe-

JI.OBaJI

6bI

<paKT

CXO,L{HMOCTH

HTepaUHOHHoro

npouecca (20.3).

O,nHaKO

HaHTH

TaKylO

ouenxy

.nJIHIIKnll

necaxa

HenpOCTO.

3TO

06CTOHTeJIh-

CTBO

BbIHY*.lI.aeT

Hac

npennpausrs

HeKHH

06XO~HbIH

MaHeBp,

pe-

3YJIhTaTe

xoroporo .neJIO

CBO~HTCSI

cpaBHHTeJIbHO

JIerKO

peanaayexok

oueuxe

HOPMbI

nexoroporo ztpyroro

oneparopa

Ha-taex

CO cneny-

lO111eii

uenosxa

T02K)(eCTB:

11(1

'CnA

}

s(n+l)!1

11(1

+'C

}

Kns(n)ll

11(1

'CnAt)-~-~X

X(I

'l:nA

} (I -

'CnA

sen)

1~·(1

}

-I

(I -

'CnA

)

(I - 'CnAI)X

X (I+

'CnA1}-1

(I+

'CnA

)

sen)

LPn

(I+

'CnA

s(n)ll,

(20.5)

r~e

(I+'C

) --

I(I - 'C

) (I -

renA

(I+

'CnA

)

-I

,UaJIbHeHmee

npeocpaaosauae

TO)K)l.eCTBa

(20.5) -

6y)l.eT

onaparscs

cnenyrouree

sauenaaae:

eCJIH 0

'C'

'C"

epEelJ,

11(1

'C'

AI}

ep!1

11(1

'C"

At}

cplf.

(20.6)

HepaBeHCTBO

(20.6) HBJlSleTCH

npSlMbIM

CJIe.nCTBHeM

<pOpMyJI

(20.2) H

.1JeMMhI 1

zraaaoro

naparparpa,

T02K.neCTBO

(20.5) c

yserox

MOHOTOH-

aoro

xapaxrepa

H3MeHeHHSI

napaxerpa

't'n

(CM.

§ 19) H

HepaBeHCTBa

(20.6)

nopoxcnaer

uenoxxy

HepaBeHCTB

n(I+

'CnA

s(n+I)JI

liP

nil-

(1(1

'CnA

s(n>

liP

nil·

H(I

+ 'tn-tAt)

s(n)lI·

HH1Ke

noxasaao,

xro

IIPnll~O<

n===O,

1, 2, ... , (20.7)

oTKy,na

CJle,lI.yeT

11(1

'CnA

s(n+l)

~611(1+'Cn_tAJ)

s(n)"

11(I+

tAt) s(n-I)

..•

11(.1

'toA

)

s(I)/I.

(20.8)

113

HepaBeHCTB

(20.2) H (20.8)

nOJIyqaeM

(n+J)

11(1

s(n+t)

JI(I

'CoAl)

s{I)II.

139