Смелов В.В. Лекции по теории переноса нейтронов

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(11.18)

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F(x)

=1:~o(x)

+.q(x);

a H b-

rpaaauu

nJIaCTHHhI.

OTMeTHM,

lITO

CJIYQ8H

a=-·oo

b=+:oo

HCKJIlOlIalOTCH,

TaK

nrc

n~aCTH'Ha

MO-

JKeT

6bITb

nOJIy6ecKoHellHOH

HJIH

6eCKOHeIlHOH:

nepexeaaoa x.

ECJIH

,nJIOCKOClB

YOZ

BBecTH

nOJIHpHhIe

KOOp,ll.HHaTbI

'C nOJIIOCOM B

TOIlKe

(Yo,

Zo), TO

yPaBHeHHe

(11.18)'

rij)HMeT

BHA

BBeAeM

HOBylO

nepexeaayio

Tornanocaenaee

ypasaeaae

eute

HeCKOJlbKO

ynpocrarca:

b co

CP.

(x.)=

+5F(x)

Sexp

(~'11~

-.X

dv,

a 1

(11.19)

Ha,KOHeu,

xopouro

H3BeCTHhIe

B aCTpO<}>'H3HKe H

H){epHOH

tPH3HKe

cne-

uaansasre

<pyHKUHH

[12, 13]:

E (

Sexp

(-~x)

x~o

(11.20)

n03BOJIHIOT

nepenacars

ypasaeaae

(11.19) B caenyromea

q>opMe

HenpH'HUHTIHaJIbHhIMH

H3MeHeHHHMH

CHMBOJIHKe)

'Po

(x)

[~s'Po

(e)+q

(~)]

£1

(~I

xl)

(11.21)

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aeonaoponaa,

HO-

TaKO·Ba,

xro

Ma-KPOCKOllHqeCKHe

CeqeHHH

~8.

3a-BHCHT

TOJIbKO

OT X, COBCeM ;HeCJIO}K'HO'

nOJIYQHTb

o60611leHHe

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(11.21).

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Hy2KHO

nOBTO'pHTh

ssnne-

OIIHeaHHhIe

npeo6pa30B8HHH,

OTnpaBJIHSlCb

CTau.HoHaptHOH

<PO

p'Mbl

y·paBHeHHH

(11.13).

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'illHCh,QTO

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npaaeneao K

BH,llY

boo"

Ix-xol

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(x.>=+

F(x)dx

S'V-1ex

[ -

(x-

:f:~:'!1

)d«]dv,

a I 0

JteJIaeM

eme

OAHy

nO,ZlCTaHQBKY:

%--a(X-XO}

X-Xo

CaOBa

nepeoeosaasas

HeaaBH'CHMhIe

nepexeaaue,

OKOHtIaTeJIbHO

nonysaea

er.

(x)=.+

[Es(e)er.~(~)+

~)]

(

~=:

I jE

('1:)

d'l:)

de.

a x

('1:)

d'l:/(e

- x)

ecrs

cpenaee saaseaae noasoro ceseaaa aa

HHTep-

Bane (x,

~).

OOOaHaQHM

3T}'

IBeJIHlI'H,ay

(x,

~).

Torna

nocneaaee

ypaaaeaae

CTaHOBHTCSI no

<lJopMe

no-rra coenaaaiomaa C ypaaaeaa-

(11.21):

er.

(x) = +5[E

(e)

er.

(e)+q

(en

(E(x.

Ie- x

de.

Bonee

nOApOOHhIe

CBe,lleHHSI

HHTerpaJIbHhlM

ypaBReBHSlM

Teo-

pHH

nepeaoca

I1'pHBeAeHhI

paoorax

[3-6,

12-15].

rnABA 3

OCHOBHbIE METO,UbI PEIIIEHH5I

O,UHOCKOPOCTHoro

YPABHEHH5I

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§ 12.

METOA

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PaCCMOTpHM

O,nH'H H3

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peureaas

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nepeaoca

MeTO,n

crpepasecxax

rap-

MOHHK.

Ilpeacne

scero

uenecooopaaao

npnaecra

onpeneaeaas

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Jlexcaanpa,

npHcoeAHHeHHbIX

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UHH

Jleacannpa

H~.

cdiepasecxax

<pyHKUHH

[12,

16],

6a3e

KOTOlpbIX

CTpOHTCSI

OIIHCbIBaeMbIl1

HH)Ke

MeTO,n.

nOJlHHOMLI

Jleacannpa,

onpeneaeaae

nOJIHHOM8

Jlescaanpa

n-H

CTeneHH

Pn(J.t) MOJKHO

II'PHHSITb

cnenyiouryio

Q>OpMyJIy

(tpopxyna

Ponpara)

Pn(p.)

2!n!

d~:

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_l)n].

TIOJIHHOMbI

Jlexcaanpa

C'BSl3aHbI

pexyppenraoft

Q>OpMyJIOH

(n +1)P

(,.,)

(2n +1)P

(p.)

+nP

(p.)

= 0, (12.1)

n03BOJISlIOI,lI.eH

nO'CJIe,nOBaTeJIbHO

BbIqHCJISlTb

HX,

HaqHHaSI

C P

(

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o(p,)=l,

PI (J.t)=J.t].

OTpe3'Ke

[-1,

nOCJIe,noBaTeJIbHOCTb

nOJIHHOMOB

Jleacannpa otipasyer

IIOJIHylO

opTOrOHaJIbHyIO

cacrexy:

{a,

eCJIH m

=1=

Pn(

) Pm(p')dp.==

_2_

(12.2)

2n+

eCJIH

m===

2. npHcoeJlHlleHHble

4JYHKIlHH

JIe)l(aHJlpa

p(m)

(,.,) MO>KHO onpene-

JlRTb

xepea nOJIHHOMbI

J1e>KaHApa:

p(m)

( ) == (1 _ 2)m/2 d,"l p .(' )

n p, p,

dfJ-

m n p, ,

OTKYJJ.a cnenyer,

liTO

p(O)

(p,)

Pn (p.)H, nanee,

p(m)

(p,)

°npa m> n. (12.3)

npHCOe,l{HHeHHbIe

<pyHKUHH

Jlescaanpa

CBSl3anbI

pexyppearnos

<pop"

MyJIOH:

(n - m+1)

P~l

(p.)

-1£

(2n +1)

p~m)

(p.)

+(n

+'m)

P~':)l

(I') == 0. (12.4)

3TO

TO>K,l{eCTBO nepexonnr

aBTOMaTHqeCKH'

B (12.1)

npa

m=O.

(12.6)

Ilpacoenaaeaasre

<pyHKUHH

JIe2KaH~pa

npa

<pHKCHpOBa'HHOM

TaK)Ke

06'pa3ylOT

nOJIHylO

OpTOrOHaJIbHylO

cacrexy

orpesxe

[-1,1]:

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rp~m)(fL)p~m)(fL)dfL=

._2_

(n+m)!

(12.5)

J 2 1 ( ), e CJIH k

=:=

-i

n-m.

COOTHorneHHe

(I2.5) nepexonirr B (12.2)

npa

m=O. B

llpHJI02Ke-

'HHH

rrpHBO,l(HTCH

psn

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ynos-

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rrpacoennnenasre

<pyHKUHH

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y(m)

(p.,

.~)

MOlKHO

onpeneaars

sepes

. n

npacoenaneaaue

<PYHIQlHH

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otipaaou:

y~m)

(tJ-,

cfJ)

=:=

p~m)

(p,)

sin

m~,

0; }

y(-m)

(tJ-,

cfJ)

===p(m)

(lL)

cos

m1J,

n n

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cacrexy

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211:

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21t

(I +

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=-=

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=:=

= n;

o BO

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CJIyqaHX.

{

npa

i=Fj·

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i j

=:=

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=:=

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npe,lI.nOJI02KH'M

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HaH

TOtIKaX

nOBepXHOCTH

crpepsr

e,nHHHtIHOrO

panayca, T. e.

-f(M),

MEx2+y~+z2-I=O.

nOCKOJIbKY

Ka2K,naH

TOtIKa

M crpepa-

qeCKOH

nOBepXHOCTH

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O,l(H03

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e.nHHHq:HbIM

panay-

COM-BeKTopOM

II'pOBe,lI.eHHbIM

na-rana

xoopnanar

TOq,Ky

M, TO

BMecTO

3anHCH

(M)

M02KHO

ynorperiars

raxyio:

f===tr(O).

E'CJIH,

na-

KOHeu,

ueHTpOM

crpepu

CBH3aHa

HeKOTopaH

cqiepasecxaa

CHCTeMa

KOOp,l{HHaT,

TO Ji(O) B

nrore

6y.neT

aaaacers

.nByX

apryuearoa:

J.t

H ¢

(Il=cpse).

CJIe,l(OBaTeJIbHO,

<pyHKUHH

f=f(O)

MOJKHO

IIOCTaBHTb

COOTBeT~T·BHe

pRJ(

cI>ypbe

IIIO

c<pepHqeCKHM<pYHiKUHHM.:

f(0) ~

Anmy~m)

(p.,

cfJ),

n=O

m&::-n

(12.7)