Смелов В.В. Лекции по теории переноса нейтронов

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H JI.eJIeHHH:

vEs===vosN;

vEf===vofN.

(7.4)

§ 8. OAHOCKOPOCTHOE AHClHDY3HOHHOE YPABHEHHE.

rPAHM'IHYE YCnOBHSI

HeHTpOHbI

,llH<p,<pYHJI.HPYIOT B

aetuecrae

pe3YJIbTaTe

paCCeHHHH

HJlpaMH

aTOMOB.

THnHlIHaH

-rpaexropna

HeHTpOHa

HMeeT

saraarooo-

pa3HbiH

BH,lJ.

H COCTOHT 113

npHMOJIHHeHHbIX

OTpe3KOB

pa3JIHlIHOH

JlJIH-

HhI,

COe,ll.HHHIOI.U.HX

MeCTa

OT,lI.eJIbHhIX CTOJIKHOBeHHH. KaLK,lI.aSl H3 HH,lI.H-

BH,llyaJIbHbIX

TpaeKTOpHH

06phIBaeTCH

TaM,

rne

HeHTpOH

JIH60

norno-

maerca

cpenoa,

JIH60

BbIJIeTaeT

npenenu

06'beMa,

aaasrroro

paccxa-rpaaaexsm

seutecraox.

,llH<p<py3HoHHoe

ypasaenae,

BbIBOJl

xoroporo

paccxarpasaerca

HHLKe,

onacusaer

nosenenae

60JIbIIIOrO

QHCJIa

HeHTpOHOB,

xorna

HHJlH-

annyansasre

oco6eHH9cTH

orneasasrx

rpaexropaa

«ycpe,lI.HHIOTcH».

PaCCMOTpHM

csasana

nexyto

BcnOMCraTeJIbHYlO

sanany.

nYCTb

Be-

utecrso,

B aKOTOpOM

H3YQaeTCH

,llH'<P<PY3HH

HeHTpOHOB,

aanOJIHHeT

see

npocrpaacrso.

Ilycrs,

,lI.aJIee, B

06'bel\1e

.9TOrO

semecraa

pacnpenene-

RbI

HeHTpOHbI

HeKOTopOH,

sooome

rosops,

HepaBHOMepHOH

nJIOT-

HOCTblO

n(r,

[n(r,

t1)==n(x,

y, z,

t)].

Haxoaeu,

ClIHTaeM,

liTO

cpe-

OTCyTCTByIOT

KaKHe

,6hI

TO HH

~6bIJIO

HCl'OllHHKH

HeHTpOHOB

"',

B:QI,lI.eJIHM

MaJIylO

nnoutanxy

B nJIOCKOCTH XY, npoxonsuryro

lIe-

pe3

uasaao

',KOOp,lI.HHaT

(pHC.

8).'

Iloncsuraex,

CKOJIbKO

HeHTpoHoB

rrepeaner

xepes

nJIOIUa,lI.Ky

sepxnero

nonynpocrpaucrna

HHLK-

nee

nexoropsra

'MaJIbIH

npoxexcyrox

BpeMeHH

OT t ,lI.O

t+~t.

,UJISI

KpaTKocTH

H3JIOLKeHHH

e,llHHHtIHbIH

aKT

nepexona

HeHTpoHa

nepea

nJIO-

nianxy I1S

sepxaero

nonynpocrpaucraa

B HHJJ<Hee B

HYiKHbIH

npo-

MeJKyToK

BpeMeHH

(t,

t+~t)

Ha30BeM

cross-co6bITHeM'

HJIH,

xopoxe,

Cvcotiurues:

OTMeTHM,

xro

Ka*,lI.OMY

C-C06bITHIO

06SI3aTeJlbHO

npen-

IIIecTByIOT

pacceaaaiourae

CTOJIKHOBeHHH B

KaKMX-TO

TOtIKaX

npOCT-

paHCTBa

(eM.

pHC.

8).

ECJIH6bI

3TO

6bIJIO

TaK,

3TO

03Ha-lIaJIO

6bI,

liTO

«npOCKOllHBIIIHH

OKHO

lL\S»

HeHTpOH

npOIIIeJI

npensaparens-

6eCKOHellHO

,nJIHHHbIH

npHMOJIHHeHHbIH

nyrs

6e3

enanoro

CTOJIKHO-

BeHHH

H~paMH

BemeCTBa.C

noxomsro

i<POPIMYJIbI (5.2),

rne

non

HyJKHO

nOHHMaTb

cyMMapHoe

ceseane

3aXBaTa

pacceanna,

JIerKO

y6eJJ.HTbcH,

tITO

BepOHTHOCTb

ocymeCTBJIeHHH

TaKoro

HBJIeHHH

panna

HyJIIO.

nenonxe

pacceasaiomax

CTOJIKHOBeHHH,

npe,llmeCTBYIOIUHX

C-CO-

6bITHIO,

.nOCJIeAHee

CTOJIKHOBeHHe

TOllKe

(CM.

,pHC. 8)

MO}KHO

aa-

3BaTb

«YJlalIHbIM»

TOM

CMbICJIe,

sro

HMeHHO

OBO

,npHBeJIO

K ocyme-

CTBJIeHHIO

C-C06bITHH.

:It Ilpennoaaraerca, 'tITO 'HCTO'tlHHK'H,

06ycnOBHBWHe

HaJIUlIUe

'HeHTpOHOB B cpeze,

y>Ke

nepeCT3JIH

tPYHK;UHOHHpOB3Tb.

113

BCerO

CKa3aHHOrO

cnenyer,

xro

KaLK,nOe

C-C06bITHe

HMeeT

BepXHeM

noaynpocrpaacrae CBOIO

e,nHHCTBeHHylO

TOtIKy

rne

HeH-

TpOH

HCnbITaJI

«ynanaoe»

cOYJI.apeHHe.

3TO

oanasaer,

lITO

nOJ(CqeT

otimero

·KOJIHqeCTBa

C-C06bITHH

MO>KHO

3aMeHHTb

nO,nClIeTOM

'lIHCJIa

«y,nallHbIX»

CTOJIKHOBeHHH.

TIOCJIeJ(HSISI

aanaxa

OKa3bIBaeTCSI

CpaBHH-

TeJIbHO

npOCTOH

COCTOHT "B

TOM,

nro

BCeX

pacceaaarounrx

CTOJIK-

HOBeHHH,

npOH30IIIeJ(IIIHX

BepXHeM

noayrrpocrpaacrae,

HyLKHO

OTepHJIbTpOBaTb

«y.rr.aqHble»

CTOJIKHOBeHHH

nOACtIHTaTb

tIHCJIO.

PHC.

8. Cxexa

DpOJIeT~

Heli-

TpOHa

qepea

3JIeMeHTapHyro

naotnazncy

lAS

DOCJIe HeCKOol1b-

KHX

pacceaaaiomax

CTOJIKHO-

BeHHH

L1sr---r---_~~

PHC.

PacnOJIOJKeHHe

3JIe-

xearapaoro

06'beMa

:AV -H

3JIeMeHTapHoH

nnoutaznca AS

,UJISI

.naJIbHeHIIIHx

paccyscneaaa

BBe.lleM

cnenytotuae

orpaaaseaaa:

1) B

paccaarpaaaeaoa

cpene

ceneaae

aaxaara

npeaeripescaxo

MaJIO

cpaBHeHHIO

ceueaaex

pacceSIHHSI

(~a~L'$);

pacceanne

H30TponHo

natioparopaca

CHCTeM€

KoopAHHaT

XYZ,

HeHTpoHMoLKeT

OTJIeTeTb

pacceanaioutero

uearpa

(snpa) B

JII060M

HanpaBJIeHHH

paauoii

BepoHTHoCTbIO;

3) cpenaee

BpeMH

MeLK,ny

.rr.ByMH

nOCJIe.rr.OBaTeJIbHbIMH

paccenaaro-

mHMH

cOy)J.apeHHSIMH

('t=ls/v)

HHqTOLKHO

MaJIO

cpaBHeHHIO

ape-

MeHeM,

TetIeHHe

KOToporo

3aMeTHO

H3MeHHeTCH

pacnpe.rr.eJIeHHe

HeHTpoHoB

B semecrne.

HHLKe

6y,neT

aseneno

erne

O,lI.HO

orpanaseane,

eCTeCTBeHHbIM

o6pa-

30M

aurexarouree

'M3

annpOKCHMaUHH

nJIOTHOCTH

nOTOKa

HeHTpOHOIB'

KOHetIHbIM

OTpe3KOM

pana

TeI1JIOpa.

nYCTb

reneps

V-

npOH3BOJIbHbIH

MaJIbIH

06'beM

BepXHeM

nOJIy-

npOCTpaHCTBe

(pHC.

9).

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coynapeuae

06'beMe

,~V

MOLKeT,

KaK

JIerKO

BHJ(eTb,

npoH30HTH

TOJIbKO

BnpOMe>KyToK

BpeMeHH

OT t'

no t' +

(t'=t-r

/v),

nOCKOJIbKY

.llJIH

npoxoscnenaa

paccToHHHSI

06'beMa

V ,lI.O

TIJIOIUa,lLKH

;~S

(CM.

pHC.

nearpony

-rperiyercs

BpeMH,

pasuoe

[o,

Oouiee tIHCJIO

aKTOB

pacceanna

06'beMe

HHTepBaJIe

BpeMeHH

(t';

+~t)

'COCTaBHT

(L\V

;L\t

MaJIble!):

Esrp(r,

t-r/v)AVAt.

3TOrO

KOJIHtIeCTBa,

BBH,llY

,llOnymeHIDI2),

HaI!paBJIeHMM

nnouian-

paCCeeTCSI

L1S

cos

V·

4 2

EsC?

(r, t - r/v) a

D.t

'itT

(8.1)

(8.2)

HeHTpOHOB.

TOJIbKO

3TH

HeHTpOHbI

MO)K·HO

paCCMaTpHBaTb

«xaanana

TaMH

B ynaxasre». Ha

nysxa

(8.1),

anyuiero

Han'paBJIeHHH

TIJIOma,ll-

~S,HeHTpoHbl

MoryT

BbI6bIBaTb

pesyasrare

H pa!CCeSI'HHH, H

no-

rJIOmeHHH.

Onapascs

pesynsrar

(4.4),

MO)KHO

aanacare

lIMCJI!)

HeHTpoHoB

'nYlIKa(8.1),

,llOCTHrWHX

npOMe1KYToK

BpeMeHH

OT t

nJIOIUa,llKII

CJIe,ll.oBaTeJIbHO~

npomenumx

sepes

aee:

I1S

cos 9 / { V

4.'ltT

Es<p(r,

t-r

v)exp

-(Es+Ea)r}L\

At.

3Ta

BeJIHlIHHa

ecrs

lJHCJIO

C-C06bITHH,

06YCJI~BJIeHHbIX

. '3JIeMeHTO"1

nOJIHOe

lJHCJIO

C-C06bITHH

nOJIYlJHTCSI

<;yMMHpoBaHHeM

aupaace-

HMH

(8.2)

scex

3JIeMeHTaM

V aepxnero

nonynpocrpaacraa:

110BepxueMy

noaynpocrpascray

cos 9 / V

-2-

exp

[-(E

)

(r, t - r

v)d

(8.3)

BeJIMIIHHa

(8:3)

- a.n..aHTHBHa5.f <PYHKII)ISI

nepeaeaaoa

06JIaCTH

~(8S,

8t).

ECJIH

0603Hal.lHTb

npOH3BO,llHYIO'3TOH'

tllYHKIIHH

OOJIaCTH G

(pasnyio

lJHCJIy

HeHTpoH<JB,

nepecexaiouiax

1 ce« e,I(U-

HHlIHYIO

nJIOII.I.a.ll.KY * B

HanpaBJIeHHH

-z)

HBBeCTH

paccxorpeaae

crpepaxecxae

KOOp,llHHaTbI

(dV

sin

HdrdBd,¢)

, TO

nOJIYlJHM

1C/2

5dr J

Sd6<p(r,d-r/v)exp[-:-:-(Es+Ea)r]cos6sin8. (8.4)

o 0 0

BeJIH1JHHa

Ha3hIBaeTCH

nJlOTflOCTblO-

rosa

nearponoe B nanpaene-

HUU

-Z~·

,llJI5I

,lla,,'IbHeHllIHx

onepanaa

HHTerpaJIOM

(8.4) BOCnOJIb3yeM.cSI

TeM

cyutecraeaustx

06CTOHTeJIbCTBoM,'QTO

(8.4)

non

3HaKOM

HHTe-,t

rpaaa

HaXO!lHTCSI

3KcnoHeH·U·HaJIbHbIH

MH01KHTeJIb

exp

(~8+

~a)

r].

3TOT

MHO:>KHTeJIb

6bICTpO

y6bIBaeT

pOCTOM

(~s+

~a)'r

,(T.

Mepe

y.naJIeHHH

nnomanxa

~S),

noroxy

OCHOBHOH

BKJIa,ll

neperexa-

HHe

HeHTpoHoB

nepes

nnotuanxy

Jl.alOT

pacceasaroutae uearpsr,

pacnonoaceaaae

HenOCpe,llCTBeHHoH

·6JIH30CTH

IlJIOmaJlKH:

L\S.

~'v\O}l{HO

CQHTaTb,

l.ITO

paccToSIHMHX,

npessuuaioumx

TpH

,llJIMHbl CBO-

60,lI.HOrO

npo6era

[r>31,

1=1/

(~s+~a)],

epyHKUH5I

exp

[-(~s+~a)rl

cnanaer

npaKTH1JeCKH

HYJIH

1(6oJIee

GeM

B 20

paal).

ECJIH

nJIOTHOCTb

nOTOKa

HeHTpOHOB

,pa3JIOLKHTb

pan

TeHJIOpa

OKOJIO

Ha1JaJIa

KOOP-

nnaar,

TO B

3TOM

pa3JIOLKeHHH

,llOCT8TOlJHO

coxpaaari,

JIHlllb

CTOJIbK()

QJIeHOB,

CKO&TIbKO

HyLKHO

JI.,,lISIXOpOlIIe~annpoKCHMaUHH

OKpeCT-

* 3.lleCb H

IJIH)Ke

B CJIOBa THna

«qepe3

e)I.HHHtlHYIO

lJIJIOma,l(Ky

sa 1

ceK»'BKJIa.rlH-

aaerca

CMhlCJI:

1J nepecsere aa

enanany

IIlJIOma,zUI H na e.ll.HHHllY

BpeMeHH.

(8.8)

(8.9)

HOCTH

HyJIH.

Teneps

MoiKHO

C'<pOpMyJIHpOBaTb

HeKOe

JlOinOJIHHTeJIhHOe

orpaanseaae

'JlH~~yaHoHHoH

TeOpHH.

HMeHHO,

norpeoyex,

tITo6bI

paccToHHHHX

JI.O

Tpex

',llJ1HH

CBo6oJlH'orO

npo6era

nJIOTHOCTb

nOTOKa

HeHTpoHoB

XOpOII.IO

ormcsmaaact,

tIJIeHaMH

HyJIeBOrO,

nepsoro

no-

pH,llKOB

Teit7IopOBCKOM

paaciIOJKeHHH.-

H3,

T3KHX

paCCTOHHHHX,

CHJIy

.llOIIymeHHH

3),

MOLKHO

,nOJlOLKHTb

<p(r,

t~'lv)==:;::;cp(r,

t),

TaK

'tITOHarna

uenosxa

ynpouieaaa

i6Y,lleT

BbIrJISIJleTb

CJIeJlYIOIUH'M

o6pa30M:

rp(r,

t-r/v)=rp(r,

t)=rp.+X(o~).+Y(~:).+Z(~~)

••

(8.5)

rne

HH,lleKC

«0»

03HatIaeT,

lITO

COOTBeTCTBYIOIIiHe

BeJIHqHHbI

6ePYTCSI

aasane

lKOOp,llHHaT,

T. e. B

MeCTe

HaXO)Kj(eHHSI

3JIeMeHTapHOH

:nJIO-

manxn

~S.-M01KHO

HeCKOJIbKO

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JiaJIbHeiimylO

aanasy,

eCJIH

aCHOBe

nonynteaaa

1) nOJIOLKHTb

+a,= E

• (8.6)

Jlerxo

snners,

qTO

tIJIeHbI,

conepxcaurae

x H y B

pa3JIO}KeHHH

(8.5),

naror

npn

narerpnponanaa

HyJIb.

TaKHM

oopasoa,

a3MeHHH

(8.5) z

rcos6 H

nO,llCTaBJIHH

(8.5) H (8.6) B (8.4),

nonysaex

[

21t 'It/2

j - =

rpo

Jexp

(-Esr)

dr 5

dlfJ

cos6sin

6d6+

000

z/2

]

+(:;)05

rexp(-Esr)dr

dlfJ

5cos'6sin6d6 =

(:)."

, 0 0 0

(8.7)

nJIOTHOCTb

TOKa

HeHTpOHOB

HarrpaBJIeHHH

'nOJIytIaeTCSI

TOtIHO

TaKHM

}Ke

o6pa30M,

HCKJIIOtIeHHeM

Toro,

qTO

HHTerpHpOBaHHe

np9BO,lI.HTCH OT 1t/2 ,llO 3t

(T.

e. TOJIbKO

HH1KHeMY

nonynpocrpancr-

BY).

BeJIHtIHHa

OKa3bIBaeTCSI

paBHOH

(0')

1+=4-

61:

PeaYJIbTHPYIOmaH

nJIOTHOCTb

TOKa

HeHTpoHoB

jz .B '110JIO}KHTeJIbHOM

aanpaaneaaa

OCM z

paana

paaHOCTH

Me}K,Z(Y i+ H

i-..,

TaK

tIro

•

.•.•

(0,)

1z ==1+ - J- = -

31:

iJz o·

3 a M e tI a H H e 1.

cpopMyJIaX

'(8.7) 'H

1(8.8),

naiomax 'O.llHOCTOpoHHHe

nJIOTHO-

eTH

TOKOB

HeMTpoHoB,

TOqHOCTb

orpaHH~eHa

~JIeHaMHHYJIeBOrO

TIepBoro

fiopa.llKOB

pa3JIO)l{eBHH

(8.5).

O,!{HaKo

B rpopsryne :(8.9) TOtIHOCTb OKa3b1tBaeTCH

60JIee'BblCO-

KOM, a HMeHHO

BKJIlOtIa5!

qJIeHbI

aroporo 'nopanxa B

pa3JIO)l{eaHH

(8.5).

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TOM,

xro

tIJIeHbI

sroporo

TIOpflJI.!Ka rrpn

HHTerpHpOBaHHH

;J.aIOT

JIH60

HyJIb,

JI'H60 asrpasceaas,

B TOtIHOCTH coananaioume .llJUI i+ H

noroxy

B3aHMHO

noranraromaeca B ssrpa-

>KeHHH (8.9).

a M e q a H H e 2.

HaJIHtIHe

3KCUOHeHUHaJIbHOrO

MH01KHTeJIH

!IIO,ll

3H8KOM

HHTe-

rpana

naer

B03M01KHOCTb

pacnpocrpaaars

peayasraru

-KOHetIHbIe

06'beMbI,

JIHIIIb

6biHX rpaaana

6bIJIa

yaaneaa OT peccxarpaaaeaoa naomanxa He xeaee qeM na

TpH

AJIHHbI csoeonaoro

npo6era.

Iloayaeaaue

pesynsrarsr

COXpaHSIIOT CBOIO

CHJIy

iH rrpa HaJIH'lIHH B cpene pacnpe-

,zr.eJIeHHOrO

·H~.TOtIHHKa

HeHTpOHOB,

eCJIH .TOJIbKO 'IIJIOTHOCTb

HCTOqHHKa

MHOrO

MeHbWe'

nJlOTHOCTH

·paCCeHBalOIII.HX

CTOJIKHOBeHHH.

HaKOHeu,

pe3YJIbTaTbI

6YJJ.yT CnpaBe,ZI.JIHBbl H B cpenax C

JIOKaJIbHbI'MHHeO,llHOpO,ll-

HOCTHMH,

JIHilib

ObI

9TH

HeO,llHOpO,llHOCT,H 6b1JIH

ynaaenu

accaeayexax

TOqeK

cpems

MeHee

tIeM

1Ia

TpH.

,zr.JlHHbl

CBo6o,!l.HOrO

rrpooera.

,lleJIO,

qaCTHOCTH,

TOM,

sro-

BOJIH3H OT OOJIee

HJIH

xeaee

KOHIJ,eHTpHpOBa'HHoro

HCTOqHHKa

HJIH ·"CHJIbHOrO

IIOrJIOTH-

TeJIH HJIH B'OJIH;3H

rpanaua

JJ.ByX

$H3HtIeCKH

pasaux

cpen

'nJIOTHOCTb TIOTOKa

HeHTpo-

HOB

pe3KO

MeHHeTCSI c

paCCTOHHHeM.

TaKHX

ynacrxax .u.a)Ke

MaJIblX

paCCTOSlHHSlx..

pa3JIO)KeHHe

(8.5)

MO)KeT

OKa3aTbCSI

CJIHIlIKOM

Z r

,py6bIM,

noroxy

ttiyner

nOCTaBJIeHa

non

COM-

HeHHe

nparonaocrs

Q>OpMyJI

(8.7)-(8.9).

113

scero

cxasaaaoro

B .u.aHHbIX

3aMeQaHHSlX-

caenyer

TaKOH

npaKTHiJeCKHH

BbIBO.u.: na pac-

CTOflHUflX,

npeeuuuuoiuux

TpU

OAUHbt

CB0600HO-

ZO npotieea OT CUAbHOZO

UCTOtlHUKa

'UAU

nOZAO-

TUTeAfl

UAU

epanuuu,

¢OPMYAbt

(8.7)-(8.9)

iiator

npaxtuxecxu

oOCTaTOtlHYIO

TOtlHOCTb,

npu-

tleM

TOtlHOCTb

¢OPMYAbt

(8.9)

eutue TO£tHOCTU

¢OPMYJt

(8.7) U (8.8).

ECJIH

reneps

nOMeCTHTb

nnoruanxy

npOH3BOJIbHyro

TOqKY

npOCTpaHCT·

opHeHTHpOBaTb

npOH3BOJIbHMM

06pa30M

(pHC.

10), TO

JIerKO

y6eLI.HTbCH,

qTO

IIJIOTHOCTH

TOKa

HeHTpOHOB

aa-

Puc.

10.

3JIeMeHTapHaH

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