Смелов В.В. Лекции по теории переноса нейтронов

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sropoe cna-

raejsoe

TO}KAeCTBa (14.15), HaX0lJ.HM

[Y(kd

l /2) -

Y(k)

)} +(1)(k) (x

)

ok!)

, (14.18)

Xk+l/2

o~l)

2~k

S(x -

Xk)1

(1)" (II)

[~(x)]

dx,

Xk-I/2

J1s

(14.17)

caenyer

aurerpapyexocrs

epymmHH

Xk)!

cD"

(k)

(x)],

BKJIIoqeHHe

<D(k)

C(2)

[a, b]

03HaQaeT

:JcD"

(k)

(x)

~B~l)

= const aa

acex

npoaezsyrxe

a~~b.

113

rr. 1 ssrrexae

4aJIee,

qTO

tIHCJIO pas-

,JIHqHbIX

<pyHKnHH

«D(k)

(x)

M01KeT.

npeaocxonarsm

+1. H

OHHHHsa-

pH8HTHbI

OTHOCHTeJIhHO

acero

ceMeHcTBa ceTOK. Ho

rorna

MbI snpaae

:84 '

CIIHTaTb,

qTOqHCJIO

pa~JlH1JHblX:

KOHCTailTB

Ha60pe

{Bk')}Z=t

npe-

BOCXOJnIT

m+l

OHa,

OqeBHAHO,

·He . saBHCSlT

'OT

ceTKH. 0609H8tU1M

. . .

24m:x8~11.

Teneps

JIerKO.

HaIIHcaTb

o~eHKY

J(JISI

SeJIHlJHHbI

a~):

~(1)

<,B

k·

1 ,

rAe

===maxh

,llJISI

npeoopaaoaaaua

BhIpa)KeHHSI

axsanparastx

cxotixax

B .TOlk-

necrse

(14.18)

BOCnOJIbayeMcH

B}{JIlOqeHHeM

(14.12), xoropoe

n03BO~

JISleT

asmacan,

CJIe,llYIOIIlHe

npencraaneaaa

,llJIH

tq)(k)

(Xk)

<Pek)

(ali)

<popMyJIe

TeHJIOpa:

3 '

ep(k.)(X.k)

(±

hk)V

(aV,~»)

J,:(!!l!-)4

(d4ep~»)

(a)

~v.

\.dX

x1l,-1/2

4.2

ds:

<PCk)

,,=0

. k

rAe

ei,

[Xk

- h

Xk].

3THX

AByX

paBeHCTBJierKO

HaxOJ.\HM··

)_CfCk)

(Xk)-'(k)

(ak)

(!!!!-)I

<k~

k-I/2

- hk 3! . 2 T

(k)

-1-1/2

-!!l!- (hk.

[(d4ep(~))

k»)

]

96 2· dx

~k'

naaee,

1 Y

A 'Ck) (Xk) -

CfCk)

(ak) qr ( )h .

~(·2)

(il

- k hit -

·(k)

i-ok

rAe-

(14.19)

(14.20)

(2)

_ 1

)2 ( )

[(d

»)

(k)) ]

~96

D(1;)

Xk-I/2

ax·

tr ·

113 (14.7) H·- (14.12) cnenyer

OrpaHHqeHHOCTb

D(k)(X) H

(k)/dx

aaMKHyToM

npOMeJKyTKe

[a, b], xro

naer

B03MO}KHOCTb

asmacars

CJIe,l(YlOmYIO

...

O[(e~KY

,lI.JISI

3~2):

(2)1

B(2)h

kkk'

B(2)

=_1_.

- maxD

)

(xl

max I

d~'(k)

(x)

192~~

~~b

dx4:

•

CHoBa4>HK~HPyeM,

QTO IIHCJIO pa3JIHQHbJX

KOHCTaHT

)

He npeBOCXO-

.nHT m +1 H OHB He saBHCBT

OTceTKH.

nYCTb

B. =

maxB~~).

-Ang

~k2)

MO)f{HO

a3IIHcaTb

Tenepb

yaaaepcaasayio

oueasy

)·1

~Blh!

B.h!.

~nOrBqHO

HaHAeM

1 Y ( )_

'(k)

(bk) -

'(k)

(Xk) qr ( )h

~.(3)

Ck)

- k

hk -

(k)

k - fJ

rAe

(14.21)

nO,llCTaBJIHH

(14.19) H (14.21) B

TOiK,lleCTBO

(14.18) H

3aMeHHH

nOCJIe,llHeM

BeJIHlIHHY

<DCk)

(Xk)

paasepayrsra

BbIpaJKeHHeM

(14.16), nonyuaeu

npencraaneaae

(14.14) c

a~i),

i=l

_flJ.e

a~I),

a~2),

a~8)

6bIJIH

aseneasr paaee, a

8~4)===

- h

PI'

ll'

k (X

)].

I1a

epOpMyJlbl (14.20) H BKJIlOlJeHHH '(14.6), (14.12)

BbITeKaeT,

lITO

qr~k)

e(l)

la, b], a

noroxy

na OCHOBe

H3BeCTHOH

q,OpMyJIhI

Jlarpaaxca

3~)

M<;)JKHO

nepenacars B

BHAe

a(4)===

_ h

(d'lJ!

(k)

) e

E h

k k dx

npasex

npOH3BO,llHhIe

'I"(k)(X)

(k=l,

...

, N)

orpaaaseau

aa [a, b}

H ,llJIH

OneHKH

Mo.nyJIeH

norpeoyercs

OOJIee

KOHCTaHTa~

KOTOpbIX,

B CBOIO osepens,

MOiKHO

BbIopaTb

yHHBepCaJIbHylO

KOH--

cranry

TaKylO,

lITO

la~"

'~B4h·2k~B4h2.

I1TaK,

paBeHCTBe

(14.14)

asaa

qaCTb

OneHHBaeTCH

cJIe,llYIOIUHM

o6pa30M:

k I

«se,

~Bh2,

B==

i==I

h==maxh

(14.22)

(14.23)

rne B

(nonxepxaex

erne

paa)

3aBHCHT

ceTKH.

Ilepenanrex

Tenepb

npHOJIHiKeHHhIe

paseacraa

(13.15) B

epo:pMe

crporax

paseacra

HeH3BecT~bIMH

npaasnaa

lJaCTSlMH

C%k+1/2

~k+1/2

[CP(k) (Xk)

-t-

CP(k)

)] -

[CP(k+t>

(Xk+t)+

CPCk+J)

(ak+l)] ==

C%k+l/2;

I'Ck+t)

(Xk+l)

fCk+t)

(ak+t)]-

Ilk

[CP(k) (b

) -

- CP(k)(Xk)]

~a+l/2

(k=I,2,

...

,N-I).

JVm

oneaxa BeJIHtIHH

t%k+l/2

~k+I/Z'

HCnOJIbSyeM

CHOB8

4>OpMyny..

TeA.

Aopa

H B

pesYRhTaTe

HaH~eM:

akf-I/2=

+[h

'P"

(k)

(Xk+I/2)-

,k+t'P"

(k+t)

(Xk+I/2)]

rne

'I'(k)(X)

naercs

q>opMyJIoii

(14.20), a

(i)

=::;;;::

, r.ii) I~ R • 1 2

~ao;

rk+l/2

~ro'

, ·

(14.24)

(14.25)

(14.26)

(14.27)

nOCJIeAHHX

oueaxax au H

IPO

3aBHCSlT

ceTKH.

HaKOHen,

npH6JIH2KeHHbIe

paseacraa

(13.20) nepenaurex B

BH,ne

'h)(a

)

'h)(X

)

_ M ,ct}(x1)-fP{t)(a

)

- g = y ·

o 2 • hI

o.t

L '<N)(xN) +

'CN)(b

)

fCN)(b

)

- Cf(N)(XN) _

2 h

._g-y

OneHHM

HeH3BeCTHbIe

BeJIHtIHHbI

H y.

H3BecTHOMY

yJKe

penerrry

oe3

TpYAa

HaiiAeM

(14.28)

rne

3aBHCHT

ceTKH.

3aKJIIOtIeHHe

nacrosmero

pasnena

BhInHweM

ypaBHeHHSI

,l{JlSI

norpeWHOCTH

'PU~)

(Xk)

- 'Pk; sa

. 'P(k) (al) - 'P

ak;

}

sb - ffCk) (b

) -

~bk'

-1,

...

Bsrxaras

H3 (14.14)

paseacrso

(13.16), noaynaea

(14.29)

AHaJIOrHqHO

H3 (14.23) H (13.17)

HaM,neM

(Sk

sk+t)

(sk+t

sak+)

(sk

- sb

ak+I/2;

(sk+l -

)-llk(sb

-Sk)=~k+1/2'

k+t

1, 2,

...

, N - 1.

(14.31)

HaKOHeIl,

,HS'

(14.27) H (13.21)ycT3H3BJlHBae·M

(14.32)

Floayseaaue

ypaBHeHHH

,l(JIH

norpeumocrn

arparor

Ba2KHYIO

POJIb

npn

aH3JIH3e

CKOpOCTH

CXO,llHMOCTH-

npH6JIH:>KeHH,Oro

pemenus K

TOllHOMY

Mepe

YMeHbllIeHHH

napaxerpa

ceTKH

h=max

hi.

3TOT

sonpoc

nonpooso

paccxorpen

cJIe.nYIOIIleM

paanene,

3. OcHoBHble

CBoHcTBa

pasaocrnoa

cxeMbl (13.24).

Oueaxa

CKO-

pOCTH

CXO,ll.HMOCTH.

YCTaHoBIIM

HeCKOJIbKO

Ba2KHbIX

CBOHCTB

CHCTeMbl

~a3HOCTHbIX

(cKaJIHpJIbIx)ypaBHeHHH

(13.24), xoropsre

nOTpe6ylOTcSI

.pH

HCCJIe,llOBaHHH-

CXO,lI.HMOCTH

.--npH6JIM)KeHHOrO

peurenns K

TOqHOMy.

Crpopsryaapyea

3TH

CBOHCTBa

B rpopue

JIeMM,

xoropsre,

KCTaTH,

npen-

CTaBJIHIOT

caMOCT05lTeJIbHbI:H

narepec, ,·TaK

KaK

MoryT

6bITb

HCnOJIb-

30BaHbI

npn

aHaJIH3e

MHorHX

npyrax

pa3HOCTHbIX

cxex.

Jl e M

3. ECAU 8 CUCTeMe (13.24) XOTJl

6bl

oono

<Ok OTAUf/,nO

nyAJl,

.TO

3Ta

CUCTeMa

oonosnauno

paspeiuusca.

HarrOMHHM,

lITO

Bee

CJ)k

.HeOTpHuaTeJIbHbI

[CM.

(13.25)], a

Tpe6o-

BaHRe

«He

Bee

CJ)k pasnsr

HyJIIO»

ecrr,

npHMOeCJIe,nCTBHe

yCJIOBHH

paapeurauocra

'(14.1).

c!IeMMa

6Y,lleT

,nOKa3aHa,

eCJIH

MbI

yCTaHOBHM.,

nro

OAHOP'OllH3S1

CHCTeMa

k=l,

...

, N,

(14.33)

HMeeT

TOJ!bKO

JiyJIeBOe

peurenae.

npe,l(IIOJIO:>KHM

nporanuoe:

O,ll.HOP0,llH3H

,CHCTeMa

HMeeT

HeTpHBH-

aJIbHOe

peineaae.

yMa,,1HH

o6mHoCTH,

"MO:>KHO

ClIHT3Tb,

xro

cpena

QHCeJI <PI, Q)2,

••.•

,..(J)N

ecrs

nOJIO)KHTeJIbHbIe.

CJIe,ll.OBa.TeJIbHO,

HaHOOJIb-

uree

QHCeJI

q:>k

(nycrs

OHO

oraeuaer

annexcy s)

T3KJKe

nOJIOLKH-

reasao,

T. e.

(14.34)

,lI.3JIbHeHIlleM

oy,ll.eM

pa3JIH1I3Tb

,ZlBaCJIYQa5I:

f:Pl=q>2=·

=(J>N=:f.1; (14.35)

see

CPk

(k=l,

...

, N) paansr

MeJK,llY

c060it

nepeox

cnyxae J.1>O,

'KaK

3TO

BbITeKaeT

(14.34). ,UaJlee,

ue-

IIOlJKa

paseacrs

(14.35)

oripamaer

cacrexy

(14.33) B cnenytonme

COOTHomeHHH:

COkfJ, 0,

k=l,

...

, N,

xoropsre

B03MO:>KHbI

JIHIIlb

npa

COl

=0l2=

•..

=(J)N=O,

'ITO

npOTHBO-

peuar

yCJIOBHIO

JIeMMbI.

Tlepexona K

aH8JIH3Y

sroporo

CJIyqaH,

.0003HaQHMp,--

max<p~.

113

(1"4.34)

cnenyer

IJ.t>O.

Flyers ())l =:J1.

MO}KeT

CJIyqHTbCH,

qTOH

<P2=J.t

II T.

,l{.

Ho B nOCJIe,llOBaTe.'IbHOCTH <PI,

(J>2,

q>3,

•••

paao

HJlM

n03,llHO

nepBbIH pa3

BCTpeTHTCSI

TaKoe

<1'8,

tiro

CPS=IJI.,

(fs+l

<J1,

l~s<N.

(14.36)

ECJIU

6bI

3TO

6bIJIO

TaK,

T. e.

s=·N,

TO MbI

OKaa.aJI~Cb

ObI

yCJIO-

BHSIX yxce pasotipaanoro

nepsoro

CJIYIIaSi.

cacrexe

(14.33)

BbIJleJIHM

ypasneaae,

oraenatomee -

k=s,

nepenaurex

ero

yserox

(14c36) B tpopxe

(p.-~s+t)·+As(p.-~s-t)+m8tt=O.

(J4.37)

Taxoe

paseacrao

HeB03MO}KHO,

TaK

KaK

nepsoe

cnaraexoe

CHJIy

(14.36). H (13.25)

crporo

nOJIO.>KHTe.,TIbHO, a

OCTaJIbHbIe

aeorpana-

TeJIbHbI.K

Ta'KoMY

1Ke nporaeopesmo npaxonaa,

,ZlO'nYCTHB·

q>N-~

Flyers,

HaKOHeu,

CPl~f.t,

q>N=#if.t,

T. e.

IM8KCHMaJIbHOe

aaaseaae

JlOCTHraeTCH

npa

KaKOM-TO

k=s,

paaaox

1 H N:

nOBTOpBSI

npezns-

-nYIUHe

paccyzcneuna,

CHOBaMOJKHO

,llOOHTbCSI

-cHTyaUHH-

(14.36),

TaK

KaK

CPN=F'f.1.

3aKJIIOIIHTeJIbHbIH

aHaJIH3

paaencraa

(14.37) -

npHBOjlHT

Hac

onsrrs K nporaaopesmo.

JIeMMa

ztoxaaana.

JI e a M a 4.

Ecnu

YCIlOBUflX

lleMMbl

CUCTeMe

(13.24)

k ='6ks

(~k8

- CUMBOJt

Kponesepai

petuenue

CUCTeMbl

CTp020

nOIlO:JICUTellbHO, .

npusesc

nocneooearensnocm

<PI,

<1'2,

•••

(J>s

MOHOTOH,-

11,0

pacrer. a

nOCAeOfJ8aTeJtbHOCTb

qJs, q>s+l,

.•

,:.q>N

M-oHOTOHH,O

y6bl8a.;,

er, TaK,"

'lTD

'q>s~k

(Il:::::l, 2,

...

N).

n0 K a 3 a r e JI b C·T B o. KaK yzce

6bIJIO

noxaaaao

asnne,

onaopon-

Haft

CHCTeMa

(14.33)

HMeeT

TOJIbKO

HyJIeBOe

peureaae.

OT6poCHM

3TOH

CHCTeMe

nocnenaee ypanaeane,

OTBeqaIOIUee

aouepy

k=N.

TaKaSi

«ype3aHHaSl»

O,llHOp0,llHaH

CHCTeMa

y~e

,llOnycKaeT

HeTpHBH-

3JIhHbIe

pemeaas.

Ilycrs

{Uk}N

k=l

- O,llHO H3

TaKHX

penreaaa.

Ero

MOLKHO

HaHTH

CJIeJ(yIOUlHM

otipasoa, 3a,na,llHM' u,

npOH3BOJIbHO,

TaK,

tITo6bI

Ul>O. TaK

KaK

A1=O,

nepaoro

ypaBHeHHSI

CHCTeMbi

(14.33)

HaXO,llHM

= u

+CDtU

u..

J{aJIee

3JIeMeHTapHO

yCTaHaBJIHBaeM

(CD

maUl)!

npe,AfiOJIOiKHM,

qTO

k-I

1 -

=Uk-l

(DJ

;;;'Uk-l"

(14.38)

j=l

Conepzcameeca B (14.38)

.paBeHcTBo

nO,llCTaBHM

B k-e

ypaaaeaae

CHCTeMhI

(14.33) H

HaH,lleM