Смелов В.В. Лекции по теории переноса нейтронов

Подождите немного. Документ загружается.

n(r,

§ ·to. TO'lHOE OAHOCKOPOCTHOE

'YPABHEHME

nEPEHOCA HEATPOHOI. rPAHM'IHWE

YCnOBH~

np06JIeMbI,

CBH3aHHbIe

C nepesremeaaex

HeHTpOHOB

IBBemeCTBe

06JIaCTeif

C BbICOKOH 'IlJIOTHOCTblO

HeBTpO'HOB

06JIaCTSIM

C lHH3KOH

nnOTHOCTbIO,

6bIJIH

paCCMOTpeHbI

B §. 8 C

nOMOmbIO

TeOpH!H ,ltH<!>ct>y-

3HH.

O~HaKo

,M'HOrOQHCJIeHHhle

OrpaHHQeHH.H

,nll<P<t>Y3HOHHOH

TeopH·H

CHJIhHO

CY}KaIOT

06JIacTh

·ee

Ill,pHMeHHMocTH.

aacroamea

naparpa-

4>e

C<!>OpMyJIH·pyeM

HHTerpo-.}J:H<p<pepeHII.HaJIhHOe

ypaaneaae

,n:JIH' O'IIH-

CaHH.H pacrrpenenenas

'HeHTpoHOB,

npa

3TO'M'

CHH'MeM ace cytnecrsea-

asre

orpaHH'QeHHH

,l{H,Q)(PY'3HOH:H.OH

reopaa,

E,nHHcTBeHHbIM

orpaaase-

HHeM

(xoropoe

'6Y.lleT

'C'HHTO B

rJI.

4) OCTaHeTCSI IIpe,n,IIOJIO:>KeHHe,

liTO

BCeHeHT.poHbI

H'MeIOT O,l('HHaIKOBYIO

3HepTHIO

,llJISI

BbIBO~a

ypaaneaaa

:IIepeHOCa

norpeoyerca

CJIe,nYIOmee

OIIpe-

neneaae.

nYCTb

Fi(AV,

AlQ,

tl)

-qHCJIO

HeHTpOHOB

06De~e

OKOJIO

HeKOTOpOH

TOllKHr,

HalII·paBJIellHSI

CKOpOCTeH'

KOTOpblX.

3a'KJIIOqeHbI

TenecHoM

yrae

il\lQ

oxoao

aexcroporo

HantpaBJIeH·H.H

see

3TO

Sa<pHKCHpOBarHO B

MOMeHT

IBpeMeHH

t, Jlerxo

BHi)(eTb,

xro

ecrs

a.lll(HTH·BHaSi

4>YHKUHH

O()JIaCTH

~G=(~

AQ).

npOH3

BO,nHaSl

3TOA

4>yHKII.HH

rro 06JIa

CTH

F (4V,

I~GI-t>O

!\Q

H8'ShI,BaereH

t/Jaao80u

n./l,OTHOCTblO netitponoe.

·OqeBH,nHO,tITO

asenea-

aas paaee

;nnOTHOCTb

HeH-rpoHoB

(r,

.~)

CBSl3a'Ha c

.4>a30B08

naor-

HocTbIOn(r,

paseacraox

n(r,

t)=

n(r,

O. t)dQ,

rne (B C<j>epHqOOKHX xoopaaaarax)

d;Q=sin

ded1l'.

BeJlHtIHHycp(r,

t)=on(r,

O,~)

HaSOBeM

nltOTHOCTblO

tPaso6020

nOTOK.a

neuTpOH08

**,.

npaxea

ep(r,

t)=S

ep(r.

t)dQ.

(10.1)

: ,LlJlSl

.naJIhHeHWHx

ueneii

y,no6HOBBeCTH

repuaa

nqwo« nearponoe

(AQ)g,

nOHHMaSi

non nOCJIe,nHH'M IBCIO

COBOKYII:HOCTb 'HeHTpOHO:B,

HarrpaBJIeHH5I

cxopocres

KOTOpbIX

38KJ11OqelIhlB

3JIei\-It:HTapHoM

re-

JIeCHOM

yrne

AQ OKOJIO Ha'II'paBJIeHH'S!

BeKTOp

3AeCbIH

BCIO,ny

.naJIbHeifweM

IIpe,lI.IIOJIaraeTCSI

e,nHHHqHblM

11I0

'MO,llynIO.

HaKOHeII.,

0603HatIliM

C«

(AV,

!l\IQ,

iAt) qHCJIO

aKTOB

i-ro

npouecca

(i=a,

S HJIM

f),

auauaaeaoro

'HeHTpOHa,MH

nysxa

(ll\IQ)g B

061leMe

L\V B

resenae

'IIpOMe'*yTKa

'BpeMeHHAt.

Toraa

n,pO'H3IBOJlHYIO

STOB

.a.n.llHTHBHOH

epyHKII.HH

rio

'o6J1aCTH

G=(AV,

.L\Q,

Alt) (IGI

)

Cl>aKTHqeCKH

eure npeanoaaraercs, qTO

'HeHTpOHIi

CCTaJIKHBaIOTCSI»

MelKJJ.Y

CO-

60H.

CHSlT,He

sroro

OrpaHHQeHHSI -

npenaer

HeJIHHeHHol'

'KHHeTHtleCKOH

TeopRH.

ECJlH

rexcry

SlCHO,

KaKOM

nOTOKe

HJIH

KaKOH

DnOTHOCTH aaer pens,

6y-

nex

orrycxars

CJIOBO «<I>a30BblH».

Ha30BeM

q,aaOBOU

nnornocrmo

cOyaapeHuU

B i-M nponecce. OHa ona-

ChIBaeTCH '<pOpMy,,10H

(10.2)

YCTaHOBHTb

3TOT

,pe3YJIhTaT

3HaQHTeJlhilO

rrpoute,

lIeM

3HaJIOrHtI-

ayro

<pOpMyJIy

(7.2) ,,ll,H'<p<py3HOHHOH

reopna,

caMOM

neae, B ,nH<p<py-

3HO;H'HOH

reopan

'np~XO,ll.HJIOCb

<pOpMyJIHpOBaTb3KBHBaJIeHTHylO

scno-

MOraTeJIhHYIO

aanasy,

nepecrpaasaa

XaOTHtIeCKH

j(BIDKymHecH

HeH-

TpOHbI B

nJIoC'Kona'paJIJIeJIhHbIH

nysox. npn

nOJIytIeH'HH

<pOpMyJIbI

(10.2)

Mhl

HMeeM

.neJIO' C

aHCaMOJIeMHeHTpo-

HOB, CKOpOCTH

KOTOpbIX

opHeHTHpOBaHbI

B on-

HOM 'Han:paIBJIeHHH (B,llOJIb

nexropa

~cno-'

MOraTeJIbHaH

aanasa

CTaHOBHTCH

H3JIHWHeH.

Ilpescne

tIeM

nepexonars

K BbIBOJJ.Y-ypas-

HeHHH

nepeaoca,

paCCMOTpH~

BcnOMOraTeJIb-

ayio aanasy: CKOJIbKO

HeHTpOHOB

nysxa

(!L\,gh!

npoaner

3JIeMeHTapHbIH

npoxescy-

TOK

BpeMeHH

xepea

3JIeMeHTapHyro

nJIO-

manxy

CTOpOHy

e,llHHHlIHOH

HOpMaJIH

n (pHIC. 13)?

cay-rae

H,lI.e·aJIH3HpOB8HHoro

nysxa,

xor-

PRC.

13. Tleperexanae

HeA-

TpOBOB

lIepe3

3J1eMeHTa

,n.a

nOCJIe,nHHH He

nperepneaaer

H3MeHeHHH

BYIO'

UJlOma,ltKY

lAS

Baa';'

(HeHTpOHbI

rrysxa

Bbl6bIBaroT

npa-

npasaeaaa sexropa 0

6bIBalOT

'Hero),

see

npomenmae

xepea

nno-

manxy

BpeMH

IdrtHeHTpOHhI

nyxxa

(.~g)

3anOJIHHT

UHJIHH,np,

OCHOBaHHe

xoroporo

ecrs

~S,

otipaayiomaa

napaanensaa

sexropy

o H

HMeeT

,llJIHHyVdrt

(CM.

pHC.

13).

06'beM

sroro

UHJIHHApa

paaea

l\SV1&tOn,

T8K

xro

nOJJ.JIe)KamaSl onpenenemno

BeJIHQHHa

3anHmeTCH

aane

(10.3)

OTMeTHM,

liTO

nOJIY'lIeHHbIH

peaynsrar

TIpOnOpUHOHaJIeH

npoaasene-

HHIO

AQ~V

(IAV~SvAtOn).

ECJIH

HeHTpOHHhIH

nysox

(A:Q)o

nperepneaaer

H3MeHeHHe

pe-

3YJIbTaTe

H,lI.epHbIX

.peaKUHH,'

TaKLKe

H3-3a

B03'MO:>KHOrO HaJIHtIHSI

nocropoaaero

HCTOllHHKa

HeHTpoHoB,

aerrpepstsno

paonpeneneanoro

scexrrepeaeaaux

t, TO 3TO

H3MeHeHHe

OO'beMe

'Bpe-

L\~

nponopuaoaansao,

,KaK

JIerKO

BH,lleTb,

n~pOH3Be~eHHIO

IA'QI~

V1\t,

T. e. H'BJIHeTCSI BeJItHlIHHOH

60JIee

IBblCOKoro

nopanxa

M8JIOC'TH,

tIeM

(10.3), 'H

MOlKeT

npHHH,MaTbCH

BHHMaHHe.

Ta'KHIM

06·pa30M,

epnpMyJIa

(10.3)

ooocaosaua

H B

o6meM

cnysae.

Ha OCHOBe. epOp'MyJIbI (10,.3) HecJIOJKHO

noncnarars

nJIOTHOCTH

O)l:HOCTOpOHHHX TOKOB i+ H

j.;

a TaK:>Ke

pe3YJIbTH,pYJOmyIO

'nJIOTHO'CTb

TOKa

HeHTpoHoB

inB

'HalnpaBJIeHHH

'8e'KTopa

[3TH

BeJIHtIHHhI

orrpe-

nenensr IB § 8, H HX

aapaaceaas

xepea

'nJIOTHOCTh nOTOI):a

HeHTpoHoB

na aC-HOBe npH'OJIH>KeHHOH ,nH'<P<PY3HOHHOH

Mo,neJIH

~aHbi

<popMYJIaMH

(8.10)

(8.11)].

YqHThIB8S1,

xro

i+, i- H

in,

pe,neJIeHH10,

HOP-

MHpOBa'HbI

e~HHHUY

nnomana

e,nHHHUY

BpeMeHIH,

JIerKO

·R3XO-

J{HM

H3 (10.3):

1+=

J Onq>(r,

t)dQ.

(On>O)

J Onq>(r.

t)dQ.

<00<0)

nOCJIe~HeH

<l>opMyJIe

3HaK

M'HHYC

nepezt

aarerpaaoa

BBeAeH

~JISI

KOMneHca'UHH

3HaKa

MHHyca

'cKaJIHpHOrO

npoaaaeneaas

On,

no-

CKOJIbKY,

onpenenemno, i-

aeorpauarenen.)

arore

nonysaex

In(r,

t)=J+-I_=

Sonq>(r,

t)dQ. (10.4)

3TO

paseacrao

MO>KHO

nepenacars

sane

(r,

t) ==

(r,

t),

rae

Hr.

t)=

SOq>(r.

t)dQ

(10.5)

ecrs

aeKTopHblH

TOK

HeHTpoHoB.

(HanOMHHM,

rrro

BeKTopOM

j (r,

MbI

ysce

BCTpeqaJIHCb

~H<I><PY3HOHHOH

reopaa.)

nepeH~eM

reneps

BbIBO~y

ypaB'HeHHSI

nepenoca.

nYCTb

HMeeTCSI

HeKOTO'pa51

O,6J1aCTb V

3a'nOJIHeHHaH

HeH'I'pOHaMH.

BbI,lI.eJIHiM

'BHyTpH

3TOH

06JIaCTH

n,pOH3BOJIbHbIH

06'beM

Bynex

BeCTH

HaOJlIO,neHHe

TOJIbKO

TeMH

HeHTpOHaMH

06'beM3

xoropue

npaaanneacar

nysxy

(A'l)!}_

Iloncvaraex

caasana

y6bt/lb

HeHTpOHOB

aaurero

nysxa

npo-

Me>KyToK

,BpeMeHH

06beMe

V. Flpexcne

acero

HeHTpOHbI

BhIObI-

BalOT

nynxa

6JIaro,na

1pH

peaKUHHM

3aXBaTa

pacceanas.

Oomee

KOJIHlIeCTBO

aKTOB

3aXBaTa

paccesnaa

HeHTpOHOB

nysxa

(aQ)g

061>eMe

BpeMH

'BbIpa3HTcH,

COrJIaCHO

<t>opMyJIe

(10.2),

B BH.n:e

/!'JJ.at

.\E(r)q>(r.

t)dV

(E=Ea+E

(10.6)

E·CJlH nornomennsre

HeH'f.pOHbI

nOJIHOCTbIO

'BbI6bIBaIOT

nyxxa,

TO H3

q'HCJIa

pacceHBIIIHXCH

HeHT,pOHOB

qaCTb

OCTaeTCH

CHOBa

nyxxe

(L\Q)o-

3TO

HeHT-pOHhI,

xoropue

pe3YJIbTaTe

pacceaaas

Me-

HHIOT

narrpaanenne

,llBH>KeH'HH B

npenenax

TeJIeCHOrO

yraa

·A,Q.

nOKaJKeM,

liTO

3TOH

'u'OJIeH

HeHTpOHO'B

MO>KHO npeaeopexs

cpasae-

HHIO C BeJ11!rQHHOH (10.6).

BcaMOM

,lI.eJIe, H3 oomero

lIHCJIa

paccess-

UIHX'CH

HeHTpoHoB

aQl1t

s, (r)

(r,

t) dV

JJ;OJIH

OCT3BmHXCSI

nysxe

nponopurroaansaa

i\.Q,

TaK

qTO

3Ta

,nOJISI

OMeeT

nopanox

MaJIOCTH

AfJ.2At,

B TO

.BpeMSI

KaK.

(10.6)

HMeeT

nops-

. ,nOK

MaJIOCTH

d'Qi\~

,UaJIee

HeHTpOHbI

·MoryT

YXO~HTb

06'beMa

BCJIe,n:CTBHe

MeXaH'H'lJeCKOrO

nepeueiueaaa

npOCTpaHCTBe.

KOJItHlJe-

CTBO

HeHTpO'HOB

paccxarpnaaeaoro

nyxxa,

BhI6hIBaIOIUHX

06'beMa

BpeMH

pe3yJIbTaie

nep~MemeHHH

npOCTpaHCTBe,

JIerKO

IIO)lClIHTaTb

nOMOmbIO

epopMynbI

(10.3); 3TO KOJIHtIeCTBO

HeHTpoHoB

paBHO

I:::.Q!lt

gnep(r,

t)dS.

OOACtIHTaeM

reneps

npu6bl.l/,~

HeHTpoHoB

nysxe

BpeMH

001>eMe

V. OqeBHJl.HO

xro

HeHTpOHbJ

MoryT

npH6bIBaTb

nynox

(aQ)g H3 npyrax

nysxoa

(I:::.Q')O',

KOTOpblX

OHM

nperepne.na

pacces-

HHe C

H3MeHeHHeM

HanpaBJIeHHH

OKO'pOCTH C

)lJI5I

MaTeMa-

rnnecxoro

BblpaJKe'HHH

sroro

epaKTa

BBe,neM

paCCMOTpeHHe

sepo-

HTHOCTb

roro,

lITO

HeHTpOH,

HMeBIIIHH

,n:o

axra

pacceanna

nanpaane-

HHe J(B'HJKeHHH

g',

nonanaer

'nOCJIe

paccesaaa

nysox

(~!Q)g..

He-

TPY~HO

BH,n:eTb,tITO

O,n:H.OPO,n:HOH

H30TpOIIHOH

cpene

3Ta

'BepOHTHOCTb

P 3a'BHCHT

OT xascnoro H3

BeKTopOB

H g B OT,n:eJIbHOCTH, a

TOJIb-

KO OT

yrna

MeJK,n:y

HHMH,

T. e. OT

CKaJISI,pHOro

npOH3Be)l.e:HHH

gO'.

Kpoxe

Toro,

paayxeerca,

3aBM·CHT OT BeJI'HlJHHbI L\Q.

0603Ha'tIHM

g(r,

(!!!')

nJIOTHOCTb

3TOH

BepoHTHoCTH

**:

. P (r

00'

4&1)

g(r,

00')==

11m

' , .

IAf2I~oo

Ll&l

113

onpenenenas

nJIOTHOCTH

BepOHTHOCTH

cnenyer

TOJK,n:eCTBO:

g(r,

Oo')dQ=

(10.7)

(10.8)

nOKaLKeM

reneps,

sro

npHOblJ1b B

rrysox

(I:::.Q)o

apexa

!It

061>eMe

V H3

scex

)l.pyrHx

nysxoa

pesyasrare

paccesnna

HHX

ziaerca

HHTerpaJIOM

!J.QAt

JdV

dQ'~s(r)f(r,

0',

t)g(r,

00').

AeHCTBHTenbHO,

npOH3Be~eHHe

(r)

(r,

0',

t) dV

dQ'!lt

(10.9)

(10.10)

HBJISleTCH

yJKe

npHBbIlIHOH

,n:JIH

Hac

epOpMyJIOH

npe,n:CTaBJIHeT

C060H

KOJIHlJeCTBO

aKTOB

pacceaaas

3JIeMeHTe

06'beMa

nposreacy-

TOK

BpeMeHH

H3 lJHCJIa

HeHTpOHoB,

npHHa)(JIe2Kam'HX

nysxy

(dQ')o'.

* B

HaCTOSlmeM

naparpaipe,

T3K)Ke

B § 7 'IIO,lI.p06HO

060cHoBbIBa1OTcSI

TOHKHe

.neT3JIH, CBSl33HHbIe

cpasaeaaea

6eCKOHeQHO

MaJIbIX

BeJIHlIUH

pasaux

nopanxoa.

Jl3JIbHeHWeM

TaKoH

aHaJIH3

onycxaerca

3aBucHMoCTb

OT r snecs oaaasaer, aanpaaep, nro

H3YQaeMaSl

06JIaCTb

M01KeT

6hITb

cotipana

H3 cpen c

pa3HbI'MH

CBoHCTBaMH.

113

MaCChI

paCCeSI'B'WHXCSI

HeBT'pOHOB

(10.1-0) B nysox

(AQ)g

no-

nanaer

cnenyiouiaa ,l(OJIB:

~s(r)f(r,

,g', t)dVdQ.'J1tg(r,

gg'jMl.

(10.11)

HHTe'npaJI

(10.9)

nonysaerca

arore

cyM,MHpOBa!HHH

B'KJIa,l(OB (10.11)

BceM

3JIeMeHTaM

OO'beMa

:no

scex

ny'lJKa'M

(dQ')o'.

Haxoaeu,

,npH6bIJIb

,l(aHHbIH

Uy'qOK

MO>KeT6hITb

oriycnoaneaa

<pyHKuHoHHpoBaHHeM

H'CTOlJHHKOB

HeHTpoHoB

nJIOTHOCTblO

(r,

)

(aanpauep,

peaxuaea

AeJIeHHSI

snep

ypana).

3TOT

B'KJIa.n: B aaur ny-

lIOK

Bblpa3HTcSI

HHTerpaJIOM

JiQM

Jq(r,

t)dV.

COCTaBHM

Tenepb

'OaJIaHC

lJIeHTpOHoB

TaK,

KaK

'MhI '3TO

..nenaJlH

npa

BbI'BO,lle

yp.a'BHeHHH

,l(H;<pepY3HH:

AQJ

(n(r,

t+Jit)-

n(r,

t»)dV

=A~

too<p

(r,

t)dS+

(r)

(r,

~')

(r,

0'.

t) dQ' - E(r)<p(r, c. t) +q

(r,

o.t)

]dV}.

(10.12)

ECJIH

paseacrae

(10.12)

nepesecra

noaepxaocrasrf

aarerpan

OO'beMHbIHUO

rpopxyne I'aycca -

Ocrporpaztcxoro,

nO,l(eJIHTb

ooe

qaCTH paseacraa

A~t

R nepeara K

npeAeJlY

DpR

At~,

.nOJIyqHM

S[~~

+oV<p+E<p-q-

SEsg(r,

oo')<p(r,

0',

t)dQ']dV=O~

113

n'pOH3BOJIbHOCTH'

o'O'beMa

V 'B ·nOCJIe,l('HeM

pasencrae

cnenyer

IHHTe-

pecyioutee nac

ypasaeaae,

aasusaeaoe

ypaBHeHueM

nepenoca,

ana

KUHeTu~ecKuMypa8HeHueMBoAbq~aHa:

iJ'f(r'ot

t) +oV<p(r,

t)+E(r)<p(r,

t):...-

=SE

(r)

(r,

00')

(r,

0',

t) dQ'

+q(r,

t).

(10.13)

YpaBHeHHe

(10.13),

0YAyqH

,nH<}>4»epeHllHaJIhHhlM

npocrpaa-

CTBeHHOH

BpeMeHHoH

nepeMeH'HbI'M,

Tpe6yeT

3aAaHHSI

Ha'lJaJIb'HOrO

rpaHHlIHbIX

yCJIOBHH.

llOCKOJIbKy

aanaaae

HaqaJIbHOrO

yCJIOBHSI ,l(JISI

ypaBHeHHH

(10.13),

xax

)(JIH

,lI.H<P<pY3HoHHoro

ypaBHeHHSI,'

HBJISleTCSI

Tpa,nH'UHoHHbIM

C TOII'KH

3peHHH

ypaB'HeHHH

'MaTeMaTHqecKoH

$'H3HKH,

MbI

OOCYAHM

TOJIbKO

nocraaosxy

rpaaasasrx

yCJIOBHit

PaCCMOTpHM

caasana

rpaHHUY

pas-

neaa nsyx cpen,

soropyio

npenrroaara-

KyCOqHO-rJIa,llKoH.

Buoepea

HB.

rpa-

uaue pasnena

npOHSSOJIbHyIO

DJlOma.llKy

S H

aaKJIlOqHM

«TOHKHH»

060eM

(nonepesaaa paspes

3TOH

KapTHHbI

cxe-

MaTHqeCKH

aaoopaacea

pHC.

14) .

ECJIH

Tenepb

ypasaeaae aefrrponsoro

6aJlaHCa

(10.12)

CqHTaTb

3anHcaHHbIM

PRe.

14. Flonepenaua paapes

AJISI

Hs6pa:HHoro

o6'heMa, a

aarex

«CTH-

3J1eMeHTapHoro

06'beMa,

OXBaTIi-

HyTb»

3TOT

001»eM

nOBepxHOCTH

saiomero

3J1eMeHT rpaaams

pas-

(Sl~,

Sr-+-S,

V--+O), TO Bee

000-

neaa

llByX.

cpea

eMBhIe

aarerpanu

paseacrse

(10.12)

otipararca

ayns,

norouy

06paTHTcSI

UYJlb

nO'BepXHOCTHblH

HR-

rerpan:

limSa01f(r.

t)dS+lim

SaO,f(r

a,t)dS=

f[ao1f(r+o,

1)+

Sl-t>S SI

S2~S

+gotep(r-O,

t)]dS=O.

B CHJlY

npOH3BOJIbHOCTH

nJIOma,nKH

S -H3 noayaeauoro

paaencraa

cnenyer

paBeHcTBO

HyJlIO

nonuarerpansaoro

BbIpaLKeHHSI.

0003HaQaSl

01 sepes n H

yqHTbIBaSl,

nrc

02=-01,

nOJIyqaeM

OKOHQaTeJIbHO

(10.14)

Pa3YMeeTCSI,

3TO

ycnoaae

M01KHO

nonynars

OCHOBe

epOPMYJIbI

(10.3).

3 a x e q a H

-H

e. B paBeHCTBe (ilOJ'4)

o6e

1JaCTH conepacer O()IIl'HH MHO>KHTeJIb

On. Onaaxo coxpamenae

paseacraa

Ha3TOT

M'H01KHTe.JIb MOJKeT lJIlPHBecT'HK

HelKeJIa-

TeJlbHhlM'DOCJIeJlCTBHHM. Hanpaaep,

BOMHOrHX

npHOnHJKeHHhlX

KeTO,Ztax HCnOJIb3yeTcH

caxo YCJIOBHe (10.1'4), a

pasaoro

pona

JfHTel'panbHhle

COOTHomeHHSI,

nOJIyqaeMhIe

Hero.

B 3THX

caysaax

'IIO,lJ.hlHTerpaJIbllhlH 'BeCOBOH MiIO}KHTeJIb On

aecer

Ba1KHYIO

aarpyaxy

Hero

OTcyTCTBHe

.MozeT

npeaparars

MeTO,Zl

-B

HeKOppeKTHhlH

(ca. saxesa-

HHe a KOHue § 12)-. Koraa JKe coxpameaae aa MH01KHTeJIb On

-onpasnaao,

oYlleM

3TO J(eJIaTb.·

,llauHoe

sauesaaae B

paBHOH

Mepe

OTHOCBTCSI

H·

K CJIe.zlYJOmeMY pasea-

crsy (10.15). .

ECJIU

cpena,

npeJ(!CTaBJIHIOIUaH

COOOH

asnryxnoe

TeJIO,

rpaH:HQHT

BaKYyMoM,

TO (.B cay-rae OTCyTCTBHSI B

'B8KYYMe

BHeIIIHHX

HCTOlfHII-

KOB)

HeHTpOHbI

iH3 nycrorts B

cpeny

nocrynaior.

PaCCY)K,naH

KaK

H ssnue,

JIerKO

nonysaex

ycnosne na

rpaaaue

BaKy.yMOM:

oncp

t) "PH

(10.15)

rne e,nHHH'lJ'HaH

aopxane

.npeztnonaraerca

HarrpaBJIeHHOH

saxyyx.

B aaxmoneaae

,paCC'MOTpItM

HaH'OOJlee

-Ba>KHhIe

nacraue

<POpMbI

ypaBHeHHH

(10.13),

OOYCJIOBJleHHble

BbIOOPOM

TOHHJIH

H~OM

CHCTeMbI

K90PAlfHaT

JlJISl

.npocrpaacraeaaof

nepeaeanoa

'f.

OT~eTHM

cpasy

1Ke

npa

3TOM,

xro

aexrop

Han-paBJIeHHH

JJ;BH>KeHHH

rHeHTpOHOB

MbI

6y-

,neM

scerna

CBH3hlBaTb

HeKOTOpOH

ctpepasecxoa

CH~TeM.oji

KOOp,nH-

HaT

1. DpHMoyroJlbHaH

CHCTeMa

KooPAHHaT. B

3TOM

cnysae

BeKTopO

M01KHO onpenenars

yrJI8MH9

"1',

OTCIIHTQlBaeMhlMH,

aanpaaep,

ocea Z H X.

nJIOTHOCTh

~nOTOKa

HeHTpoHoB

aecranaoaapaoa

cJlY-

nae

OKaShIBaeTCB,

TaKHlM

oopaaox,

<pyH,KUHeii

meCTH

KOo'p,nHHaT:

~~(x,y,z,8,.,

t:).

YIIHThIBaH,qTO

sin 0cos

i +sin 0sin

j +cos 0k;

\7~=.E!.

+.E!.

0)(,

iJy

iJz'

nepenaurex

ypasaeaae

(10.13) B

BHAe

+sinO

(cosljI::

+sinljl

)+coso

+~ep=

211:

11:

=,1:.

5dljl' Sg (x, y, z,

PO.)

(x. ,y, z, 0',

1jI',

t) sin 0'

dO'

o 0

+q(x,

y, z, 6,

<JI,

t), (10.16)

1£.

gO'

cos 0cos 6' +sin 0sin 6' cos

(tJ1-

~/).

YpaB'HeH'He

(10.16)

CYIUecTBeHHO

ynpomaerca, eCJIH npoueccu

nepe-

aoca

H3YllalOTcH

6ecKoHellHoif

,l(ByM

HanpaBJIeHHHM

(aanpaaep,

no x H y)

nJIaCTHHe,

xorna

1::.,

g, lj

3aBHCHT

xoopnaaarx

H y

(nJIOCKona·p·aJIJIeJIbHhIH

caysaa).

Ilpa

3TOM

xame scero

<pyHKUHH

HCTOIlHHKa

3a'BHCHT

a3HMyTa

11'.

OqeBH)I.HO,

uro npa

3THX

npe.n-

nOJlOJKeHHHX

HCKOMaH

epYH'KUHR

3aBHCHT

OT Z, 8 H t;

E,cJIH

euie

orpaHHIIHTbCH

cTaUHoHapHbIM

caysaeu,

TO ypasaeaae (10.16)

3aU'H-

lIIeTCR

KaK

cosO

+~ep=~s

jdljl'

g(z,

PO.)

(z, 0') sin

O'dO'

+q(z,

0).

nonyqeHaoM

ypaBHeHHH

npoHHTerpHpyeM

H3BecTHyro

3apaHee

epyHKUHIO

g(z,

110)

nepeanex

nepexeaaof

J..t=cos8.

OKOHlIaTeJlb-

H,MeeM

: +

~ep

(z,

po)

5g(z,

po,

po')

(z,

po')

dpo'

+q (z,

po),

-1

(10.17)

(10.18)

211:

g(z,

po,

po')

= J

g(z,

po.)dljl'.

* B

npHJIOEeRHH

B JlaeTCSI

o6maSl

4»opMa

sanaca

Y'paBHeB'HSI nepeaoca B npo-

H3BOJlbHOH

~pHBOJIBHeilHO,H

OpTOTOHaJlbHOH CHCTeMe KOOpJl!BHaT, a Haii.lleHHh1le

RHEe

npeJlCTaBJleHHSI ypaBBeB'HSI

:~(lO.13)

AJISI

Tpex

BapHaHTOB

H;aHOOJlee

ynoTpeOHTeJlbHhlX

xoopnaaarmsx

CHCTeM

cIl0pl4aJlbHo

BhlTeKalOT

B3 sroro

npBJlOlKeBHSI

KaK

sacraae

CJIyqaH.

~"cnOBHe

(10.14)

rpaaaue

paaaena

nayx

cpea

Z H

npaaer

8HJ(

P,f(H - 0,

p,)~

p,CfI

+O,p.).

(10.19)

COOTBeTCTBeHHO

rpaaane

B8KyyMOM

z=Ho, eCJID,

nanpaxep,

sa-

KyyM

npOCTHpaeTCH

3TOH

rpaanusr

nOJIO}KHTeJIbHOM

nanpaane-

HHH

O'CH

(10.15)

nonysaex

p.ep

(H',

p.)

= 0 npa

II-

(10.20)

(10.22)

MHOrOqHCJIeHHhlX

TeOpeTHqeCKHX

HCCJle,nOBaHHSlX

nJIOCKOnapaJI-

JIeJIbHYIO

sanasy

qaCT9

H3YQaIOT

npa

H30T,ponHOM

pacceaaaa

Heii~o

HOB 'B

Jla60paTOpHOH

CHCTeMe

KOOp,llHHaT.

H30T·pOn'HOe

paccesnae

osaasaer

CJIeAYIOIllHH

'BHA BepOSlTHOCTH paCC.eSlHHSI

P(r,

00',

Ag):

P(r,

00',

L\Q)=AQ/411:. (10.21)

113

Q>OPMYJIbI (10.7) c

yserox

(1'0.21) HaXO,lJ.HM

g(r,

00')=

I/41t.

orxyna

,zvm

<PYHKIUIH i (10.18)

HMeeM

g(z, p"

p.')=

1/2.

TaKHlM

o'6pa30M,

ypasaeaae

(10.17) B

cnysae

aaorponaoro

pacces-

HHSI

npHHHMaeT

BH,l{

: +Eep(z,

po)=

ep(z,

po')dpo'

+q(z,

po).

-1

(10.23)

UHJIHHAPHQeCKaH

CHCTeMa

KooPAHHaT

OKa3hIsaeTCSI

ocooeaao

Y'u'06HOH, eCJIH 3a,U.aQ'! uusunopuvecxu-cuxeetpuiuca«.

H'MeHHo

3TOT

CJIyqaH

MbI

paccuorpaa

CeHqac.

Be-KTOp

MO}KHOon·pe,U.eJISlTb

yrnaxa

H H

11','

nepBbIH

'KOTOPhlX

OTCtIH-

ThlBaeTCSI

OC'H

nHJIHHApa,

BTOpOHHB-

J1HeTCSI yrJloM

Me>K,lly

npoexnaea 0

nJIO-

CKOCTb

nOnepeqHOrO

CeqeHHSI

IlHJIHHJlpa

JIyqOM

3TOH

nJIOCKOCTH,

npOXOJlSlmHM

qe-

pes

OCb

UHJIHHApa

TOq'Ky

M MeCTOHaXO>K-

.neHHS!

HeHTpOHa

(pHC.

15).

Orpaaanaaaacs

cpaay

CTaUHOHapHbIM

CJIyqaeM,

yoe>KJlaeM-

CR,

qTO

DJIOTHOCTb

nOTOKa

HeHTpoHoB

3a-

BHCHT OT

qeTbIpeX

nepeaeaaux:

<p=

=cp(p,

Z, 0,

¢).

OTMeTHM

.reneps

CJIeJlYIOIllee

Ba1KHOe

oOCTOSlTeJIbCTBO: eCJlH

4lHKcapoBaHbI

KOOP-

JI.HHaTbI 6 H 'i'

nexropa

TO 3TO

BOBce

PRC.

15.

KOOPJlBHaThI

JlJlSl

osaasaer,

qTO

sexrop Q coxpanser CBOIO onRcaHRSI

nOJlO1KeHRSI

HeHTpo-

H HanpaBJleHRSI

ero

.H.BH1Ke-

0pHeHTaIlHIO

.npOCTpaBCTBe

[ecJIH

nepe-

HRSl B UBJlHBJlPHqeCKH-CHM-

KemaTb

TOqKY ,M(p,

(i),

BlI.OJIb.OKPY>KHO.-

MeTPUOhIX

sanasax

CTH.

p~nst,

z=const,

TO npa

4l'uKCHpoBaaHhlX

3H8'QeHHSlx6

''''

BeK-

TOp

OYAeT

MeBJITh

CBOe

H8·npaBJIeHHel],

B S80(S0POT, eCJIH

BeKTOp

COXpaHBeT

npOCTp~BCTBe

CBO~

aanpasneaae,

onpeneamouras

ero

cOBoKynHOCTb

KOO'P.llH'H8T

(to,

1J')

H3MeHHeTCH

[yrQJI

eXOTH H

OCT8-

eTCH

nOCTO.sfH;HbIM,

11'

MeHSleTCH

npa nepexemeaaa'rosxa M (p,Cll, z)

B,ClOJIb,

aanpaxep,

TOM 1Ke

OKpy1KHOCTH

p=const,

~=const].

Tlpa 'BbIBo',n:e.

y·paBHeUHH

.nepeaoca (10.13) nonaranocs,

tITq

aa-

rrpaanenae

nynxa

npOH3BOJIbHO,

$H'KCHpO-BaHo.

CHJIy

c,n:eJIaHHOrO

BY'We

3a~MeqaHHg

'.9TO

osaasaer,

'liTO B

BhlopaaHbIX

xoopnanarax

one-

parop

V B

ypaBHeHH'H

(10.13)

,neiicTByer

TOJIbKO

xoopnaaaria

p H

Z'$YHKUHH

(()

('I>,

'2,

la,

1J'), HO

KOOp,llHHaTy

1J',

3aBHCHmylO

Ol. H

Cn OJIb3YH asrpaaceaae ,l(JlSI

rpanaenra

UHJIHH,llP'HqecKOH

CHCTe-

xoopnanar

[10]: .

U-

-.Op

+ p

iJz

aanacuaaex

iJ<p

, 1

\7ep==ape

dm em -+

•

113

reOMeT·pHtIeCKHX

COOOpa1Ke'HHH

anexearapao

YCTaHaBJIHBaeTCH~

qTO

d",,=~dCJ).

ECJIH

erne

ysecrs

pasencrso

= sin 8coste

sin.B

sin

~~m

co'S

nOJIYllHM

o'\7tp=sin0

(cos!J'

: -

Si~'"

-t-)+cosO

Ta'KHrM

o6pa30M,

crauaoaapaoe

ypaaaenae

(10.13)

npnxer

BH,lt

2'1:

11:

sinO

(COSIjI

.'-si;+::

)+cosO

+Etp=E

d!J"

Sg{P.

z,p..)X

o 0

Xep(p, z,

·B',

tJI')

sin8'd8'

+q.

(10.24)

I'paaasmse

YCJIOBHH (10.14) H (10.15)

'oe3

TPYJ(a

'MO}l{HO

sana-

cars

UBJIHHJ(·pHtIeCKHXKoo,p,nHHaTax,

eCJIH

HCnOJIb30BaTb

npasenes-

Hoe

:BbIpa}l{eHHe

AJIH

0 H

ysecrs,

qTO

KOHueHTpHlIeCKHX

rpaaauax

~HJIH'H,llpa

n-e

a 11a

nepneanaxyaapasrx

K OCB uamranpa

IIJIOCKHX

rpaaauax

n=±e

•

C4JepHq~CKaJl

CHCTeMa KOOPAHHaT

HaXO,l(HT·

eCTeCTBeHHoe

npn-

MeHeHHe

J(JIH ctpepusecxu-cuwuerpuwcux saoa«. B

3TOM

qaCTHOM

CJIY-

xae

ooe

CHCTeMbI

'KOOp,l(.HHaT

-"H

Ta,

lITO

'CJIyLKHT ,nJIH

perHCTpaUH'H

TOtIKH

npOCTpaHCTBe,

aropaa,

KOTOpylO

MbI

BbI'OHpaeM

,llJIH

OIIHca~

HHH

Han·paBJIeHHH

,llBHlKeHHH

HeH~pOHa,

- ctpepasecxae.

ECJIH

nepsaa

3THX

CIHCTeM

<!>HKcHpyeTcH

'13

'npOCTpaHCTBe,

BTOpylO,

Ka'K

B cay-rae

U'HJIHH,l(PHqecKOM

reOMeTpHH,

Y,l(06'HO

MbICJIHTb

npHBH3a

'HHOH'

nepesreaaoa

TOqKe.

H3YlIaeMOH

c4>'epHqeCKH~CH'MMeTpatIHOH

cpe-

,l(hI.

pa .9TOM

illHPOTHbIH

yrOJI

BeKTOpa·

npHHH~O:

OTCtIHTbI,BaTl>

npononzceaaa

panayc-aexropa

zoaxa

M',

rrpoaenermoro H3

uearpa

(10.25)

o nepsoa

KOO'PAHHaTHOH

CHCTeMbI.

Onacaaaas

reoaerpasecxaa

carya-

nBSI

CXeMaTHqeCKH

npe,llCTaBJleHaHa

pac, 16 B

sane

ceseaaa

npo-

cTpaHcTB8

nJIOCKOCTblO,

npoxonsutea sepea panayc-aexrop

TOllKH

H·BeKTOp

.B 'CHJIy

c4>epHQeCKOH

cHMMeTplmMOJK'HO

yrsepacaars,

nro

B ypas-

'BeHHB

(10.15)

nJIOTHOCTb

HCTOllHHKOB

c, a

CJle,nOBaTeJIb'HO,

-H

HCKO-

MaH

nnOTHOCTb

nOTOKa

HeHTpoHoB

KaK

$YHKUHH

BeKTopHoro

apry~

MeHTa

MorYT

saaacers

TOJIbKO OT

lliHPOTbI

6. B

cranaoaapaov

peJKHMe,

KOTOpblM

MbI

ceiillac

OrpaHHQHMCH,

nJIOTHOCTb

nOTOKa

HeHTpo-

BOB

6y,lleT,

·T8KH.M

oepaaox,

c1JYHKUH-

eft

TOJlbKO

,n:Byx

nepeMeHHHX

r H 6. .

,llJISI

3anHCH

ypaBHeHHSI

(10.13) B

C~epHqeCKOHCHcTeMe

Koop,nHHaT

OCHOB-

HylO

(XOTSI H

JIerKO

npeO,llOJlHMYIO)

TPY.n-

HOCTb

npencraanaer

caaraexoe

Vq:>.

3Ta

TpYAHOCTb,

KaK

H B

UHJIHH.npHlJe-

CKOD

reOMeTpHH,

CBSl3aHa

TeM,

liTO

npa

<pHKCHpOBaHHOH

opHeHTaUHH

BeKTopa

COOTBeTCTBY:~lI(aSl

eMY

nrapora

3aBH-

CHT OT

nOJI01KeHHH

TOq·KH M B

'npOCTpaH-·

CTBe

(ex.

pHC.

16).

CJIe,noBaTeJIbHO,

one-

parop

«ooH3aH»

.neHcTBoBaTb

ooe

ne-

peueaasre

:6.

YlIeToM

sroro

saxe-

qaH~H

ssrpasceaae AJISI

rpanaenra

V-q:>

HaH~JIee

H35IIUHO

M01KHO

nOJIYllHTb

cne-

.nYIOIUHM

otipasox.

nOCKOJIbKY.

JIeBOH

PRC. 16.· KOOPJlBB8ThI

.nJISIODH-

qaCTH

ypaBHeHHB

(10.13)

BeKTOp

CaBHJI DOJI01KeBRJI

HeHTpOHa

OTHomeHHIO

oneparopy

V epHKCHpO-

U8npaBJIeHHSI

·ero .llBHJKeHHJI B

BaH,

CBSlJKeM

HMeHHO

3THM

BeKTopOM

cepePHqeCKH-c~~:eTPHqHhIX

sana-

cc!»epHqecKYIO

CHCTeMY

KOOp.nHHaT

,llJISI

n'pOCTpaHCTBeHHOH:

TOqKH

(OM.

pRC.

16). B

3TQM

cay-rae

mapornaa

KOopJJ:H'HaTa 0

TOllKR

(r)

==M

(r,

6y,neT

TO}l{,neCTBeHHO

COB'na-

nars

apryuearox

6 B nJIOTHOCTH

nOTOKa

Hei%Tpo'HoB

q:>(r,

8).

3TO oaaaxaer, xro

<l>YHKU'HIO (()(r, 6)

MO}l{HO

reneps cxorpe'rs

KaK

tPYHKUHIO

TOllKH

(1',

npa

OTCyTCTBHH

3aBHCH:MOCTH

aaaxyram-aoro

yrna

ID.

BbI'6paHHOH

TaKHM

o6pa30M

CHCTeMe

xoopnanar,

onpenene-

HHIO,

MOlKHO

sanacars

[10]:

Vep=are,+,aaeo

tITo6bI

BbIHHCJIMTb

npoaaseneaae Q

V{j),

nenecooopaaao

sexrop

pas-

JIOlKHTb

TeM

JKe

oprax

erH

eo' IB

KOTOphIX

sarracaa

rpanaenr

V:<p.

113

pac. 16 cnenyer,

nro

===

e.cos0-

eosin

(10.26)

YIMHOJKaH CKaJISIpHO (10.26)

(10.25),

nOJIyqae.M

o\lm==

cos

8.E!.-

sin

0'_

Y T

08·

4-301