Смелов В.В. Лекции по теории переноса нейтронов

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YqHThIBaH

yCJIOBHe

orpaIHlflIeHHO'CTH

TIJIOTHOCTH

nOTO'Ka

HeHTpoHo;B

6eCKOHe1JHOCTH,

nonaraea

B=O, T. e.

f(x)==:Aexp(-xx).

(9.8)

lITo6bI HaHT.H

KOHCT8'HTy

HCnOJIb3yeM

sanaaayio

'B YCJIOBHH

DJIOT-

HOCTb HCTO'q'HHKa q. O'KOJIO

rinocxoro

HCTOlJHHKa

pe3YJIbTHpYlOrnaH

nJIOTHOCTb

TOKa

HeHTpoHoB

pasna

q/2:

lim iz==-q/2. (9.9)

x~o

Onpeneaas jx us (9.8):

i,,=-D

=xDAexp(-xx)

H rroncraanaa

B (9.9),

noayaaex

A=q

/2xD,

T. e.

(x)

2xb

exp

xx),

(9.10)

(9.11)

Pasyxeerca,

'KOHcTaHTy

IMO)K~HO

tTIOnyqMTb,

KaK

rr.

OCROBe

sroporo

II'p'HlHUH:na:

q/2=

SEatp

(x)

dx.

3 a M e

a H

BeJlHtIHHa

Is lJIOCJIe nO,llCTaHOBKH B (9.10)

'HaHJleHHOrO

saase-

HHSI

npHHHMaeT

B'H.zr.

ix= +

exp

(-xx)

(9.12)

OTJlHtIHe

(9.12) OT (4.4)

aaKJIIOtIaeTCH

'B K03<PcIJHUHeHTe

nOKa3aTeJlSI

CTeneBH

ooyc-

JIOBJIeHO

TeM, 'qTO B

paccaarpaaaexoa

aaaase

KpOMe

peasnaa

aaXS8T8

eers

peaxuas

pacceaaas

H ,ll'BH>KeUHe

'HeHTpoHoB

B opene HOCHT ,6nYJK,llaIOIUHH xapaxrep, a

MO-

aoaanpaaaeaaua,

\. \

.;1Si

',\

\.\

~----------~""-:JI~

(9.13)

PDC.

12. K

pacsery

nOJIS

HeHTpoHoB

nnocxoro

HCTOQHHKa

npH

nOMOIUH

TOtIeqH~X

HCTOtIUHKOB

<l>O·pMyJIy (9.11) MO}KHO

IIOJIyq'IITb

,npyrwM

nyrex.

,llJISI,

sroro

paCCMOTpHM

'Haw

lTInOCKHH

H'CrolJHHK

'Ka'K COCTa'BJIeH'HhlH H3

,6ec'KoHell-

aoro

qHCJIa

TOqelJHbIX

HCTOlJHHKOB.

nYCTb

- O,llRR HS

TaKHX

3JIe-

MeHTapHbIX

«TO'tJelJHbIX» IfCTOlJHRKOB

(pRC.

12). HHTeHCJIIB'HOCTb

ero

paaaa

qtss;

,HH'HUlHwpyeMaH

BbI,n:eJIeHHbIM 'HCTOlJHHKOM

llJIOTH9CTb

no-

TOKa

HeHTpoHoB

q>t(x) B

TOlJKe

HaxoJtHTCSI

c rrcxomsro: BbI'pa:>KeHHSI

,nnSl

TOqeqHOrO

HCTOlJHHKa

(9.6):

ql:!s· exp

(-rix)

(x)

4~D

_..o..-..;..---..;;.~

3·

Ilonuaa

nJIOT'HOCTb

nOTOKa

nOJIy'qHTCH

,cy·M·M,HpoBa·HHeM (9.13).

BCeM

InJiom8,nK8M

~Si:

co k

(x)

4:D

SSexp

~-xr)

ds=

4:D

SpdpJexp

~-lCI')

dl(l.

i 0 0

Ilepeanea

nepeaenaos

H·HTerp·HpoBaHHH

.p 'K r.

:HMeeM

2=p2+x

·r·dr=pdp.

CJIe,l(OBaTeJIbHO,

2z co

f (x) =

rdr

Sexp

~-lCT)

del'

Jexp

(-XI")

2:D

exp

(-xx).

x 0 x

nOJIy'lIeHHbIHpe3yJIbTaT

InOJIHOCTblO

cosnaaaer

HaH,l{eHHbIM

paaee

BhIpaJKeHHeM

(9.11).

3 a M e q

H H e 2. 'B paecaorpeaaux 'DcpHMepax CHOBa aapyureao

Tpe60BaHHe

npH6J1HJKaTbCSI K

KOHlleHTpHpOB8-RHhlM

HCTOQHHKaM 6J1H>Ke

qeM

2-3

AJlHHbI

csooonnoro npotiera.

3. npocTeiimaH

saaasa

ByqHCJleHHe

KpHTHlJeCKOrO

paaaepa,

n'pe,l{nOJIOJKH'M,

liTO

HMeeTCH

HeKOTOpaH

MaTepHaJIbHaH

O,ll.HOp0,lI:HaH

cpena, 'B COCTaB

KOTOpOH

BXO,l{HT

,neJIHmeeCH

eeutecrao

(nanpaxep,

ypaa).

Ilpennonoscax,

ztanee,

liTO

'HeHTpOHbI

.paccxarpaaaeaoa

cpene

,MoryT

1JI0HBJIHTbCH TOJIbKO B

pesynsrare

peaxuaa

,neJIeHHB

B ypaae, T.

nOCTOpOHHHX

HCTOqHHKOB

HeHTpOHOB

HeT

•.

ECJIH

0003Ha-

IIHTb

cpenaee

KOJIHqeCTBO

HeHTpOHOB,

B03HHKalOIIlHX

O,Zl.HH

aKT

,ZI.e-

JIeHHH,

npennonoscars,

lITOBCe

npoueccsr

MOH03HepreTHQHbI

lJaCT.:.

HOCTH,

B03HHKalOIII.He

npn

,ZI.eJIeHH·H IHeHT-pOHbI HlMelOT 'TY JKe

3HeprJlIO,

uro

HeHTpOHbI,

'OJIy2K).{aIOIUHe B

opene),

ypaeaenae

,nHq),(py3HH

(8.15) lJ1pHMeT CJIeJ(YIOUJ:HH

qaCTHbIH

BH,n:

D'\rf

Eaf

"Eff,

rae

~G-~c+

~j;

MalK!pOCKOnHlIecKoe

ceneane

saxaara,

"PH-

sonamero K

,neJIeH}lIO.

CcIK>pMyJlHpyeM

-reneps

CJIe,nyIOIUYIO

epH3HQeCKYlO

aanaxy:

MO)KHO

JIB

ecJ11H

·MO)KHO, TO

npn

'Ka1KiHX

yCJIOBHHX)

,paCCM8TpHB8eMoro

semecraa

caeaars

urap,

KOTOpOM

nOJJ:,ZI.epLKHBaJIaCb IGbI

'CTa'UHOHap-

aas

UeIIHaH

pea.KUHH?

MaTeMaTHtIeCKH

3Ta

aaaasa epOp'MyJI.H-pyeTC·H

TaK:

MO)KHO

JIB

ecJIH

MOLKHO, TO

npa

KaKHX

yCJIOBHHX)

HeKOTopOH

crpepe

KOHeq-

noro

panayca

inOJIY1JH'Tb OTJIH'q'Hoe OT HYJISI InOJIO:>KH'TeJIbHOe

pemeaae

caenyiomero

OAHOpO,ZlHOro

CTaUHOHapHoro

ypaBHeHHB

J(Hepepy3HH:

D\72cp+(vE,-E

)

= O, (9.14)

nonsnaeaaoe

rpaaauaoxy

YCJIOBHIO (8.16)?

Ilpoaaaaasapyea

cnasana

3Ty

sanasy

dnieuwecnu.

ECJIH

'V~f<~a,

3TO

oaaasaer,

qTO

B K(l:>K,llOM

3JIeMeHTe

OO'beMa·

qiHCJIO

HeH~pOHOB,

:BOCn'pOH3BO,nHMhIX B

pe3YJIbTaTe

,l{eJIeHHH

Y'Pa:Ha~

MeHl>me

'q'HCJIa

HeHTpOHOB,

nOrJIOmaeMbIX

B3TOM

3JIeMeHTe

OO'beMa.

·tIneH

DV2q>,

OTBeTCTBeHHhlH

nepeaemeaae

HeiiTpoHOB

'B cpene,

epaooraer»

TOM

JKe

Ha(}IlpaBJleHHH,

xro

aaxsar

HeHTpOHOB-

B cpene

KOHetiHbIX

paaxepos

HMeeT

MeCTO

yrexxa

HeHTpOHOB

H3 nee.

OqeBH,nHO,

xro CTaIUHOHafp'HOH

u.eIIHoH·

peaxuaa

B 13TOM

cnysae

MO-

iKeT

6bITb

. ECJI'H

~~f=~a,

TO n KaJK,nOM

3JIeMeHTe

o6'beMa

tIHCJIO

HeHTpoHoB,

aocnpoasaonaasrx B

pe3YJIbTaTe

,neJIeHHSI

ypaaa,

B TOtIHOCTH

paBHO

qiHCJIY

lJIeHTpo:HO~B,

'BbI

i6hIBaIOIllHX

aroro

3JIeMeHTa

06'beMa

BCJIe,n-

CTBIHe

3aXIBaTa.

CTau.HOHa,pHaH

uenaaa

peaKUHH

MOJKeT

HJI.T,H, eCJIH

HCKJIlOqHTb

yresxy

HeHT,pOHOB H3

cpenu,

T. e.

cnenars

cpeny

6eCKO-

HeqHO

npOTHiKeHHoH,.

ECJ1H

"~f>~a,

TO lB 'KaiK,l(O'M

3JIeMeHTe

06'beMa

lIH'CJIO rHeH1'pOHOB,

BOCnp0'H3Bo,nHMbIX

pe3YJIbTaTe

neneaas,

npeasnuaer

tIHCJIO

HeHTpo-

HOB,

noraoiuaexux

3TOM

'3JIeMeHTe

O,6'beMa.

H3,6bITOK

poacnaio-

IUHX'CH

HeHTpOHOB

nan

norJIOmaeMbItMH

MOiKlJIO

nonsrrarsca

CKOMneH-

CHipOBaTb

YTeqKOH

'HeHTp,OHOB IH3

KOBellHoH

cpenu.

3TOM

cnysae

ecrs aaneacna

nOJIYlIHTb

CT8UHO'HalpHylO

uennyio snepayio peaxumo

B cpene

KOHellHbIX

paaxepos.

,lla,nHM

reneps

crporoe

MaTeMaTUtteCK,Oe

06OCHOBaH1He

BCeM

T-peM

cnysasa,

paCCMO'I'peHHbI'M

'Bblwe.

"~f<~a.

YpaHHe:HHe

'(9~14)

n,pH·MeT

npa

3TOM

BH,l(

1 d 2

2 0

7 dr '

dr-x

<p==

rae

X2=(~a-'V~f)

/,D:>O.

Iloayxeaaoe

ypasaeaae

ToqHO

COB'll8,ZlaeT

ypaeseaaex

(9.2) , otimee

pemeHHe.

xoroporo

JlaeTCH

$OpMy-

JIoA'

(9.4).

Ilpeacae

scero

,nJIH

nauiea

sana-ra

JIerKO

aanacars

rpanasaoe

YCJIOBiHe npa r=O.

llOCKOJIbKy

'nJIOTHOCTb

nOTOKa

Heii"poHoB

,n:OJI2KHa

'ObITb

orpanasena

uenrpe

crpepu, ·H3 otimero pemenas

cpa-

nOJlY1JaeM

A=-B,

T. e.

f(r)=A

8It

';.'

*. (9.15)

ECJI~H

,nony,cTHTb,

nro H'MeeTCH

HeKOTopaH

cipepa

panaycox R, B

npe-

AeJIaX

'KOTOpOH

aanasa

HMeeT

nOJI02KHTeJIbHOe,

OTJIHQHOe OT

'HyJlH

petneaae

.(9.15) (A>O), TO

rpaaaue

~nYCTOTOH

,nOJIiKHO

eme

y,nOBJIeTBOpHTbCH

yCJIOBHe

(8.16). nO,nCTa.BJISISI (9.15) B (8.16),

no-

nyqae-M

If2xD+cthy-l/y

_0,

y=xR>O

RJlH

F. (y) == 1/2xD,

F. (y) ==

Ify-

cthy==

(sh y - y ch

y)/y

shy.

(9.16)

• YCJlOBHe B

nearpe

c~epbl

MOJKHO

C<!l3pMyJIHpOBaTb H

anase:

uearpe

c«l»epw

DJIOTBOCTb

pesyasrapyiomero TOKa

HeifrpoHoB

paaaa

HyJlIO,

liTO,

COrJlaCBO

~oPMYJle

dep

(8.11).

,A3eT

YCJlOBHe dr. r==e =

pe3YJlbT3Te

~OJlY1JHM

TO :lKe

mp3:lKeHHe

(9.15) •

<l>YHlaUllo

(y) noorrpenenau npa y

===

paBeHCTBOM

(0)=

~lim

(y). C

nOMOlI(hlO

npaBHJI8

JlonHTaJIH

JIerKO

H8XO,lUfM P,(0)

===

g-+O

OCHOBe

Hep8BeHCTBa

sh y >y (y >0) HaXOJJ:HM

dF0 (y) I + 1 " 0

-y2- sb

<...

npa

CJIe)J;OBaTeJIbHO,

Q:>yHKUHH Fo(Y) MOHOTOHHO

y.6bIBaeT,

naxaaaa

C Fo(O)=O, a

IIOTOMy

MO)KeT

npn

y>O

'np'HHHTb

IIOJIO)KHTeJIbHOrO

3Ha"qeHH5I

1/2%D B COOTBeTCTBHH C

ypasaeaaea

(9.16).

TaKHM

o6pa-

30M,

nOCTaBJIeHHaH

sanasa

HMeeT

peureaaa.

'V~f-La.

YpaBHeHHe

(9.14)

npaxer

IIIpH

3TOM

BHA

_l_~_

dr di" ·

Oouiee peureaae .9TOrO

ypaBHeHHSI

JIerKO

lJIOJIyqHTb:

~(r)===Alr+B.

(9.17)

113

OrpaHHqeHHOCTH

peurenna

uenrpe

CJIe)J;yeT A=O,

a"R3

yCJIOBHSI

(8.16)

BHeW'HeH:

rpanaue

cpensr

B=O.

I1TaK,

aarepecyiomef

Hac

CQ:>epbIKOHeqHOrO

pannyca

cymecrayer

'V~f>~a.

YpaBHeJI.He

(9.14)

sanaurerca

KaK

d' + 2 0

-2

-d

xee==,

r . r r

(9.18)

rne

x2~("~f-~a)

/D>O.

HeTpYAHO

y6e,nHTbCH,

QTO oriuree

penreaae

ypaBHeHHH

(9.18)

HMeeT

BHJ(:

ep(r)=A

Si~xr

+Bco~lCr.

Tpe6oBaHHIO

Ol'pa'H'HqeHHO'CTR peureaas B uenrpe crpepsr

M01KHO

Y~O'B

JIeTBOpHTb,

nOJI01KHB

B=O,

T. e.

ep(r)=Asi~xr.

(9.19)

rpaaaue

'HOKOMOH

Cq>ephl

r=R aocnonsayeacn

yCJIOBHeM

(8.16).

3TO

11'pHBO,llHT K

rpaacueanearaoxy

ypaBHeHHIO

(9.20)

rAe

y==='HR,

(y)==

lly

ctg

y " (sin y - y

cos

y)fysin

Zloonpeneaaa

q>yHKUHIO

F1(y)

ITIipH

y=O.

KaK

·R B n,

1),

nerxo

nOJIyqHM

F1(0)

=0.

Ilpa

H3MeHeHHH

y OT 0

Q>yHKUHSI F

(y)

* B

6eCKOHetIHOM

,npOCTpaHCTBe

H3 (9.17)

HMeeM

(r)=B=const.

nperepnesaer

paapsrsa

·B TOlJKaX Yk

k«

(k=l,

...

113ylJ

'H'M

nonpooao

nepBbIH

HHTe,pBaJi

aenpepsmaocra

(y),

T. e.

0~<1I:.

C Y'lJeTOM HepaBeHCT.Ba sin

y<y

(y>O) nonyxaea

t (Y)

1 +_1_>0 >0

dy - y2 sin" Y ,

TaKHM

06pa30M,

npOMe2KyTKe

O~<n

q,yHKUH,H

Fi'(y)

MOHOTOHHO

pacrer

OT 0 no

+00

[F1(O)=O,

F1(n)=+oo].

CJIeJlaOBaTeJIbHO,

rpa-

<p'RK

q>yHKU'HH FI (y),

HHTe.p'BaJIe

nepeceser

ropH30HTaJIb-

ayio TIlpHMYIO y=I/2'X,D>0,

npasea

B CHJIy MOHOTOHHoro

xapaxrepa

H:iMeHeHHH PI (y)

raxoe

nepeceseaae

npOH30H,AeT

TOJIbKO

O,ll.HH

pas,

I1Ta

IK,

npoueacyrxe

O~y<n

y'p8BHeHHe

(9.20) ;HMeeT e,ll.H'H'CTBeH-

HbIH

xopeas

yo<n.

Kopeas

- HalHMeH:OIIIHH H3 l6eCKO'HelJ:Horo MHO-

JKeCTB8 ,nOJIOJKHTeJIbHbIX

xopaea

Ylpa'BHeHHH (9.20) H

onpeneaaer

KpHTHlJeCKHH

paaxep

cq>ep'H'lJecKoro

peaxropa

RJ}-Yo/'X,:

~PH

R<R

TpHBHaJIbHOe

crauaoaapaoe

COCTOHHHe

peaxropa

<p(r)==0 y'CTOHlJHBO

nenaas

peaKUHH

ctpepe

oTcyTCTByeT;npHR>R

crauaoaapaoe

CQCTOilHHe qJ(r) ==0 a:6'COJIIOTHO :HeycTOHlJHBO IH UecIIHaH

,peaKUHH

He-

H36eLKHo 3apO,ll.HTCH H

npeapararca

B JIaB'HHHbIH

npouecc.

B 3aKJIIOlJeHHe OTMeTHM, ttITO

npa

r~o

BeJIHlJHHa xr He

npeBOCXO-

,lI.HT

~'XJRo

yo<n, a 1nOTOMY 'nJIOTHOCTb ·nOTOKa

HeHTpoHoB

(9.19)

(npa

A>O)

6y,neT

crporo

'nOJI02KHTeJIbHoH. ECJIH LKe

paccaorpers

JII060H

H3 npOM,e2KyTKoB

O~~k,

rne

- panayc ctpepu,

OT~elJaIOIUHH

KaKoMy-,HIH6y,ll.b

,lI.Pyr9MY

KOpHIO Yk y'paB'HeHHH (9.20)

(Yk=FYO),

TO,

paBeHcTB~:,(9.19)

JIerKO

y6e,ll.'HTbCH, lITO cp(r)

6y,neT'n,p'~Ii"MaTb

KaK

nOJIOLKHTeJIbHble,

TaK

orpanarensmae

3HalIeHHH.

3TOT

anaaas

eme

pa3

nOKa3bI,BaeT,

lITO TOJIbKO 'BeJIHlJHHa R

H'MeeT <pH3HlJec-KHH CMhICJI

(UJIOTHOCTb

nOTOKa

HeHllpOHO'B !He MOLKeT !6bITb

OTpHuaTeJIbHoH!).

3 a Me q a H

e 3. Bee

TpH

cayxas,

paccxorpeaaue

asnne,

MomBO

aarepnperapo-

aars

K3K

'eJIe,ZlYIOlllYIO

sanasy

coocraeauue

3HatIeHHSI;

:; =

"cp;

.. }

(9.21)

,-!!LI

=0.

(-'-+D

)

=0.

dr r=o ' 2 dr

r=R

51cHo, 'qTO

coocrseaaue

qHCJIa 3TOM

sanasa

CYTb

q>YHKUHH

napaxerpa

A=:A(R).

Hy>KHO

reneps OThlCKaTb

TBKoe

QHCJIO

R>O,

11'f06h1'

MeHbweH

.Mepe

O)1.HO

.H3

C06CT-

BeHHbIX qHCeJI Y.lloBJIeTBOpSlJIO

Tpe6OBaHHIO

.: { (E

- v"St)/

cnysae 1);

A(R) = 0 B cnyxae 2);

-(vEf-Ea)/D<O

B cnysae 3).

(9.22)

Ilpa

JIlO6oM

paaayce

c4»epbl

iR>O

see

CO'6cTBeHHbIe 3HaqeHHSI aanasa (9.21)

orpaua-

TeJIbHhI.

3TO

oaaaxaer,

QTO

Tpe60BaHHe

(9.22) B

cnyxaax

1) H 2)

'peaJIH3YeTCB,

a ,llJISI CJJ}"IaSl 3) npHXO)1.H'M K

CJIe.u.YIOII.J.eMY

ypaBHeHHIO

.Il.JlSI

onpeneaeaas R:

Pa3YMeeTCB,

3T0

ypasaeaae

CBOAHTCSI K

ypaBHeHHIO

(9.20).