Neutron physics

•

-

[ "E. .r..--.=.0;:;;;.- 1 ---+ E E,{I - - ' . )

",1_3 .. E,(I-ooeO) 1- •

I

•

~

(15.2.26)

l

fOf 1(1. M EJI, -O, II" again a pproaches th e diHuaion length definM in Eq . (5.2.15); Eql. (5.2.24 ) l nd (5 .2.26) ftllnain the l&IDe. In particular, one comel to th e Nome concluaiona regarding the validity of elementary diffuaiOD theory &I in the C&lI6 of iIotropic IC&ttering. 6.2.3. The Neuwon

nux In the Vlclnlty or a Poln' Sonree

We next canm t'l f an isotropio point IOU I'<.'e in an infinite medium end in · veatigate how th e neutron flux behaV6fl at Imall di.tanoee from th e lOurce. For ilOtropic IClattering in th e laboratory . y. w m. the integral tranaport equation take. the form fP(r) =

f

,-&,1"-"'1

[r, IP(r') + Q ~(r'») -h (p

•

r )i d Y •

(tl.2.27)

Th e integration over the d-funetion ca n immediately be carried out and yieida

.-"" , + f E. lJ>(r') 611(1' , - &'1"- "1. r )' If Y .

(P(r) - Q 611"

(5.2.28)

The filllt term repreeente the contribution to the flux of th e neutronl which arrive It t he poin t r with out having made I oollision. At large di.tanoee from the IOW'O(I. the inteption of the eeeond term will again lead to an uymptotio IOlution which becsuee of the . phoried sym met ry h.. the form I

.-.r

lJ>"", (r)- - .-

with" taken from Eq. (5.2.21). In order to obtain IlOlution for all r, Eq. (6.2.28) muat be IOlved ; tbia can eaaily be d one by I Fourier tranaformltion method. I

ct. Sec.&.U .

.. w. will DOl m.cu.. thiI method of .mation here

(d. • -8., n HoI'l'llUJl'). bat ooJ;y nmmariM the reeultl • They are

~.,

P1.I.ClUK, and

~ (r) -lJ)""""l(') + ~.M (rl

<JlI.. (r )_

1....291

O~ . . .- -. b I. - . -

(6.2.30&)

w..~

I I

r.

p- .~;

"'"

I,' -~

..- r;r;

=

,....

....!l..- ' - &"'1..., r.) .

(6 .2.3Ob )

1 s.s, 1

3 1)

Here • ia a alawly varying fue ctio n which ia of th e order of magnitude of unity ; it ill plotted in Fig. 6.2.3 for lJel'eraJ valu(lII of

E,il; .

u

'."h':!;;-±--t-:"""+ + -l

"t--t--t--t--t--t--t--ht--H , , + --!• ,'--iH"--'!----!--+-+-++ • S I J Ii'' ' ""1.1.L %101"" .. .. r..•u _ I

-

I

~r, ."

" '=.. ' I J . S 7 .r Ii'''....1.1.'. ~
Plotted in Fig. 11.2•• ia the tot.aJ: On .. wen .. both . u.m.ma.mh on the rig ht.. bDd .kle of Eq. (6.2.29) for EJE, .O.l. h can euily " _n that cfIu._, (r) eoo.tribut.el noboMbIr to t.be 0 . . oD..ly ill the 1mrnedlate neighborhood of the -oarce: for larp ~ the U7JIlptotio IOlut.ioa domm.ta. Forth. dm.6n oe '" at which both pan..,. equa11ylup, da. oomplet.e oaloulatioD ti~ .. -

~ ""rJ.I'" • ZiI or

(5.2.32)

Acoordin.II: to t.hiI relatioa, for E.<E•. "' .0.2211;.. On the other hand, the

x..

t.ex&Dr different. for IIVooI' ab.orptioa. U. for e:umple, X. _ 0.9 Le., EJE,-lO. " -11.48 x IO'lIXa . i. e., t.be uymptotio ltate iI Dever reached in

iAmation ill

ally praoetic.l . . ..

'!'be q1l&Dtit, P introduced in Eq. (6.2.30b) baa...peciallignifiean08. To eoe _hat it ia. ~t UI uk for tho ftWDber Z of beuVonl which &nI abllorbod per IoOCODd

~\ 01

toM StatiaaarJ, &r..u'.tndepeedent 'lnMpcn Eq_tbl

froDI the U)'Dlptolio diIItributioa. It.

w

Z-JE.t1J~dV_ iE.t1J..(r)' '''''' dr • • = QP-'/ r, -'" dr-

•

I

(....33)

QfJ·

Since t.h, -.ot&l. number of neutrool aheorbed por 100 mut be equal to th, 8OUf'Oe avength Q. fJ gin- th, fraction of tho Deal.ronl froQl, the ~ which 'IItel' the uympt.ot.io dimibution. If EJE.< I, then u 0In euily be verified,

....

• r.

(

P- l - ' -l;

, ,)

IJE.- OaU tbe nout rona entel' th e uymptotio diatributioll. One can aJao a pply all th_ oonaiderationa to media with weakly ani..otropio _ttering aDd obtain ql1lJ.itl.tinly the a&lDe reeulta. In this ...ay, important ooDe!uioD.II can be draWD. about the validity of oJemoutary diffuaion theory : In an infinite, WMk.Iy abeorbiD&; medium. e1omontary diffwQOIl t.heory app1* beyond a diAance of 1 to 2 moan free path. from an Pouopio point -oaroo. It gi"M not only t.ho nriatioa of the flu" but at.o iu abtoJ.ute 't'aIoo within an error of the order of magnitudo of EJE,. Thia atatem6Qt aIao bolch for arbithriJy lIbapeel Ml'IU'CelI .uch u .urfac:., etc . ...hich can be \hought of .. eeeapeeed of point llOuroet. If we now noti ce th at localized Itrong abeorbel'l, u ".llu.llI'faoo. through whiob n.eutrona can eecapo from tho medium, can be I'Opr-ntod by 111itl.b1y dittribut.odnegatin point 1lO\1I'OM, .... then ... that in any ww.kJ.y abeorbing medium thoh exUta an uymptotio DOUtron field whiOO can be cI-ibed by tho elomo.ntary diffu.ioIl equation beyond a diet&D.ce of 1 to 2 IDOUl free path. from t.e. u

1O\1I'OM 01' boundary ewiaooe. Thu .... han UI1nIred t.be ~n of the 't'&Iidity of tho elementary diffwRoD. equation oxcept in the cue of . patially diatributod ~, which appNl' in Eq. (3.2.18&) u the IOI1l'CO derLlity S (lr) . That the oooe!~ lIta\ed abo"e regarding th o appUoability of elementary diffwQon theory &lao hold wbon the medium oontaina homogeneously dittributod 1lOW'CN can be &bOWD. in tho following way : Let uu uume that .... han found an eDOt IlOlution ' (t' , D ) to tho m.oaport equation in a medium in whiOO there it a homogeneoua llOW'OO diatributioo 81''''. Tben " (t', D ) _P{r , O) _SI'JIIE. it a -lution of tho ~free In.ntpon equation . Eq. (IU .23) bolda for tho flux ~ _ Jr (t'. O )lO in tbe uymptot.lo range i therefore V' ~--' lfI _ _ r; S (3.2.36)

1oeaJi"A"d

..

which beoon188 Eq . (15.2.16 b) in the

limi~EJL',_O. h oidl for 4>-"'. + :. . Thia

proof, ...hich baa boon gi ,.en here for .. . p" tially OOWltant be generalized to Ilowly varying llOuroe denaiti... 1.1..4. N,. &roD l1 u

1lO\1l'Oe

deIllity , can alao

at Ua. 8urbee 01 • 8eaUertDc .Mham

Now thU _ bow in wW range oJemontary diffwQoo theory it nlid, .... mud. innRigato tho boundary oondiUotia which apply at the interface benreon a eoattering medium and a n.cuum 01' .. black .. abeorber. Let u oooaider a medi um

.. which emndt in the z.wrection from %-=0 to z ... eo . nd from - (IO ta

1.

+00 in th e

y. andz-direetiOtll. Let there be. plane aouroe . t %_1'. > Let-the h.u.lpace 1'<0 be empty ; uu. mer.n l lb.' .. Mutton which _vea the medium through t ho pIaae * _0 e&nn~ retwn. If we t&ke E. _ Oand ueumo botropic IIUt~ring. tbe t~port equation beeomM

coa l

• :~ + E.F_{ E. ! J'1in{J'd {J'

•

(6.2.36&)

and mlUt be -olved wit h the boundary conditio n

O<' < n!2 .

F (D , %_0)_0 for We

e&n

euily .peci fy the general form of the uympWtie

E. =O and "-=0. for z>l/E. but < Ze _ hav e ~.M

T .. · ...

0

(5.2.36 b) ~ution ;

ainoe

(ts.2.31)

i.e., where d iI ubitraryl. If the Du ia to be known for &1.1 zo, Eq . (5.2.36. ) muat be eolved with the bouDd&ry condi tion (6.%.38 b). Tbe detailed tlN\uumt of thi. ma them atically demaDding problem of ~ th eory, eeued th e Milne problem. can be found. for eDJDp&e, in the book of DaV18Ol'I ; there the foIlo1ring rNWW anl given :

(5.2.38 . ) (6.2.38 b )

.od

l#l_,=- o.~ {E. (E. z) -O.821l E,er,zol}.

(5.2.380)'

Fig.IU.S . ho_ th e vari&tion of th e flu in the neighborh ood of the l urfaee. The ..ymptotic ,t.ate iI clearly rea.ebed alter .bout one mean free path . The quantity ~ i. called the c:dmpolated clltlpoillJ• •Inoe t be enrapolation of t/J» f .,.aniabe- al- .% --~. Frequently. ~ ia aJao can ed the cnrupolcat'oa~. but thi. J!...iDoorreel. The ennpolaUon IenstJa ia defmed for an ....bit.ruily ahaped

'Wiaee"

1=

(,,::;;:~

(6.2.39)

where the diffehlotiation hi in tbe Dormal clirecl-ion. Here. and. in many practical ~. d and A are identical ; however. we will &leo encounter u llIIptionfl. Th e Milne problem OM al-o be trMted for an .beorbing medium. Fo r the aaymptotic part. of the IMllution {6~ ·401 ·Tbepn«aJ~"~"

•n. end . . . fill. •

lz+l),"'_oIlooet&M_~aodIM

. _I .

pHaM.d ; Eq. (U.JSe) .. all pro~ d_touC£nf. wlroWlo _de...w.&om ~ • .-eobIUoa b, __ thaa For &be ~. ~ fII. e.ppeadi.J: Ill. v-iMlt

~

..

&po

0.3".

.. hold•• The utrapolated endpoint d i. plotted &II a function of EJE, in Fig . 6.2.6. For E.<X" d II again O.7HKIE, . It ia important to note that in an abeorbing medium thl'l extrapolation length 1 and the e:ltrapot.ted endpoint d ate not the Mme if d ill not < l/x.

In the CMe of 1VMk.Iy ani8otropic ICI.ttering. we obWn the MmO rwul~. Ia n that the te&ttering Cf'()lllI .etion iI everywhen rep laoed by the v.n.pon m:. lleCtion E", ,,,,E,(l - OOlI e.). In puticul&r. t he flItrapolated end point in a weUJy absorbing medium i. given by d_

O.71Oi _ E::!IO& E. (l-;:;}.> Er,'

(5 2 " 1)

. .

Thu in the applicati on of elementary diffIWon theory the boondary OObditJon at. the IJIU'f.oe of .. ncuum or a black u al-orber i. th.t th e neutron Dux vaniah on the extrapolated boundary i If " ia th e outward normal, th en

tl' {f',,+d. n )- O.

"

(5.2 .42)

I

u

~

/

'UOAfJSfJI

F1II.u.a..n._ ........ u.._,ot .pluo \Iou4o.rrot. _ ~ - - .--_

-

-

-

..,.pIOdll eohUoloo.

. . . .:

.....ue.

D:'

...

"''''"' _ ....... . ., ~

Thi . reeult hold. for both plane and mildly euned . urfae:- ; for etrongly curved . •urf_ it ill better to UAe .. boundary condition of the type of Eq. (6.2.39). Data . on th e ertrapolation length for .pberN and eylinden are given in Sea. tI.3.%. h 1. in.tructin to _ what bell_riot for t.be neukon Dux at the fono- fro lll .. nNq . pplieatioD. of e1emeD.tar1 diffualOll theory. We Ran from

bound.,.,. .

the defining equ.Uon for Lbll neutron. CUlI'I!lnt denaity, Eq . (IU .8). which we reeolq ~~

J _J. _J-

(fi.2.4h)

..

into .. oomponent J. in the dUectJon of Lbe datribution aD' and .. compon ent Jin Lbe Opp;*U1 di rection . Here

r :::z21r J F (~. OJ CIO" lin '4' o

•

.,.

J- - -bJ F (~, Il)

CIO' ~Iin'

. 46.

(fi.2.43 bj (5.2.430)

,

.. u aooording to the approxim.~ of , (••

elementary cliffulQoa theory ...

I

I I ". a) -.;;-¢l l .)- ~ . . _

_Ca' 1..

J--

- ,-- er; "7i'"

J-

. (61+

- -,-

1

••

ex,. h

DOW _

I

'

'

Th .e ndatirma aN euct. if the oonditions of applicability of elementary diffuaion theory are fuIfiUod. i.e., ill. our 0Me. if we U'O M9eral mOloD hee pathl from tb e bouDdaty. U ODe aaum. for the lake of argmnent that elementary difhWOD theory hoIda riPt up to the IRU'faoe of the mediwu. \ben one moat take J· (:r ) _O at (fi.2.4.h )

._0

.. the bouDdary CODditioo.. With the help of Eq. (5.2.44) it {oUo... th at

. ,.) )__. - :sE". ' 0.'" - - zt..(,../4i

'

Thu we obtain O.661/Ee. for the nt.rapolation length , which it ideot.ioa1 wiUa the enn.poll.ted endpoint in thia _ . inaloNd of the eJ:act value of O.71040fE" .

U .i. Impro..e4 A,prol1matSollll for &be SoluUon 01the 'l'nnIpon Equation Enet eolutioo. of Lb. tranIpoR equatJoo i, only pouible in .. fe" DUN (e.g., for limple pometry) &nd enn \.beo IOmet.i.m. only at the 001\ of oonaiderabl. matbemt.tie&llabor. w. bn. ~ in the lut eect.ioo that in -..kly ab.ot'hing media the DeuWoa field

CaD.

be d.eribed by elemen&&ly diffuaion theory beyood

..--wn di.t&Doe from ~ ADd bcKuldariM j thia theory it euct in ibe limitina: _E.Jr._O&Dd~t. &Dapprommat.iOD.whicb.beoom.~"11

wone

the larger EJE,ia. 10 maDy pnotioal_. E.JE.
~~ +.2;1. - E•• '-+B.(:r) 1+1 IIIIlH 1 11111 -1 s"+1 ~ + 11+1 ~+ Jj -

':+1 1.11._

1

...

~

""" Ii.

l _ l.2 ... . , N _ l

+ E,F. ,"",E••FJI"

-7'~8~"" .z;, :;--. _ be<Et for \1M appUoMiort oI~~ , . . _ ~ wi&b

Et.- u...~ for u..ditfuaiaG~

GIll

(6.2.48)

~r..iD

• /'+J- __" iI\IM DUDl_ oI ~ 1I'tUab. ".....v.a- DIII' ..... - . d boa bo&h fl.. 8M. 6.1.1.

__ l

ThiI is a . ystem of line&r diHerential equaUoru for the J; (:r) which mut be 80lnd in oombination with appropriate boand&ry oonditi~ ; the fon:ll1l1Mioa. of the bcnmd&ry oondft1orYl ll .. ~o di1ficnIbr of tbe ~llUIthodwhiob . . w:W not eorudder here. S,..A PfW'O%if'MJliml { M aAotl 0/ lX«:rtU OnIi1t4lu i W ICK' S M dW} . In tbiI method the tI'&nAport equation 11 &8a.in eeduced to a .pklm of line&r diffenlntlal equatioM . However, th ie red uction is not aooomp1iabed by n panding , in a IfIm of spherieal harmonica, but rather by replacing th e integration over the angle f} in t he t ranspo rt equ a tion by a summation over a disc rete num ber of ~tiollll . I n orde r to Wlderstand t his method, let us consider in th e interNt of simpli city an iaotropie&lly IIlC&ttering, eouroe.fn!e medium with plallfJ ' ym lM tr)', introduce p = COlI (J . . the direction ..ri.. ble, and repl&oe th e vector Dua F (%, A ) by F I%.p ) =2'l1 F (%, A ). The tran-port. equation then beoomN p

..

~~ +E,F= } E.J

-.

' lz.p' ldp' .

(6.2.47)

If Ineteed of t he eonti nuOUll direction vlUiable p , di8Cretedirectiona ~ , i _ I , .. . , N , &l'fI int rod uced and F l:r. Pi) denoted by Jj(:r) ,the integral equation ClI.n be re pla.ced by t he eYllt.em of differential equatione

~

aF.

1

N

,,:-+E1Ji- "j-E. L

<-. g. F. (z)

(5.2.48)

where the g.. are l uita bly ch~n _ ighu. In WIc..'1 method, the direetkMa P. and the _ ight faeton g.. &l'fI ch<Men 80 \hat the t-\ pollIible approJ:imation to the int.egral is achieYed ria. the OaOM integration method. In
6.3. Thermal Neutronl If one introdUON neutron. into an infinite, non.ablorbing D:lfJdiuD:l, they will rema.inthere an infinitely long timo. Alter a timo, a true equilibrium will.-.riJy be efot&blilbfJd bet~ the neutrona and the thermal motion of the _tWing atoml : the neutron energiel will . .umo a M&:n'I!Ill diatribution with the klD:l per&o ture of the IIlC&ttering medium. In pra.ctiOll, I Uch an ideal ease dOOl no\ OCCUI' ; aU media. ClI.pture neutrona and &l'fI finite . SinQ8 aU practieal .IOtll'OllI emil neulrona with energWi higher th lUl thermal ellfJrgieI , • OIlrtaintime e1&JIlII!II before a eeutecn is "thermalized". Tb u.e there iI a certain probahility that th e neutron il los\ by .bsorption or leakage before ite complete th ermalization ; th.t meanl, howover, that true equilibriuD:l II neve r rea.ohfJd. Tbi. would be eo oven if lhe IOUl'CO oou1d emit. DeU t.ronIo with loll eqwlihrium distribution, booaUM all croaa MCtiona are energy depeDdent. Do_ ver , in a good moderator th&t. is adfic:iently larp and 11.._ 1 ~lN.

PtI,...

.. OIllr weU..I)' ab.orba. the Del1troo. eneru dinribation at;

the eed of lobe d enting_ down ~ 4ppoziJlltdu an equilibriwn diatribution. Si.ooe i\ t. OOIloeptaa11y u.fuI. in UliI leOtion '" .hall Rudy lOme propertiN of AD idMlited thermal Deut.roD field in which the DeUtr'oM h a.. .. true equili bri um diAributlon. The pnMnt IMUlto. ClAn be looked upon .... tint approzimation ; in Cbapter 10, _ eall.tudy the propert.* of thermal ntlutl"ou in more detail. 5.8.1. HarweU DlItribuUon 01 EnergIN Aooording to the Maxwell diAribuUon law , the nwnber of neutron. dn per om' with tloergiell between E and g + d E 11 (6.3.1 )

-

..!!J') U _ ...!...e-./ar·l fI- ~~

.beN . _ J. (8) 48 Ui the

•

•

Vi

(~) deDllity .

Vir

iT

(63 2) ..

1& foUo_ that the . ..rage .nerBYW (6 .3.3)

(6 .3.4)

(6.3.6)

; ~ -lT .

(U.6)

FmJoently we write IT -Ef' I , 80 that ;

~-E2"

When T _293.e ·K (20.• -0), Ef' -J:T _ O.0263e..... and "1'_ 2200 m!teo 'fb4, _..rage 'Velocity ia

(8.3.7)

"._ (....8)

. The di8tribution of the neutron flux i8 given by . ~ (E) "g _ e-.I•• _E~ dE " E'l' E'l'

er

(6.3.9)

ss; ... ... .,

. ("I~ =2t-("'.')' (-"-)· ~

H.~

•

t1J =

J

•

,. (v) "ettl=n1) =

f;,

(6 .3.11)

fltlt'

ill the total flu:I. Fig. 6.3.1 moWl .(8)/75 and cJ)(E)/cJ) aa functiODl of EIEf" It il euy to _ that compared to the density, the flUI ill alwaY' Ihifted to higher energiea. It ill worthwhile to derive lOmegeneral formulaa for UP~0D8 of the type (EIEr)1 over the density and Ow: diatribtltiona. AveragiDg over the deoaity

,,"raging

giV8lJ

"

(6.3.10)

~" "I t\.

11

, • ,

'\ ~

"Ji<

I~ z

""

s

£/~-

t-, • s

J'lc, " 1. Tblllilalnr.u 4Wl(bQU<:a 01. . . . . fIu .... _tJr

•

•

E)'- J . (E) (-!.)' dE=-.!..J z1+1e-' dz "", nl+f) (8'1' ,. HI' Y;; 1'(1) • • while averaging over the DUll: gives •

Herer(ar.) _

,

(6.3.12)

•

, - Sf' )' dE _ J ~+le-·da: _r(I +2) . (E'l'E)'... J~(E)(E • • •

(6.3.13)

f ,.-lr'dH.. the gammafunotion I, Bomlllpeoi.al valuea of the gADlllla

fun"" en ere •r(ll~ JIn; r(ll ~l; r (,)_ JlnJ'; r(') ~l ; r(f) -3 JIn/" 6.S.!. BeaeUOD Bat.ee In Thermal. Neotron l1elQ

H & IJUbetance with the crose eect.iOD X (E) iI plaoed in .. thermal neutron field, the number of reactiOtll per em' and per 100 ill given by

•

Is.s,")

.~J E (E)
• Thia reaction rate oa.n be written .. the product of the total neutron Dux

~ and an average oroeeeeetlcn Xi the latter i.I obtained from. an average over the eDergy distribution of the Dux:

" ""X~; "1:-

J• I (E) (:,.) ,-./•• ~~ .

•

In the important Ipeoial cue of • l/-.beorber. I. (B)-E.(Br )

({I.UG)

V¥.

and

• 'I'be pmma f\moUoqI. kb1aWed IDJ4DQoBIID.Ul«m: TabJelofmper J'aDodoaI. Bwttprt: TeubDer lMO.

"

100

it follo. . from Eq. (5.3 .13) tb.t

" ~I r., (i'rl.

6001 -

(6.3. 18)

The . . . . . . CllOeI Mation of .. I/_ t.orber i. tho am&1ler by .. factor of }'if2 0 .886 than \be m. ..,tion at lbe eoergy EY': t.hiI h-.. to be '-'ken into aooount. _ bon the cro. -eet.iou gino in '-hIM are appli ed to the rm al DOUUODI . 1\ it flUy to _ frual Eq. (6.3.8) that Uti. aYel'a£O em. 8llcl.ion it equal to the croeeetiOQ at the

."nap neutron eeIocity It_ ~ $Of': t.hld ,, -E.4> -E. (i )Jt ·"

(:1.3.17)

From thia it folio. . tbat for .. ginn neutron de naity • • th e reaction ra te of .. l /" .ab.orber d..- not depend on the ene rgy E,. ""iT I . Oocuionally. one ~ad.t worlu in which th o neutron nUll: ie not given, .. it I. here, by IP_I" hut rather by 'UI ; the reaction rate of .. l ftl-abllorber i, th en obt.a.ined by uaing the croee Il&Ction eceeeeponding to thia velocity. ThUll tI can be arbit.rarily oho.en ; in the IO-C&lled WMteott convontion (el. Chap. 12), for nample. tI ia &.I••r2200 mIMe, i .e. , the m08t probable velocit y .t .. temperature of 293.6 -K. enn when the actu al &empeRtUnl T i. different from th i. n lue. Th. l'NCltion rate of .. l/.... t-orber i, then u .E.{tlJ . 10 that t he orose eeetion . Ubi, velocity, which i, what illpecified in moat ta blel, C&I\ be ueed direc\1y. U th e e~OIl. fI' .... R' , whichil uua! here, iI used for th e fiuJ:, \h en for any temperature it follo_ that

tI._

r..- J; V~:! X. {.'> , H a c:lr.- 8eCli0ll. doea not follow the 11v-law, _ defined .. foUo_: Let ,,=E."', wi\h X. ""'P(T) Jf

(6.3.18)

frequently use the g-faeto r,

v=~.i X. (tt. ).

(3.3.19)

From thil equation, it fono_ tbat • ~T&) , (T) - - - -1- JW E.(B) (Tlir

'fie

m .e

z:.(~

•

n

c-·

,& . IU • ........ aT

For a l/_beorber, g lT) -I, ... oau euily be . erified. U one m.tMd of the OUll:. the ftIaCltioo rate iI given by p - .... E.(".>,(T).

UIIeI

(5.3 .20)

the denaity (6 .3.21)

The ll'.fador e&zl be obtained ... a function of tem perature, if Eq. (6.3.20) i, integrated ppme..uy 01' numerically Ofti' the experim ental . ariation of \he croee Iectoion at low eoergieI.. H the croee eecti.on can be re~nted by the JhoeitWJgnI!II' formula in t!WI eneru range, the integRtion e&n alao be carried out analytically. W~ ginl detailed tablel of Il'(TI for many element-. I Enala lbe _ 01 aD arbltrary.~R{B). ta • .-clioa rat.e IKI\ &ep.r.d GO lbe .peatnIlD; f.. it i.I a1••1. thI_ that

¥-

f II(~'~""_K f .(~4&'_.I"II.

of .. I/....bmrb.r doM

101

6.3.3. The Elementary Dittuslon EquaUon tor Thermal NeuUon. U the energy distribution of the neutrons in .. medium dcee not depend OD poeition , it. is alwaya possible to deeceibe the difluaion process by an energy_ independent one.group equation in which th e ceoea aectiou are suitable "veragee over the epectrel distribution. In particular, we e&n derive an element&ry mi. MOD equation for thermal neutrons. For thiB purpcee, we begin with the energy., epece-, and angle-dependent tr&n8port Eq. (5.1.17)• ..00 IlIIlIUDl6 for IimpUcity plene eymmetry, isotropic eourcea, and isotropic scattering in the laboratory system: cos(}

aF{~~. K) ~

I

+ L'(E )F (x , n . E )

.. ""

JJI. IE' ......

E )}'(z,

n', E') Bin {}' dO' dE' + .~

(6.3.22 )

8 (z, E) .

eo The further treatment of Eq. (5.3.22) will 00 exa.ctly like that of the energyindependent equation in Sec.6.2.1. Aga.in we 86t I

3

F (z, 0 , E )= .-; ~ (z, E)+ -..- ; J(z. E) If _ substitute thi8 in Eq. (5.3.22) and integrate 00II (J dD we obtain

0000

008

(6.3.23)

{J.

over dD and cace over

~

lIJ1~; E) + E. lP{%. E) _S(z, E l +

f I, (E ' _8)4'(z,

E') dE'

(6.3.Ua)

o

and

.!. it ~itz (z, E) 3

+ E.(E )J( z EI -O t

,

(6.3.Ubl

•

By combination of both tbeee Clquatiol1ll, thClnl ari_ an energy-dependent dif· fue.ion equation

- -3E~jT VI (1) (z, E) +Et~)4'(z, E ) = 8 (z , EH

I

f I, (E ' _&)4'(z, E') dE'.

(6.3.26)

o This equation is still free from special _umptiODS concerning the spectrum. In subeequent chaptere, we will frequently use it in various fOl1ll8 to C6lculate the spectrum in epece-dependent alowing-down and thermaliza.tion problems. We now make the &8lIumption that in a weakly abeorbing medium the flul: 4)( z, E ) can-alwaya be written in the form E t;P(z, E) = (1) (z) (iT)1

e-·'u·.

(G.3.26)

If we 8Ubatitute this in Eq . (6.3.26), we find that an integratioD over all ClDClrgieel yiCllds - D~(1)(zl f. (1)(z)= 8 (z) . (4.3.27)

+

~

H$l'IIIIM hu beeD made ohhllfac$ t.hM I E,(B"_K)48 _E.(r) IoIld Ienn <»nWning t.bb~terinr _ -moo.. 0 1

t.h1lll

~

\be

-

'"

Bore 8 (zl _[8(%,&)/''',

•

-• -

- J('U ' ) ' -GIl)" 1 - . It!' •• i f'

(3.3.Z8)

~ - f (II~ )r.(8)e -.II'f' ::.

(6 .3.29)

D_

•

TblY fobe effeotl..... ditflaaioll ~.ten D -.nd

r.

for thermal Deutrorla are found. by a..,...m, tM eDelK1-depeodeodent parameten o....r the Ma:nrell eDergy

diatri1nrtioa..

Izl UMI e-e of uU8otlopio ~ io. whi eb the an1ft£' COlIine of the teat-

teriD& aaP depead. oaf,. weakly OD lbe

,.. !(iT) •

eM!U. _

haTe

(6.3.30)

--1-

J~(.;.-.Itf' : :

when A" II \he thermal oeatroa tranlport ~ thea Jiftll by

IDNa

free path. The diffUIioo ~

II _ -"'A" ,-

(6.3.3))

y- ~

J,. -11J:. -t.be 1De&n frM path for ab-orptJ.oDI. Fl'equeratly. t.t. diffu,Goa equt;ioD .. writtea DOt in te.m. of lfl bel, in term, of ,,-lfl/i; in I«ma of" it. t.

(with

(6 .3.32 )

or wilob the

di//wio- ClOtWIkI'" D.=Di _ -~!- V(em l/leC) and tbe av erage lifetime

.pn.t. abeorptiOll " _l"~

D.V- ,,-a/4 - - S.

(6.3.33)

Cb.ptel' &: BererenMi

_peai&Il,.,

QeD....

AIuula. E. l ~ st., UI-ID. ~ X.lI.. PoD. BOftXUlI. aDd O.~ : Int.rodl>Ot4oa to . . TheorJ of Nev.troa. ~ .oL L 1.0. A.1..uIc.. N. ll.: I - AJ&mo- BoIeo.tUio ~ INS. D4Tl8OII. a t N TraMpgn n.ory. ~I a.z-doa. Pr- IN7. &1,aro•• I .. ., .. I nMrt. CiDkiqae de N...v-Ra. . . . ParioI : ~Uai~ . " - I M l. WanDG, A. II.. MIll K. P. WIG".; n. nyaiall 1'beor7 of NMIvo.. Clata RM«.on. 0U0ar01Wc.ro UloIftnItJ ~ 11158. _P.OiaUY o...p. IX. n-pon 'l"beoey MId00· fllIiaD. of )L .. _pUe NeaVorlL

.os

.......

I

Bona. W. : Z. Pb,.u. Il8. 601 IINl)1 U .. M (l kI) CTJ-pon~). ~ ~ J . , ft,... a.-r. 71,. 6&f, (1N 7). x..&. c., Ph ya. BeY. 71. &8 (JM7). K.o.aus. R . E. : Phya. Be... 71. 47 (IM7).

Tr.t -t of \be XibIe

h:>b1llm. G., Nld W. BartI&L: Ph,.. Ron-. 71, MO ( IN T). WKIlfUao. A. J(.: AECl),UOG ( l~) (Tra.-px\ 'Theory Uld the Diffuaiocl. Eq...~). Ko~. W. : ORNI,2334, 23M (IN7) ('JlMI P,.Ket.bod). ~JI'. B. G. : LA.letl (1966) (1'IMo BN"Metbod). Wwa. G. C.: Z. Ph)"lik III, 102 (1963) IWKDt', Method ). Ca n-. &Ild O. Rowu,,~ : J . NuoL E-v AA; 1 (1901) (Variou AppNaima" Met.bo4lI for the TMatment of 1M 8~. EDerv·lDdef-I-' orr-pan EquAt.Ioa.). W&n'OOTl'. C. H. : AECL-llOl li MO) Ie . - 8ectiou for 'l"bermal N....u-.). ~

B."

c..

6. Applications of Elementary Diffusion Theory In t.hiI chapter, we ahaD . nend \he .1ement&l'1 diffuaion \heory lhat ..... developed in Sec-. 6.2. and 6.3 and apply it to lOme import&nt 'pecW problem.. W. abaU mainly be intereeted in UJe diffalion of thermal neatroDi in weakly abeorbing "good " moderaton, ncb AI 11,;0, D.O, graphite, beryllium . beryllium o xid., and nrioua hydrcgeIlOUl mat:.eri&lJl (panffin, plexiglaN. eto.). In theM media, the ooodition. for the appliOl.tion of diftualOD theory a:e . .n fu1filled; ... long .. we Ita,. ODe or two mMn free p&tba •••, hom bouDdariM IoDd 1OUr'OllI, th e reeulta of difhWon theory will be quite aoourat.e. W II tpeoifioa1lr .-ume that the M Utron ~ . zu.it neutro~ with .. dtermal tlIJm'g1 dimibatlOll.

6.1. HmDllon of the General Theo..,. 8.1.1. IJemNl.Wf DerI"ado. or net', 1£. A ccording to Ouast'Olf. and EoLtTlfD, FtOlt'. law and elementary diffuaion theory can be derived with out knowledge of the general uu.port equation . Let. 1U oonaide1' an iIot.ropioally _tiering, weakly abeorbiDg (E.<X.1 medium. a nd calculate the ourrent or DeUtrona which penetr&tee the element of surfaoe dF per lteOOnd (Fig. 8.1.1). In the tim plaoe,

.

J-dl_

:~

. ,... .

x.JJJ

'- th e Dumber of DeUtnlM m-miD,g from

(6. 1.1)

+. to -I in al8OODd;fwthermon,

••• • J."" _:= E.JJJ

....

'- the num.berof neut.roDJ .trer.ming from - I to in a Taylor Mel around the origin,

(8.1.2)

+* per IeOOnd. U _deTelop4lJ

.....

(

)

... and l1Ib.t.itute into Eq•. (6.1.1) and (6.1.2), an integration yielW l

1- -+ <>(0)+ .~ ( ~:)... 1__ .!. <>(0)_ _ 1_ (~) " IE" B..._0 Flcc'sla. follOWI immediately from th_ equa.tiona: J =-J' - J- _ _

.

l

3~. (~~) .

(6.1.4)

(6.1.6)

I

I

I

/,

;

II I

""I

, I, ' I I III

/1- /

U".1et.

D-lf3E., 1'1(1:1:'.1&.. CUI be written in the familiar form J _ _ D·"",<>.

(8.1.81

For Eq. (6.1.6) to hold, the linM.r approximation (6.1.3) muat. be .. goodrepresenta. tion of the nUl: onr an int.e"aJ of Ieve raJ mean free path" l.e., (6.1 .7)

'06

.

th e cue in a strongly , bBorbing medium. U we . ., one CIQ • VIZ

-di.,J

... ~ .,.i. UlroIqIl

+

E./J) ...

00'" wri~

.. neutron t.la.Doe 101'

---

_

S (r)

IAroacII ~

(6.1.8)

we im.mediately obtain th e elementary diHuaion equation

(6.1.9)

6.U.• DlttuaIon Parameten for Thermal NeutroDl Tab le 6.1.1 contains t he diffwdon parameten of vanoua moderatorl for th erm al neutron.l (T ... 293.8 OK ). 'l'h_ ValUM have been obtained from vari oue expe l'ime n o tal data " hich oeeuionaUy diHer from one a nother rather strongly .00. therefore e&nnO\ be corWderod very aoeun.te. The abeorpt.ion ern- lleCtion- of graphite -.nd beryllium can be greatly Ineeeeeed by hardly notioeable impuriliM ; th e v&I.. given here &nI for pure mode ratol'l. If we are dcU.ing wit h .. neutron faeld with .. temperature oth er than 293.6 OK, we un ,euil y correct the ab80rption emu lIo&<:tion, but we mUlt newly determine the diffuBion COll3 tant . In th e di8CU!lllion of method. of meaaurement fOJ' L and D, we will learn of v.JUelI meaaured at oth er t.emperaturM.

.-

.

......

.

-- ", ,...» , .00

0'" 08..

...., ...., ,...... ZoO>

. .'M , ."

1M )( 10-· 0.0 1 0JI1 x l~ cee 0.'10 )( I ()-' 1.00 0.... «su x 10-- 62J BeryUium • • • 0.40905 1.10 x l o-- 2U BoryUium Oxide 0.4071 6 .30 x 10-' " .S U l Pu.ft'iJI • . • . 0.109 t.te x 10-- Il.l9 0 .32'1 0.8'7 1.18 0.136 1.73 x 10"'4 ,.so Mil Do AI .• 1.06 0.198 1.105 x 10"'4 40.16 0.... Ziroon ium H)'drid-I 3." 0.233 U2 x l o-" s.Il3 0.... I Dipbenyl oD:1e aM dipbeaylillized in. mio of I :0.30. The • IrOn'

11,0

~E)~ ~':.

PIo= ...

.

1.10 1.10

,.. ,.86

'07

...,U,.."

1>.- 1)i

C_ '-'-l

Io-l/ir.(i)

e-•

%.13 x I 0"'" 01>1 u e x lO" 6.'lll )( 1(t -· U 3 x lO1 l.a xlo-1.13 x 101 U6 xl0""' 1.17 x 101 1.eo x l()-l 2.70 x l O' 1.'78 x Ie>-' U xl'" UtxlO"-' 40.92 x l O' U8 xl0""" 6.79 x l O' UO xio-'

0.351 )( l()l 2.01 x 10"

-lUna poiD$ I. 11.1 *C.

... .3. fte NeatrOD nu Near.. Point SollJft: The 8IplIIeaDe8 of tlIe Dilfualoa. Leartb Th ediffuaion length introduoed by the relation L _ YDII. laM u impJe ph,.caI aignificanoe. Let. U8 oonRder the neutron fllu: fl6U' .. point. aounlO of nrength Q in an infinite medium. BeeaUAe of the IpberioaJ. Iymmetry of the lyetem. t.he diffuaion equation takee the form

~ + !.r., ~- ~ tl~ 1) -O

, ...

,

r ~(r 4t) - I1 4t - O .

(6. 1• ' 0)

... Itt geoenJ. ~luti OD i8 (6 .1.11)

,,_w,

Since the _troD. flus eann~ b"OOmti infinite .. B _ O. Tbe ooutant.... can be determiDed from the ~ , \hat. la the R&tion&ly etate all U1e QeutIoU emit\ed from. the ~ mud be abeorbed 1D &M m«lium . Q=

J

2'.4'(1') tl Y=E.

_'l'I'E.A

J-

•

,-OIl. b r' d, (6.1.12)

. - rflo,dr _""E..ALI .

•

Thu finally .

J-~

o

fP(r) _ v~iJ

0 - .-- 4,.D , - Oil.

, - OIL '

.- .

(6.1.13)

Nen, _ _k the ... . . ~Oll , from t.be MlW'Oe u ...hiob .. _ won ill at:.orbed. The probability that .. Detltron it. ..beorbed bet_D ,. and r +dr ia E,.4l'(r),"'rldr/Q. Tbu

'.-h-i::c.~(')""d'\

(8.l .14)

- -iJ f ,.,-t/"d,_2L. o

-

The mean . quanKl diatauoe frouJ. the

Fi -

IlO1lrn8

d which .. neuuon 11 captured i.

-i;. J,., - rll'dr _6V .

(6.l.1 lS)

o Thua the diHwU.on Jencth .peoWeI within .. numerioeJ. factor the dil.t.&nce bet_n t.b. pt..oe "here .. neutrOn fa produoed and the p1aoe where it ia abaorbed. Thia averase diataDoo moat not. be oonfueed lrith the aVeraj8 diatanoe act-ually tJ"aTe,* by the neutron j the lat.tor ia &qual to the mea.n free path for abeorption. and becauee of the DflUtroD', ziS-zag motion ia nry much greater th~ tb e mean ~tion between the sito of production end the . Ite of abeorptloD.

""rage

6.2. Solutlon of the nitlu,loD Equation In Simple Cues U.l. Slrnple 8f1JUDeCrical SourcellD InDDU. Media Potu Sovru: See Seo.6.1.3 PIa"" lk:Nn;c: Let the M>uro', which liM in th e Ca', y).plane, emit Q neutronl per em' per - . 1. 'Iben for , 0+00, ...

I

-... - V 4>-O. 'I1liI equatioa baa $he lOlutioo

4> (d _

, - ·IL for

,>0

,-11.

' <0.

for

107

Th u. (8.1.1 )

Lint S01I.ru: Let tho eouree, which lielI along tho ....xiI of & oyllndrioal 00ordinate . y.tem, emit Q ncutrona per om per eeccnd . Then for ,+0. I

• Thi. equation ct.n be dyed in terma of t he modified Beuel fun ctiOl1l of urot.h ord.r, 1, (r/L ) aDd K.(r/L ) I, The M function. are plotted in Fig. U.l. Since I.(r/L)dinrpu rJL_ co, we find

,

f E.4J(r) b

r

d, -2n A E.V

o

- h' A:

til

z_

u

U

n. ...... .... . . - . , ....... _

•

Q-

, '"

I.J

""LLL

Q)(r ) _.A K, (rILI

~

<,

J• ';'-K.(-j;) t0

E.Y . (8.2.1)

Splwrlt4l BW Source: Lett the IpberiOlJ lIheU haft tho radlua,' and ,mit Q neutron. per IeOOnd ; thua the llOW'O& denaity per om' of lurf&Oe I. Q/41f "', We take , - , /£

(1» (,) -..4 - , -

,

Q)< (r) .. B IJinh (r/L)

(f'> r') (r< r')

for th e IIOlut ion of the sph erically l ymmetriC diffuaiOllo Eq. (6.1.10). At r _,' both IIOlut ioDi muat be equal. Le.,

A. -"/" _ B IIi.nh ("1£ ). Furthermore. _

. t r = r':

CUI .~

the

MnlI'Cll

Itrength in t.erma of the CUJ'nlnt deMit,.

'IB b folio". from

u-e two re1aUon. t.ba* A-

f~;"

sinh

("IL I (6.2.3)

for all r. Cylirwlricol BTw1J ~ru : ut. \b e eoueee of radiu a" emit Q neutron. per eeecnd and om of 1engtb; tbe aurlace density of the ecerce i8 tbua Q/2n,' . Let th e axis of the cylinder define the .t.aIi, of a of cylindri cal eee edineeee. A. .. IIOlution of the cyliDdrially Iym metrio diffulion equa tion we tak e

.,.tem

fIl,. (r ) _ ..4 K. (r/L )

Ir > , ' )

fIl dr) -B4Ir/L)

(r < r') .

At r _,' the t.wo M)jutiona mut be equal, l.e., AK.(r'/L ) _ B41,.'/L ). Ap.i.D _

NoD

relate \be ecaree atnngt.h to the

(IJ;..I+IJ< I).-*, -

_2"Q-r"

CU rTent

deneity at \b e IlOtlI'Ce:

_D(!.·'-_!.·. 4, II,>.). _, .

From the. relatiou it folJo_ that.

A=

J~D 4 1,.'/L)

(I)(r) '"

'l~b 4 1r' /L )K. (r/L I

(6.2.4)

~l'l - 2~1i K. (r'/L )I.(r/L )

'.!.I. General Sautee DlItrlbuUoIll: The

DiIfllIloD Kwuel

In practioe, thermlll neutron IIOIIroM do not have the forme auumed in tbe aboft eumpJe. ; mlMt often, _pati. U, di ltributed neutron lIOuroN are p~nt (alo1red-down n6utronI). HOtrever, u will now be shown, we ca n alway. replace diRribution by ...yltem of IIOIll"C!M of the kind ~ above. In an infinite medium the flux at the point.rdue toa unit point lOUn:ll!I (Itrength: 1 Deutron/_) 11\ r' il given by

.. 1I01lroe

I . - Ir - P"J/£ lII (r) _ •• D I.. '-1

.

When a l patie.Uy diatributed IIOU.rDll iI JK-nt, _ can oonaider t be lOutCeI in t he

volume element. II Y' to be a point. IOUrDll of It~ng\b S(r. II Y' ; thei r contri but ion to the Ou at. r ill 8 (.., .. r

IIIII(r l - - .1ii>-

. - Ir- ... ,£

- \,,- "'1--

IntegraLiOllo o"er all r' liv. the DIU[ &\ r :

f 8 (r') :.DI. - t1 tl YO . - ...- ...VJ,

4I'(r) _

(8.2.6)

100

The u preuion . - Ir-.-}I£

O,,(r . ...) =

-' :;-DI.,- .....I

(6.Z.6)

it called the point diffuaion kernel. Eq . (6.2.6) permit. the caJeul.t.tion of the neutron fiux of an arbitrary aouroe diltribution 8 (,.') . Frequentl y the -ouroe diattibution bu lOme lIyDlDlet.ry ; then in place of the point difflWon kllrnel (6.2.6) another diffuaion k&mel adapted to thU Iymmetry may appea:r. Under 80m" circumJt.anoN tlW replacement can conaidllrably ilUnplify t be int.egra.tion over th e b rnel. If t.ho aouroe deMity depend. only ~ ~ (pI.ne Iym metryl . then th e ecuece dis tribution can be replaced by plaDe lIOtlI'(lN of It.r'ength S(r') d z' and the DUI: expeeeeed ..

JB(z') ,'-:ii:;L+ ..

4>(1) Th e diHuaion kernel in th i.

ClUle

-.

- lo-"lIZ,

dz' .

(6.2.7)

hi

0.,(:,:')=

. -10- 1'1/£ 2E.L .

(6..2.8 )

A Ipberie&lly . y mme tric IOUrctl dUtribution S Ir') can be replaced by a .)'Item of . pheriw l hella of Itrengtb ", 11r" dr' S (r ) ; t be Dux i, th en given by

• 4>(r) _ ! 8(r') __ L_ {e- I' -r'I/£_ e- 1H f"IIL} 411' , 'Idr' . S" D rr"

(6.2.9)

•

In th. e-, th e diffu.sion kerne l it

o

.

(";I

L_ (, - It-r'I/L_. - Ir+,,IL) . 8J111 D,r'

'

(6 .2.10)

An lUially Iym metric 1IOUf'Ce dmribuUon can he repl&ced by • •yatem of eylindri eal abell ~ of . tre ngt b b r'elr' 8 (r') ; tben 4J(r) _

wit h

•

f 8 (r' )0 ,.. (r, r')b r' elr'

•

1 lK. {r,L)4(r'/L) G.u(r. r') = b D 4 (r/L )K. lr'/L)

(6.2.11)

r > r' r
(6.2.12)

Finally, the eouroe IttnlngUa may depend only on UIe ooordiDate. (r'. ,,') of • point in a plane. The IIOU.ro& dinribulion can then he replaced by a ayatem of line eo~ of IItrengtb 8 (r'. p') r' elr' tip' IIlId tho fiu o~ u (6.2.13) with

(6.2.141

e-

y,.a+ r' 1 2rr' ooe(p , ')

ia thediatanOlI between the eoW'C8 point (r', , ' ) andthe poin t (r, 9') being oouid4nd.

"'

»'

-, 1\

t

K

<,

~~

1\."--.

,;

r-, r-, <,

-,

.

\

r-,

-,

"f\. -,

»

»

•

• ,

I

I

.,., ut.

I.,

IJ r{I.(RfL-l) -

!

I I

1\ 4.4

•

Z J 1O,t(K/L - S) -

.I

I

I

r/UR/!. - (1) no._ _ au ..-.4 • ""'" _

I

,

!

•

\

tot. . . _ W 01

Pro m lZ)(R) ... O it follo W! that

t

Q

c -R!J.

bE.V l iDh{RIL)

andtha Q

~(,) "", ""X.Vlinh (RIL)

....

(-"if) "

(6.2 .15)

I Thill ~ _ lD\roduoed Ira 8eo. .u.~ for ~ oeuir0D81 \be ~ foDow ~ Ilmple a""llzll_ &b. ~..., 4iNib'llloblo 01 the fiaL It ~tlI 0Illy .. tim appoxtmatiM, wIUah '- . . . kd lD .-u. wKh .. 1tVoqI,

-.r-deperld«l'

-u.erma _ -u...llb 810. w. ril mum to t.bt qUMt.ioD of \hI aztnpo!Moed«ldpoin' b~

baa-.pWIO.

11l

Fig. 6.2.2 thows thU eolution for IJeveraJ. nJuee of RIL; it leta ua _ the effect of the lUrla.oe on the flux diatribUtiOD. It is instructive to calculate the fr.ctiOD B of the nentroM abeorbed inllido the .phere. B ia given by

(6.2.16) I For R_L, 15% of all the neutrons are abllorbed in the Iphere, while for R=lO L, 99 .9% are absorbed.

PlaM &u.ru at fM O~1Utt' 01 a Slob: Let the .lab be infini\e in the z- and y" directiona r.nd let it have an effective thiokneM equal to 20. Let the llOUrce lie in the plano %=0. We take

IlL

,-lolllO + t otU. (JjL).

It follows from the boundary conditions 4>(a)=4>(-II)=O that

,=

Q

- -'L

-2E,.L· -;;O;h(II/L)-

Q . 11'(11) = 'Z.'L~(G/Ll ainh

("-1' L 1) •

(6.2.17)

(6.2.18)

(r
In'iniUly Umg Oylindrical BMU 80wu I'II/inik Oyli~ 0/ RaIU". R:

0/ Badiu

r'

O~

wA a'll

.AI befOl'O. we can derive diffu.Ilob IterDelI for the trMtment of general 8OUl'08 dimibutiODll from theee eolut;1on8. l8tricil1 ~ the iDtep.l1IhouId omJy mead to B-41 hotrner, If R '- D06 too ...n,L..teBM_IIM >L.ibe_ .. ...u.

UoL TIM Ne.VOIl DlI&rU•• tklD ID a Priam " Pile" In man, experiDMInt., media in the form of priama or cylind ers - pUN. _ ea1Jed. - aN.-d. Next , let a oon.Iider .. priam infinitely long in U.e ..~ 1l'hich eontaina .. point aource located at tbe poin t z , y, .... 'The rele....nt diffusion

equation ia

y. 4J-

-iJ- 4J _ _ -~- 6 (2:- z)6(y- y' ) 6(2-

: ')

and mUllt be -olved under t he boundary conditiona:

III addition, the Ou mat vanlah in th e limita s _± ..... :r.t us deve lop q, in a Fourier lMiN:

~ I 2:,,.. " ) "" ""

~

/' . - ,

I"s 81D . -. ,", . ,...",,~/,. I..) &Ul' -.-

nul e~on alwaY' fulfilla the boundary cond itions in the

(8 .2.21)

2:- and y-directiona.

NoW' let ua ellp6nd th e 6-functiona : 6 (z - z') 6 (r-

J ... - -: .-

..JJ..

Y)_ L J". ain '~s sin

"

.

fAr

6(z - z' )6{r- y' ) ain ' : .!.ain . : ' ds d y

I

(8.2.22)

4 . , ,, %,, . .",. . a6 ~ -• ~ -,

If _ IUbltitute Eqa.(8..2.21) and (8.2.22) into the diffuaion equation. carry out

the iDdieated diHerentiationa with fN pect toz andy,multiplyby iNn J: s sin ...; ' • and integrate over the cl'06IIaection of the pile. we obtain th e diU6rential equetlcn d'~,.

-- ' -

-

'"'l, . . . _ _ 0' '1 . .'" 1 ...'"'Il,. - -VI- +.. [-0'P +-I......

""i1-

~~

~'

4

~

. _"'''_11 r;oi (1- : ' ) alD -I" %,, _.lD

Do'

(8 .2.23 b)

for the Fourier component. fI\.. (I ). Fomaall" thi. equation ia identieal with that for t he n UI 4J... u. at

~~ lin lin . : " Ioc.ted at I _I' in medium who. difflWon length i. Lr..; it. .-olution i. th a

': %"

.. plane IORl'ClO of Itnn,th

(8 .2.23a )

an- from In

infinite

'11%" . -"'·16'1" ·-- ' - ,_",•. - .... ..

~• ~

I) I'J.'1 . ... .c =- -~ D06- IU\ -0 'I n

Finally the.n ~I

Z"

)

,I

--D.' ,.

10 ~ r.

.

'11%" . . "y

k.l-...IUl . -~ -.- '

"'11' . (8.2.24) 1lD. 1lD-.-

- 1I- ..I1z.... · I1Is .

II' According to Eq. (6.2.23b) the re la.u. tio D lllngtha~. decreaae with inoreuing

1,1#; thua for large diatanoee from. the -ow:tle the Du Ie given by t1J {z , II. ;I)_lin lIa~

.m ~y . -IlI -"It.. .

In other word., the Dux ft.n. off up0n8nti..uy in the ..~tion. jut u m the eue of a n infinite medium . Howenr, inItoad of &be dlffuai on length, &bere appeal'll & relau.tion Ic~ ~, -L{Vl + n·Iit iTa·+ l/bI). whieb a.oooun~ for t.he lr.tenJ.1oa.k.a£e of Mutrona I.Dd whieh i..ma11erth. .malJerthe~eectionaJdi. meraiona of the pOe. z Eq. (6.2.24) with Q_ l give- the diffu-ion kernel for the calculation of the neutron flux due to an arbitrary ecuree diatribUti OD in an in . finite pile. In orde r to calculate t he neutron flux in a r finite pile (Fig. 6.2.3), we once agai n I t&rt with a Fouri er Iif!ri6ll . Eq. (6.2.23) must the n be colved with the boundary condition 4)". (s) _O for <1 = 0 • z a nd , =-e. A. we can eaai.ly ChBCk with th e help pUo of the method developed in 8&0. 6.2.3, th e " ub it

J'lI."" . . .-.-..

~

()

...,. . :1 -

4Lt,.Q . ,-*' -zr.r-alJI •

-".. . - D..

~ I )

.

lID

~ •

u.c.~ O . In' . ..,. lID

•

am

ail:iht...,!.t.. ) .= _.. (.c=.!....) ;IIlb{I1/Lc.-l ........ Lc,.

.... (.-1-,.--;-..)

- r M(q'Lc.. l

(6.2.U )

. . .: _ L I., '~ .J

IILII.Q

-..

U th e IlOOlCO Ie tofficiently hr from the upper 1WUoe, the n fM from the the nux it given by

~ ( !#, y. %) _

'IV

When

'z'. ;;:

' lin

~!# am .~' G T

IOW'Oe

_'_L('-'J 1:;7" .

BI.lIU

2, the einh C6Jl be replaced for aU pr a(lf.ioal plUJIOIIEl' by

i ,__.1/£,,:

thU8 tb e effect of the boundary on the flux distribution it no longer pa lpable at a distance of 8O\'eral relaxation longtha, and th e flux fall. off exponenti.aIly jut .. it d08llin an infinitely long piJo.

6.3. Some Special Diffusion Problems 6.3.1• • edit. lD Centad: The Al beit.

._0

Fi g. 6.3.1 Iho_ .. eimple arrangement of two different media.. At then ill & plane lOuree which il .wrounded on both ooea by llabe of thicknoea I" Adjaeent to th_ . labe are .llbe of another l ubltance of UUckn_ IIl -I" W. tau

~rla:) ,.,. 'u O-C

"

(/JII(: ) "" 11 Booeka$lWIrta, Mealroll Ph yelcJ

,

e -·IL,

+',ooeh (%/L , ),

ainh{rll~YI )

(6.3.1&)

(6.Ub)

•

,1< for the neutron flul: . This form of the flul: already fulfillI the boundary OOndit.iOD that the flul: vr.niah at.* =lu . To detonnino the oonat&nte we apply tho condition that the nux and cumlnt be oontinuo1lll at the interface between tho media :

',and'll

(8.3.2)

Thia lead. to

1 .-" ILJI.1nh (!!!-Lu" )- _2!1I o.»b (!1l-:- 1)1 E.JLlLii Lu .1!!!... ~ (~L) ..... ('ll -") + PL.inh (.iC). iob -(-'" - (' ) ~l ~ ~l ~ ~ ~, ~Q_ 2

•,.

. 'p1I ooeh Lu

(8.3.3)

J 11 (JI..) ..... (!.1J) +!!L,inti (.'L)oinh ('"- / ) L, L II L, L, Lu

UJ

Fig. 6.3.1Ihows the v&riation of the Mutron DUI: in .. typicU C&lI6 . For oompuiaon, the flux for the cue in which no reflector eur rounds medium 1 is .1&0 plotted. We can see that th e neutron flux is Ineeeeeed 8nrywhonl in medium. 1 by the reflection of neutron••1. the intorfa.oe. The eUoot of medium. II can very

tion ooofficient, be describedthe by alO·called reflec_~~~~§~~~~~~~~~§~;; eMily albedo. at the 1-11 interface. 'Ihe albedo

fJ

is the ntio of the

cUl'l'enLJ -nowingfrOm Ii to 1 to t~t J+ flOWlJlII: from I to

..

II. H J - and

p'--lU .I . ",. ....II'OA tlIu: UOIID4. p..... _ _ \a.

- , - - ) - - (---) ill

p =

\

~

~~" + !," ~" l 2 d.

_ _ _ ___ ~D" 11"'"

2

III

.-1,

:ra:re

now ex-

~ in terms of the properties of medium II, th en (d. &q. [6.2."))

-

l+'D,, ('~/~) (d"'IIII) · '" .-lJ

'" l-lJ 1_2D,, _ _

(8.3.0&)

H we denote the reflector t.hicknesa 1,,-1, by b, it. foUowe from Eq. (6.3.1 b) that. 11 and thue that ( 11"'1. ) __ .L 00", '"

._1,

L"

(.!..) LII

p~

1- !~!! ooth __ L" 1+ 2Dll ooth

LiI

(-'- )

LII

(..!...) . L II

16.3.61

116

For & thick reflector, b>L, coth

(~l)

1- 2Du1Lu

fJ = l+ 2Dll1Lu

"",1 and (6.3 .6)

/IO;Il-4Du ILu '

U the Albedo of a refleCtor i. known, we need not aolTe the diffuaion equatiOD In tb reflector in order to calculate the fiux in medium I ; inatead we eepleee the reflector by the condition that. tho ntio of the inwud and outward current. at the inoorfaoo be equ&! too fl. Then Eq. (6.3.6) holds in ,lab geometry ; however, Eq. (6.3.6) for thick reflectors al&o bolda in spherical and T.tlJo e.a.r. TAll Aa..4001 Y4rioKf~. for T~ N~ i .. PI4tw o.o..mr eylindriC6l geometry. Table 6.3.1 &hows that quite thin layers of 1 _ _ 1_60... 1_1IO<m heavy water and graphite reflect well. while HIO because of its 0.811 0.811 0.811 comparatively large absorption i. ~o . .. 0 .947 0.022 0.863 . . not &8 good a reflect.or of thermal D.O Beryllium 0.911 0 .910 0.007 0.881

.......

..."

"'81

neuUoIUI .

0.""

Graphite.

0.923

0003

0'"

6.3.2. A Strong Ab80rher 1D • Neutron Field! Let the neutron flul: in an infinite medium be COll8tant and. equal to t;J;, . U we now introduce a piece of • etrongly absorbing lubstanoe like e&d.mium into the neutron field, it will act as " negatiTe source e.nd depresa the neutron flul: near it . U it ill a IIpberical ebeorber of 1• r&diue R, the neutron field at a distance r from ite center is givon by

•

'1>(r) = 4'>0- ~.,

,_,/1. .

Tbe COll8tant A can be determined from tbe boundary condition at the surface of the absorber. According to Eq. (6.2.39) ~(R)

(il ~/il ').,_R

-A

-

(6.3.8)

•

o1\ "-

(6.3.7)

-

"""" --.....::

, ,

OJ

"!i- • J

S

na. U.L Tbo_polaUoll ~l
Tbe eJ:t.rapolation length 1 depende on the radilU of ounature of the abeorberj Fig . 6 .3.2 shows Aa.s a function of the r&dius of curvature for lphel"6ll and oylindere (DA.V180)f). For R-+oo, l approaches O.7104/~; the limiting valoe for RIOlO if! 4131:, . Theee ClllVe8 are ezact for monoenergetio neutrou in a non-abeOrbing, isotropically &eatt6ring medium i they oan &leo be used. a.s a firet approximation for thermal neutrons in a weakly absorbing medium with we&kly anieotropio 8C&ttering ill/I. if! replaced by~. It then follows from Eqs. (6.3.7land (6.3.8) tbat for a epherical abeorber

A =- 1

~.ReltI£

(6.3.9)

A[~ +ij+1

See alto 8eot. II.J aod 11.3.

"

and thut

ltl(r) -ltl.[I-

1

.l(-}+ { )+l

..!",.-.)/Lj'

(6.3 .10)

r

'l'be Dumber Z 01 neutrolla captunrd by t.~ .beorber per eecced can be obtained. froa:I Ute ClUI'nlDt lot r _ R:

fl" _011'<1>, 'D(~ (I +1)i)+1 . ., .-.

Z _"1I'D (~)

.l - + ~

R

(6.3.111

L

For R L, Z becomes 11 R' . ltl. 4 DIL. We can euily make .limilaranalyait for an Infinitely long oylindrical.. blIorber. For .beorben which do DOt. haTe th. . lIimpie . ymme trieaJ. fOnnl. the calculation of the DMItroa ftu.x i.I • ohaUeDging mathematoioaJ. problem.

Chapter 6: Befereneel

........

1"-.1..: (M I . G. B...u.al' ): NeatnlQ PhytiaL. AEC'I).HM (1061), etpeaia1!, cha p. Vll : 1"- DMribaUoo. of &ow Nea~ iQ . Kedi..m. GL&.IftOn, and K.c. EDunrD: 1"- Eameoc. of N.....,. ~ 1'beor1. N... Yor\: D. V... Nw\nad IW. etptoiIJI, cb6p. V: 1"- Diffu8 ioa of N.atroo..

s..

8poda1

P -'l

~ G. l m :1'lMo 8oieDot Nld ~of Nitaleu' YOl II, p. 77. a-bridp: J.c:IdilIoIt.W""lMt. 80IatD 01. \bot W~ P. Jt.: N~'" No.1, JO liMe);'" No. s, ... (IN t). DUfaUoa Eq.atiocl. -.1IDd I . LIICu:n: ~ (1M31.

l"ll,uo. o-tazac.. ANL-6800 II~ ), p. Ul7. } E.r:vapo1atloa Leagt.b. for 8pbent aad 8. A.. Jtvaou.n11l:1 )(T-ll' (IHe). aad Cy~

.uo_. B..

~

D•

7. Slowing Down A..• rule, oeutronl produced in bUeleu rMCtiODI haft energiol far .bon the thermal eDergy When ncb faat. neutrona oollide ..nth the .tom' of • -u..ring medium. lOBI of energy oooun lIimultaneouely .nth the diffUliOD ~ The oolllidClnl can be eit.her eluti.o or inelut.io. A.. long AI th e 1lIIutroD -e- it poeat.er \han .bout 1 •• , \be lItnI.ok atoml 0Ul be ooDlide~ m. and at; ~ before \be oolli.ion. 'I'hiII i. DOlongtlr the _ at. Io_ r energi_. ...hero the ohemioal binding of the .\om. of the lIC&ttorer and their th ermal motion affect. the alo'Winl-down prooI!IlII. 1~ i. the aim of .Iowing-down theory to det.ermine the IJ*lO aDd elllfl1'lYdiltribution of t.be DOnUon nUl: &riling from a given dil tributioD of - . - . In ...haI. followl.. _ IbA1l t~t. three diltine\ Alpee'U of thiI problem : To begin with . in t.hiI chapW .... Iball oonQder 11o'Winl down in the .~indepeDdent. cue, i.e., in an infinite medium ..nth. unifonnlr diltribut.ed

ra.nr.

117

eourcee, We ,han take into aooount only el&6tio oolliai0n8 With free atom. \hat are initi..Uy at l'1lIIt. neglooting both the inelaatio acattoring of fut DentroM and tho effoota &88OOiated with the ohemical binding and thermal motion of the .tama of the IlC&tterer. In Chapter 8, we ,han treat. the lpe.oe-dependent caee under the same limitatiOlUl, i.e., we shall study the aimultaneoua diffuaion &nd lI10wiDg down of neutrons in .. finite medium containing an arbitrary distribution of lI01U"06l. Finally, in Chapt.ef 10 we ehflll.tudy the therm&liz.&tion (If neutrone, that b , the moderation of neutrons in the range of very low energie6, where the chemical binding and thermal motion of the atom. rout be taken into aooount. In thoee light elements that caD be used all moderaton, inelastio acattering plays DO role at energies 1688 tban lIEIveral Mev (d. Sea. 1.4,2), and we need not take it into account in studying the Blowing-down proce88l. The transport equation for the Mowing down of neutrons in an infinite medium with uniformly distributOO. eourcee wu introduced in Sec.li.1.2. It is (I.+I.)(E) ~

s

f I.(E' ~E)(E')dE'+8(E)

•

(4.1.16)

and repreeente a limple balance equation for the ll6utrona in the unit energy interval around E ; On the left sta.nd8 the number of neutroWl Joet per em' and 1&0000 by abeorption r.nd _U.ering. whilo on the right at&nlh the number of neuHoW gained from the eourcee or by m-seatt.ering from other enetgiM. Before we treat thia equation any further, we ehall next calculate the el'OfI8 eeeaon I,(E' -+E) with the help of the laws of elMtic oolliaion.·.

7.1. Elastic Scattering and Moderation 7.1.1. CGl1lIlon DlDamJea Let WI ooneider an el..tio oouw.on between a neutron of m&88 1 and enerzy E 1 and a free atomio nucleWl of mll&ll A that is initially at relit. Our main goal 1& to calculate the probability that after the oolliaion the neutron hu an enerzy bet1V06u E. and E.+/lE•. In addition, the angular diatribution of the neutrona after the oolliaionis of intereat. For the oonaideratioJUI that follow, it it neoeaaary to oonaider the oolliaion both with te8pect. to the laboratory frame of referenoe (r...y.tem) and the oenter.of·mBU frame of referenoe (O-tlyttem) . The dynamioe of the 001. liIion are shown in Fig. '1.1.1. !Alt U8 introduoe the following notation :

vl -

t\-the velocity of the neutron before the oollieion v. =the velocity of the neutron after the oollieion v. =the velocity of the center of mBU D. =the aeattering angle v. = th e velocity of the neutron beforo the oollillion v. = th e velocity of the neutron after the eolueion op=the IIO&ttering Mlgle.

I

in the L-ayst.em.

I

in the C_eystem.

'The ~ioo" &l.tcgetber diffeteRt. Ul"" ~ which oootam DO lDOlWator r.Dd Ul which iDe1utio ~ by hM'f)' IUIClSei MriouIJy alfeot.e the ~ I Eq. (5.1.18) &lao hoIdIlOf the .~y.'t'erapI flllll: UllN'lybollWlpDeOUlmedlUUl with IlI'bitnrily diItn"buted .IOIU'Oe8j howe't'l!ll'. the medium muft be .0 IMp thM DO neutrolul - . p i through it..m-.

1lO

AI. ia ....ident from Fig.1.l .2.

001"

(7.1.1)

"+".'

(7.1.2)

lit- ~ +.,:, +2v.t>. Il, ODe " . ... ". roe

h loBo... from the theorem of mom entlUQ oonten'aUon that tbe velocity of th e center of m. . iI ~ (7.1.3)

,

I{~~

-

_ _... ~

I

-

~

~ _ /- "

".-m.

':T

..-!a..

Tho velocity of the DeuU'on C.._ aru before t he ;8COIliaiOD ie

m the

"1-".. - AA+ I ":!. '

(7.1.4)

In the C-8yete m. th e nu<:Iem hM the velocity". of the eenter of m&llll in tho L.ayat.em. Furthermore, in the C.•yatem. tb e momenta of t be neutron ami the nucleu. are equal and oppo-itely di rected . After t be oolliaion, th ey mUit. et.iU be equal a nd oppollit.ely direoted . Since the kineti o energJ' in the C..yat.em i. a1lIO OOIIaernd during an elutio oolliaion, t he P'O'l.-
~ U..t.Tlroo _W--:• • ~ ... ~

"I"

_Mal

(7. l.lI)

h no.. foUo_ hnmediately from Eq. (7.1.1) th at th e energy E. =~2 of th e neutron .tt.er the col1iaioD. in tbe L-.yatem i. E .-

K "IHA 00lI¥+! I

(A+l)1

(7.1.61

Funhennonl, the _ttr6rins: angle in th e L-ay.tern he giveR by ~

co. . -

.4_,,+1

y.;tI+IA_,,+i" .

(7.1.7)

Nut _ in\t'Odaoe the aunli&ry q'lWlUt,

• - (~=~)' A+I •

(7 .1.8 )

Thea Eq . ('U .8) beoome.

{;- i ((l +Ill) + (l-IllI _,) .

(7·1.91

The maximum energy 10M oooun in a head -on oolliaion, Le., a ooJliaion for which , =<180- and _ , _ _ I j B. then eq",at. .81' The _rgy cannot fall below.EI .. the ""ult of a tingle eludo ooUwon. For hydrogen lIl""O, and (&'>- _ 0. Forlarp .d

•

CI""'I- -.f =tf ·

(7.1.10)

'" 7 .1.B. EDetg1 and ADgular DlltribnUon

Let f/{E1_E. ) dB, be the probAbility that a neutron of energy 8 1 before the collision acquires an energy between E. and E.+dE. after tho collision. Since E. ill ec nneeted uniquely with the Boatteting angle" by Eq. (7.1.9). g (E1-+E.l dE. =p (ooe \1') 4008"

(7.1.11)

COB"

where p(oos If) d is the probability that the 00fIi00 of the IlCI.ttering angle in the oenter-of·maaa system liee between COlI "and 0081p +d eoe 'f'. Aswe&lready have seen in Sec. 1.4.2, for most good moderetore the IlCI.ttering in the O-llystem is ieotropic below I Mev; in eucb .. ceee, p(OOII f) = conet. =1. Then. however. I

g(E1-+E.) ,.,. T

c1 0lM¥

---n;- -

I

(1 01)8,

(7. 1.12)

for II.E1 ;i E. ;ii;E. , For E.> 8 1 and for 8.<<<81 , f/(E1-E.)=O. The probability g (E._E.) given by Eq. (7.1.12) will 00 used later for the solution of the tranaport equation, Eq. (6.1.16), where I, (8 ' )g(E' _ E)

(7.1.13)

will be lIubstituted for I, (E' ....8 ), Just &8 the energy diBtribution after th e collision can be e&lcul6ted, 80 can the IIUlgular watribution of the neutrona alter the oollilion in the laboratory Iyltem be calculated. For mOflt tralllport prcbleme, only the moan 008iM of thelC&ttering angle is of interest; for isotropio ecattering in the O.lyetem,

DOIo_...

j!

_

A001,,+1 , _,1 dCl()l¥'= ,I. ¥At+IA OOI,,+1 ... .

(7.1.14,)

The forward direction a more 8trongly preferred the lighter the Itruck nuc1eU8 ia. For heavy nuclei, 2f3AFtlO and the scattering in the laboratory 'y8tem ia llfIU'ly isotropic. When the scattering in the O.l y. tem iI not isotropic, the ez:preuiona for V(E t - 8 .) and p {COl') become more complicated. For weakly aniIotropio ecattering, (1.1.16 ) p(COlI f) "" i (1 +3 0i5iili Cl()If)

(1.1.1611.)

(1.1.IGb) 7.1.3. The A't'fIragI Logarithmic EnelV IhleremeDf;

Knowledge of g(Et_E.) now alloW! U8 to ealoulate the average energy of a neutrou after a oollilion. It ia given by

.,

.,

...,E._ f B,V(B1-:+E,ldE. = f ••,

d,

(lR,dE, - .) B'~ - ~ :I (1+«) .

('I .U'la)

lJO

The . . . . . energr _

per collilioD y then

.aA_B,_A._Z, _,_.

"'TW

1- •

-;;---"'il""

(7.1 .l1 b )

The avet'ai'! ~"'tc energy 10000 l hi given by

"

E-'":'-ID"'-'"(~~ -l'"(~~'("'-"'J d"' 1

(1. 1.18)

- JIn (~J (I ~~116. _1+ 1:. lD ct . d ,

The average logarithmio energy decrement i, thw aD energy.independent COlUltant . Thi.1UggMU th e introduction of . Iogarithmio energy ecale for the elowing-dcwn ~ (d. Sec. '702.1). ! it equ.a.l1;o 1 for hydrogen. and for Wge.A. can be quite -U a pprozimat.ed by (7.1.19) (Cf. a.lIo Table 1 .1.1.) Knowledge of t.be QU&lltit.1 1 make. it JK*i ble for w to eetimate the average

num ber " of oolliIiona neoeuuy to moderate .. neutron with an init.iaJ. energy Eo to the illn ergr B. OeuI.Y. nE

- In(!;): """

ID(~K)

(7.1.20)

Table 7.1.1 oont.&iDa «. E. and " (101' mode~\1oD from 2 Mey t.o 0.0263.Y) for . nnJ. DUM . The . .loel of" obtained from Eq. (7.1.20) are oo1yappronmate. 1IiD. t.hennaliu.tioD .ff~ ..t low ene rpe. and IlOD'botropiO eoatterin& in the C...,.nem at. high ebllrgiel have not I-n takon into aooount. What ia more. Eq. (7.1.20) iI. 0011 aD approllimatioD. for It; in aD.net. oa1oulI.tWn of the number of

~

~

the &DM'IY

diaVibq~

.. .

..

·· .. .. .. .. ·· .. ,. · . . . . · . A •

• (I Mn_O.0263 ..,.} a.ooontina: to Eq.(7.I.2OI ••

• 1 0 1.000 It

I

D

••

, ...., • ..,J6 0.111

.

n follow. from Eq. (1.1.1Sa) in &be

of the neukoni dtlring the aJowing-down

mun be \&kea. inw acoouu.t (D_ M.ustll . Ktltt...,s)•

0....

" that in

U

. ........

..... 1

0....

.,

• U,. 0....

.

the cue of

, lJ

o.ne 0.158

D

It

M" o.reo

I

,

...

0.' " 0. " -

I '00 '" --.kly..m.otropic _ttering 21'12

O...,.nem E- E....... -3Dl'11~.

-f+(1~.)I bur:J.

(7.1.21)

We ClaD Mtimate the effect of aniIotropic ..,.ttering OD the alowin&:-down pt'O(ll'M with Eq.I'U.2I) . U the differeDvu acattering Cll'u. aeotJon of deuteriWD. at 0 .76 MeT that ia giYeD. in Fig. 1.4.4 ia appro.lim Wd by Eq. (7.1.16), .. Taloe of

IS' 0 .067 ill obt&i.ned. for l5i5i'"¥. h t.ben folIo_ tha\ l _O.tI62 tIOU1pared $0 l-.. 0.726. The differenoe 1. . mall, a.Dd from now on we will not. \&ke ani..toVopio

_Uering intoo aocount.. Ho wever, {OJ' very aocurate calculaLiozw, the uUotroP1 of _ ttering a t high energkle mut be iDcluded. . We now int.roduoe t wo important. parameten of Uae dowing-dOWD proo8M : the ,Wwt1lg..ao- JIOIlIn a nd. the wwdmJti"9 rotio. A moderator . 10_ !leUtrou down better the largor l is . In addition, E. mUl l. be large in order that neutrona collid e aa often ... poeelble . There· fore ,the quantity l E, is called th e T..ble 7.U. 8"""'"". ~ P_ OM JlodoWir&g R
e

nlgion .. t hat.. tret.cbee from about 1 81" to _ "eral ke't' j at higher energiee EE, decreuN (wi t h

11.0 .. ... np (pure) • • E.). 1>. 0 (91i1.8 __ ..)

........, ....

1.00

1.10 1.10

1.J6 _ _1

0.11. _ ...... 0.118-.- 1

n ..,. ....,

I.OJ o.oeo_-I ite meaning. In addition , BerylliIlQl • • • • •.85 0.168 01Q- 1 .. good moder.toor .hould capture only wellkly, Le., E.llhould be 'mall. Therefore, .. better meaeure of tbe mcdeeating properdee ie the quAntity EE,/E• • the ao-called modor.ling ra.Uo. The moderating ra ti o i. usually e val uated wit h the averago capture crou &oetion for thermal neutrona .t. room t.emperature .

At lower energio. . the quantity ll~

'" 1<'

7.2. Slowing Down In IIydroro. (A_I) In tm. ..ction. _ IhaIl , tady the tran-pon Eq. (6.I.I IJ) ill hydrogen ; t.M _ttering er'C* MCtion, which it ginn by Eq. (7.1.13), t.akM a ~oularty limple fonD. for .A. _ 1. It will denlop th at in thie euo we ea.n euily obtaiu closed eolution- for 4) (E), while in tho general. cue A=I=l , we can onl y obtain approxim ate 1801utioDe. Firat, however, we shall introduoe IIOme addition&! oonoepte th at a.re CUlItomary in th e keatment of Il owing -down probleml . 7.2.1. CoUWon Denl1ty -

8l0wing-DOWD DtlDl1tJ -

Lelharv

The coUiftmt d.tuilll ,, (E) it defined. .. t.he number (X. + E.>4){B ) of neutron. which undergo oom.ion (lIC&ttel'ing or abeorptlon) per em ' and -eoood. UIIina: Eq. (7.1.13), Eq. (6.1.11J) CAD be writCeQ in terml of ,,(g) .. l oUo_ :

,,(E)-

f••r.;r.

•

(7.2.1) a

,, (E' )p(r_E)dr +8(E').

The elotoirtg-dovM ckMity q(E ) it defined .. t be n um ber of neutl'ODa I10wing down pcutt th e energy B per om' and eeeced. The probability th.t a neutron of initial energy .1" > E baa an energy E" < E after the oollisioD it

...•

GIE' ,EI -J.(E'_E" )dE" . lla b ~ &hI;.ppw limit 01. iD.~ IDIIA be &hi; IUcb-' _ 6 > " '4'

iIIMsnJ. ID_ ..... ut.I4lo 64 ill ............. ba _

(7.' .2) -V

.Q. n.

'"

Th o Ilowing-down deMity i, th en q{E )=

J·t.~lrl Y'(E')O (E', E ) dE' .

(7.2 .31

•

U foUow. froIll Eq . (7.1.12) that the p robr.bilit y OlE'. E) i, siven by

•

flr

f

0 (1:',8) _ II 11).1" or

OlE', E) ia 1«'0 for r
tho<

r ~EI«-

_,

z-.r

(1- _l r

'

(7.2.4 )

It follow. from Eqe. (7.2 .1) to (7 .2.4)

r.

-• 6. _ -E.-+E, - - .IE)- BIE) .

(7.2.6)

Freque ntly . tho Ilowing-down prooeu il deeoribed with the help of a logarithmic energy 1Oa1e. Th e U~ of l uch • .eale j, euggeeted by the oonatAnoy of th e average logarithmio energy 10&8 por oollillion. We use 61 • variabl e t he eo-eelted Itlhargy , d efined by

_- Jo( ~._).

(7.2.6)

B eN Eq ie an lU'bilrar'y ""ferenOCl energy ; in moet pnctiealapplica\iona. EQ i. Uken to be th e hlgbMt olWlf'l7 .~ in the ~ apeetrum. Then . _0 at tbo beginning of the tlowing-down ~ and during mOlhlration i.nc:reuN eontinuoualy . Eqt. (7.i .l) to 17.2.6) ONI be writte n in tenna of the lethugy ..

1000..:

" tv ) ,."

J• r.r.:.r,

,,(V' )I("' __) d. ' + 8 (111).

(1.2 .7)1

l _lII U /&)

(7.2.8) " +"(1/011

0 (.·• • )_

J •

g(.' -+." ) 4" ",

(7.2.9)

('U .IO)

lla bydropa,1obe Ioww Jimtt. of inUpa&XIn mlld be .. _0. 1.a 11M...... DloOdtrMon. the iD. . . .ioao . . . . . . .:Rend to __ 0 if s <mll").

.ss It ahould be noted t hat (7.2.13)

Naturally, q(v) =g{E) .

7.2.2. Cale.a.uoD of the 81owlD«. DoWD DenallJ &Ild lb. Eu"lV 8,.ecrv.m :&ca.uae rx _O in hydroge n. the equation for lb e collieion density writte n ,,(8)=

J" zo.;r• .,~

•

dB" + S(E) .

eNl

be

(7.2 .14)

A diffenmt iat equ ation fMulte from differe ntiation with '"I*t ~ E : ~' ("l

dE

_ _ ~ .!.!&!. + E.+ E,

g

4 8 (~>-

4& '

(7.2.16)

Thia equation can be int.egra.ted wn'Xltly . but it provee convenient firat to rewrite it in terme of th e Ilowing-down denaity and then to in~te it . It followa from Eqa. (7.2.3). (7.2.4), .. nd (7.2.14.) tb a t

(7.2.18)

f (E )- E [. (E )- 8 (E )] lJO

t hat (1.2 .17)

of no a beorptioo. From Eq. (7.2.17) we ha ve

Let u next consider lbe. _

" •

Q(B J- f 8(E' )4r .

17.2.18)

ThIH ..hen th ere UI no abeorption, the . Iowing-down dcmaity a' energy E i, equal to t he number of neutrona produced per em- and Moc:md at.energiee . bo"e E . If the source density I, monoenergetic. 8 (8) - 8 ·6( 8 - EQl. th en for E 'S,/lQ' q(E) o;;; 8 .

Thie result clearly holds for all non-abeorbing moderato n ; it i. phy6ically ohvioUi and follow. im mediately from the general Eq. (7.2.ts). According to Eq. (7.2.16), the ooIIWon denftity in hydrogen ill given by

.IE)- '': +8 (E) eo tb at when S(EI""'S ·6(B- BOl

,,(8)- .

+ 8 ·6(B- 8 0)

88

1P(8) -

E. .

•

8

+ y

•

d(E - BO) '

I

(1.2.19)

Th Ui a t all ener;iee lIIl&1Ier than th e IIOUroll enotgy g,(B) _ 8/E.B. Since the ICattering CI'OIe eection of hydrogen ill oomtant in the energy n.nge from 1 e. to 10- ev, th e flu:.: of moderated neutroDi folloWi a I /B.law there.

." Nut we turn to the general cue of .Jowing down lrith abeorpLion. We . h.ll be interMt.ed in either pure hydrogen or in .. mixture of hydrogen and .. bea-.y ..beat bel' whOM nuclei do not contribu&oe to the moderati on . Wh en the scurce denaity ia monoeDergotic, i.e ., when 8 (E) _B·6 (E'_EQ ) . dq _ - - ~~ _ ,t<El dE

E.+E.

r.

din,

..n · -

E. +r~ "g '

(7.2.20)

•-.t;r.+&-;--r J dE'

In q(E) -lnq (Eg) _ if8<E. , NOwq(E.I -(- -!·· --)

g 1

"

E.+E,.. · S ' IOtb d

(

Eo )

f (EI = .~+ ~

r.

.;S .e

-,

- . r.

...

-r..-.z; -r -

(7.2.21)

-j~:r; '~

P(E>- (r..+-.r;).....

(7.2.22)

it c.Ded \he

_ t a U UOtJpt probability. p IE ) giVN Ule probability that. .. DeUtron produced with energy EO will DO\ be .beorbed during moden.tioD to th e

(z:.x.:

onere E . 'I'bo factor Eo )•• ia the probability that alOW"ll8 ne utron will not be .beorbed on ita fint colliaion . Freqnently. X.<E. at th e l!IOW't'lD energy. 10 that OM limply b.. (7. 2.23 )

For .. BenenllOW'Oe dietribution the IJowing-dO"fD deMity it

r.. -j r..-i" 8 (1:' ), r

....

E. - ,.. -

dE',

(7.2.U)

(7.2.26)

(U 26 )

!Non. of,

In .. at-bel', u.. au will depan bora t.be JI'W'l' I/E.law beo6_ 01 Uie cleo..with cIeereMiDg E . Purthelmore, there.m be .. fl.u depn-ion DMI' MOb of the,.ariou~.

'wo

W. eh&l1 now calculate th. p-f&cWr utillg Eq. (7.2.23 ) for inabuotJ" apoaiaI Fint we cotWd. r a hypotb.ticalabeorber whoee abeorption oro-tlMlCltion E. il infinite in th e interval EI :;;';E' :;;';E. and ie zero eUlewhere. Then for 8 <81 and Eg>E. , CfoIlM.

" -f~:-

,-I·' _}.•

(7.2.%7)

'!'hu, IJthough the abeorption tlf'a. eeeuon i. infinit.el;y Wge, neutrona are l10wed down to energiee < E I j and indeed more are elcwed doWD Ule narrower the " dangel'OWl zone" i.. Thit lituation an- out of the fad tha' DflUtl'ou are moderated from enClrgiel!l > E 1 to ClDergi.. < 8 1 in OM colliaion j thUI th e;y jump over the dangoroul zone. Th_ ooneiderationll are important for und entanding th e p-factor in an a beorber whOll6 Cl'O/lIJ IeCtlon hu " ry high nllKlna.D.oe. . U t.be ratio of the ecattering Cf'08I eeetion to t he a beorption CroM eeotion i. ind ependent of energ;r and equal to P. the n

-' I 4r f 1+ 1 -r p (8)- I · -

( . ) I ~" Ig .

(7.2.28)

Th e neukon flul[ ie then given by •

IIl (E)-

(')liT X; EO 8

c 1

IHI') I'

rv

F' ,;;

(. ~

~-

(7.2.29)

!fI (E) ie thua proportional to g- 1I11+ ~ . i.•. • it; in~ more alawl;y than l iE with deoreuing energ;r.

7.3, Slowing Do... in Bea.". llIodla (A +I) 7.a.l. Non.Abeorblnr Media Let UI nan MIIume that t he energy of t he neutron eouroe ie again E g • Th•• if 8 = 1. (7.3 .1)

Reduction to a lim pl. diHerential equation ie DOt poeeible here . ai.noe the eoerv IJao oot'W'I in the upper limit of integration. B owenr. b;y uain.g • procedure developed b;y PLt.CU I , one can eo1n the integral equ.tion ltep-...;.e, i.... fin\ in the intervlJ «80 <E<Eg • the n in t he interval «IEO<8<1&80, etc. The rMUlting expreWolY are complicated, and we content ounelftllJ with Mowing the P1r.czek solution 'I', (u) for ..4 _2, .. , and 12 in Fig .7 .3.l'. In the interval

O < u <31n "~. • i.e.• «180 < 8 <80 , th e oollWon de nait y exhibitol o-eilIatiou ("Placuk wiggJe. ") around th e u ymptotic nlue lit. Th _ oecillatiorll are eon, rMM'tcw1 wit.h t.he flo(!' t.! lat • neutron ClalllOllD . ' mOl' an one'lt1 (l - I&) g 'in any Tbe ~

~beIo

DlU~ whicb. han alrNdylllede. oollWan. ill ~ \0 .&I ~ -.tn'bam. 1I{.) 01 U.

only tbo:""* to obW!llOIulJoll _pMrt.e 1IOl1Iu..., is I

In

\lIKIOUided IOUnlO

~ DlU UocI8.

126

Slo win g Dowa

ODe collision . A tfuoontinuity oooun in th e collision denaity at E =«Eo• einee 8OU1"OO neutrone can at moet red the energy «Eg on their first collision . n can eMily be mown that ".(/&) =I1'E uymptotically. The elowlng-down density 9'(.1'), which in .. non.ab&orbing medium i8 equal to th e lIOlU1:l& denaity 8 =1 when E -:;i,Eg • iagivon by

q-

,.j

,E_«E'

,, (E) {1_ a' E' d E

". -..If,f -(~=:)~-, ~::- - -;- 11 + I ~ ;-ln «1=1.

,•

,.. •, ,.. •

•

•

,it ,7

(7.3.2)

By simple substi t ut ion in Eq. (7.3 .1) it can 6aIIily be shewn tbat. , (E)_lIe E ill .. solution .

We ha ve now derived th e important reeult that in th e ....ymptoti c .. ~ in a ncn -sbsorbing medium th e elowing.down den,ity q(8) = 8 and tbe flux 41(.1') &ro connected by th e relation

-....

(7.3 .3)

. ,

"• , , r "'VI , , •• "An~­

..-

Since the ecattering

I-

.

CI'088

a&etioNi of mOllt

mod6rato.,. are oon8t.&nf. be tween 1 (IV and eev er.t key. the flux hAll a IJE-llpootrum in t ll i. "epithermal" region . FreqUtlntJy. one writes

f1l.pl = El-~. (7.3.4)

s

For a given .lowing-down denaity, <1>' 1>1 i. larger the lInaUet the slowing-down power of tb e ".. 7.1.1. Tbt PIaaoIr. ...1. _ Ibo moderator ie. All tbeae reaultll obviouilly a1IIo ooU_ d 17 Iw _ ......... ......._ oI - . . .""(II..... vt..,lIIoo hold for hydrogen, u we U D easily IMle by com• aad Willi .... 100. Ill!. ) parilKlD with Eq. (7.2.19). If inlltead of a moooenerget.i o source at E ""EQ there ie a source distribution, then

'or

f 8 (E')'r(E)4E', ,'-

. IE) ~8(E) +

(7.3.6)

Here Y1.. (E) i. the oolliai.on deMUy that ari_ trom a ecurce ot unit .trength at Jr, in other worde the Placzek eotuti on diaoUll'ed above. If f .. (E) is replaced by iteuymptotio value I/€E, Eq. (7.3.6) beccmee ,,(.&') _8(E)+

l~

,,J 8(E') dE' .

(7.3.61.)

For energiN which are .maIler than the IOWMt energy in the source . pectrum B V (E) - E.I'

(7.3.6b)

'17 Fi«;. '1.3.2 abo..... the nux lfI (.)= E
.-•.

!' H+iH++-t+-H

then defined .. \I'(E ) "" lfI (EIl: I 'I == !P(g )E• •

... . ... L f {;. ,,(E') l::::;i;'

and in the ..ymptotic range

• JEd er 0 1E) - r -E,- Ol'-' II ..,r .

3; SH-H"Nkt-H+l

(7.3.7&)

i -I •

q-

(7.3.'l b)

se:

1_1 .

U we &Nume that all the _Uering (lr()IIII that the ntio.

I, JE.

al'e

-rnonA 'Vary aimlluly with energy,l.e.,

independent of til. energy, then ,, (8) _

-;l. it ..

tolut ion of Eq. (7.3.7.) . The oolllltant OI.n be detonninod fro m Eq. (7.3 .7 b) :

E.,

1!, q= OOll8t'L -E" E, . I-I

(7.3 .8&)

I

U wo lntrodace the .vo~ logarithmio llnergy decrement of tho mb tufe (If nu oloi

•

I: E.ll,

l _ i::l.-E,

(7.3,8 b)

it followa that

IUD) '!'b Ull in a mixture tho slowing-down powtlre of t ho individual oompono nte oomhino

additively, 7.a.2. AblOrblng Media: W1GNU'. A..pprol1matlon Lethargy ia t he moet oonvoniont varia ble for the tl'ee.tment of . Iowing do wn in abeo rbing media; if theee ill. m onoonergctJo lOuroa at v _O. t.hen for 'U > 0 Eo - 1_ _ 0()

J ·~':E; "("') · d.' . .-1__" _« J I «

,, (1/.) =

(7.3.91.)

_ - 110 (1.1-)

9 (1/.)=

E. ~':E. ,, (v ')

1-«

d 1/.' .

(7.3 .9b)

_ - Iii
Sf •• --

"-

~+x. p(v).

(7.3 .90)

". R-nt.ly. Bona hu ginn aD anaIJtoio .HI1UOIl of u-. eqna.tiobl for arbitrvy E. and E,. . Ilia method of 1OJ11\ioa. fa DUlheJDatiwJJ1 diffioul\; and Je.d. to nry oomplio&ted espnMiona. W. renrict ouneJ.,.. ben to the w.eu.iOD of .vera! importaa~ appr'O%imate methoda. 1M U8 im~ .. monoenerget1o8OUl"Oe of unit Itrength at ",-0 In .. moderating medium . The Ilowing-down denaity g(.} at v is then equal to th o rNOnanoo _ po probability p lv) at .. The Dumber of neutr-on- at.orbed per cm' and 8&0 during modentioD to le\bargy '" is 1-7(-). No.... we take .(o)

c:> .!._ E

0

I

pt- )

(11) _

I

(7.3.10)

fOl' the collision deraily. Thi, i' to be undeeetccd in the lollo,"ng 8Onae : U th e ab.QrptJon were &ero, l iE would be t ho u ymptotio oolliaion d enli ty nwultlng from .. eow"Oe of unit Itrength . Bowe nI', during moderation to lothugy _, I - p (_) neuWDI an ab.orbed. 'I'bit a b.orpt.ioD e..n be fonn.uy repreeenkd by . negat;ive IOUf'Ce oI .tnlogth I- p(v). The ..ymptotio oolliWon denaif;y of to m. aource ia

I~!!!. and mua\

be lubtracted, hom the

in the at-noe of abeorption. Uling Eq. (7 .3.10), Eq. (7.3 .De)

eo~on

deDJlity lIE ....hich provail8

immediately be int0gr6ted :

C&Q

:~ --_ I;~:E.) :(01, \ p(v) :. ,

!

l(E,,+.tJ

(7.3.11)

.

In term. of the energy tb&llCl equatioDl belXlme ., r" a p (8) _ _ • f(E,,+ tOi - ..-

-f

&

E

f , B)

v-( 1-IlEa+.t;)B'

(7.3.12&)

(7.3.12bl

The deciain approximation in Eq.. (7.3.10-12) ia the aaeu.mption every...here of the uym ptotic nlue liEfOl' the co1liaioD delWty. i.e.. \be neglect 01 the .. Pla.czek wiggIN " i.D the firat. fe... oolliaion interTala _ W0 can, te fact, aeeume that the IOW"OfI ia to be found " TOl7high ooergiee and. that the ablorption fint a ppean at mucb lower ooergiee. 110 th at tbo non."ym ptotio oeoillationl atieing from the ecurce play no role. Ho.... ver. the oollieion denloity arUing from t he " negati ve IIOW'ON" &leooxhi bit. non _ ymptotio OIIOinationa. and .... must write more euotly

•

,(.)ZII -;'+ J·~:~ ft.. (. ) rl. · . •

(7.3.13)

Here ' .. (11) ill the oolli8ion donloity at 16 in . non.abeorbing med.iwn due to . uni t .auroe at 11' . If ill replaced by it. ..ymptotio value lIE, Eq. (7.3.10) &@;ain ,..wt.. We will return to Eq. ('7.3.13), ... hich ..... fint formulated by WUlfBKBO aDd Wlon. and inde penden tly by CoUOOLD, in th e Den MetlOll.

ft.'(.)

.

,

SIo1JiDg Dowu In He&ry Media (.of + 1)

Eq• . (7.3.12.) and (7.3.12b), "hich are oonaequenON 01 Eq. (7.3.10), are therefore not eDOt;: they are frequently cNled the WIONU approzim&tiou. Only lor hydrogen are they eDOt;, .. O&D be _n by compaNon with Eql. ('7.2.22) and (7.2.26). Aa "e have previo\Wy MeA. in hydrogen then are no non.uym ptotio oecillatiolY in th e oolliaion del:Wty DIU' the eouroe. Then are two men important OUOII in which &p. (7.3 .12a) aDd (7.S.I%b) are quite accurate. One it the ClUe of very -...k abaolption, E.<E•. In thi8 ceee, the de'riation of the ooIliIlIion density from the colliaion demity in a DOD.-abeorbins medium illO amall that.the nou.uymptotic oecillat.iolY ca.n. be neglected.. One often writa t.hen

"tr.. w - n; -r

J

(7.3.14a)

JI(8) - ... ~d

"Ej

(7.3.IU I

"'(EI -,E,E ' OceuionaUy,

t.~

equatiOl18 &re al80 referred to .. the

FUIII

approximation.

The ot her, by far more im portant Ca&e i . that 01 a beorption by Iharp, wen. /l(Iparated re-onanOll8. U th e width .dtI 01 th e re-onaooe b. unall com pared to th e colliaion intern.l ln (ll«1 and if the diataneo bet'Wtltln n.IODAD<XlII it /l(IvenJ colliaion intervala, Eqa. (7.3.12a) and (7.S.12 b) reF -I. aD exoeUent app-oximation. In thia cue, the oolliaion denai.ty inaide the reeonanoe regioD i8 .

· f

X.

1-1-- ,.')

' .(t1I ___ "Olio}".:r;+E;,,(t1') --,-=;- tltl' .... , (t1)

(7.3.16)

Le., it i8 nearly equal to th e conatant oolJiaion denaity that woWd be ~nt if the reeonanoe were not. there. Thi.I 00mM about bouGIe the contribution of ooUlQoDAI in the retIOI1&DOfI region to the inf.epal can be negJeet.ed. (ih,
1-

t(.r.~ E,l . 'The

'l'b.:l,.

pro'-bili\.y tbat the _troD. i. .tleorbed in tb. lim _

f r,lPt. 14.--}! r..z::.r. 4.... 1f"· ..z:. ... -T! T r.+E. i.

Tbe

~Wty

_

pel

1*1 t.b.iII

'Moe 18

4 + ZO· ..

4.' .or beoaue.d -
••thtllOgtloD eratiocl

that It -

cspWre in ..,...] ...-ive _

. tblprvMbilit)' tbatduriDf mod. _

M

(U .I·I

•

.

,

'I'be oonditioM required for the a pplicati on of ~ formulu are thUi n the r _II fulfilled in maar practbJ OUN ; Etp. (7.3.12.) aDd (7.3.l ib) are widelyUMd for the caJoulation of .lowing down in homogeneou. medl.. Eq. (7.3.121.) t. .Iao \.be Marting point in \be definnioa of the ruotIdllU iMgml, .hich will be further ~in &c.7 .•. One can conatruet a 011. in which the Wigner appro:lim au on leadll to • result. For th il P~.let UI.-ume t hd I. ia infini te in an energy interval E1:iiiE' $.E. and uro el.,.,wbere . I t tben folloWI from Eq . (7.3.1fa) tb atlor E< E I

,alae

eMil,.

E, )'" . , (8) .... ( E~

Let 1U now ~er t he . peciaJ e-o E) =-«E, ; th en it faUo_ that p "", «1 1I ( ....41-· for large .A ). In f.et., bo_ nr, all t be DeutronI In! .b8or~ between E I eed 8 • • einoe DO Deut.roI:l eaD jump OTel' tau. interval. ThUll p mut be z,ero. Of ooune, &be _ jut oooaideM dcee Dot. ooeur in pr.ctioe. We .hall nevertbe'continue t.o be intel'Mtoed in belt« a ppro:Pm.t.ionI to tbe rMOn&nO& e&e&pe probability in ~ in which the.~ t t o . MlCltion v&riel 110. 1, lrith eoM'J)' eed cannot be nesleet.ed compand to th e /lC&tter\ng ~ lMlCtion.

7.3.3. The OotlrUel-Qrellling ApproumaUon

An in\qp'O-differentilU equation for the obtained from &
~nance

Me_PO probability ce n be

(7.3.17)

la on.!« 10 obtUD t.hia nlIRl1\ the funcUon " .. (_) hu been deeceepceed into an uymptoQc ~ and. non...ympt.oticp&Z't &lfollo_ : " .. I1l) = -t- +{"•.(V)- ~.}. Neglect of the DOO.uymptotiCl pari 1e&de immediately to the Wigner approrima . tion of Boo. 7.3.2. Eq . 17.3.171 i. tbe.tarting point of !lOvera l higb er -order tpo prorimttiOD prooeduree for cal culating the rMORanOll cecape probability. It bu MeD ~lm by WEDrBPO and WIOIfP and by CoRlfGOLD in va rioUB high-order appronmatiODl. We dMcribe nen alimple ap proximat.e IIOlution due to DUllNU. A. _ have _ in Sec. 7.3.1, the _ .....ym ptotie part. of ¥.- (1l) i. appreciably diffefoeDtfrom 1«'0 only in the int.ernl 11' < ,, < ,, ' + 3 ln (l /lll. Therefore , _ lilt by _y of appRlltimatioD

j :${.' I.)- -a••·~ ~~j{.'I·)+} ··· 1 •

•

-

~ *J{".I1l'I nu. a pproximatioa

•

-H",,',

17.3.18)

will be good if II" d1l chaDp but little in one oollWon intenal; t.hu it. bad whe athe .~croMMCtioahuahalpreeont.DoeI. Tbe

i.nt.egnJ.

00

t-o evaluated by Dusna l , who gi,.. AI

the right-hand aide baa

iu nJue

j{".(t1) - .~} d. _ ~

•

with

- I

(7.3.19&)

. Ia'Uj_1 ,, - 1- 2(1-_jt .

(7.3.l9 b)

()' -=I for A = 1, 0.584 for A =2, and 0.138 for A = 12.) Ulling Eqe. (7.3.191.) and (7.3.18). Eq . (7.3.17) beeemee

"i'. i;

Eo ,.E.HE.

=* -

(7.3.20. )

p (,, ) •

•

-J,t:;7u;-

p (t1)_c ' 01'

(7 .3.20 b)

finally

(7.3.200) Eq . (7.3.200) "'AI finot derived by another mean. by GOUTZIL and GRJ:(JLIJ(Q and for tbill reuon is often called. the Goertzel.Greu!ing appromnation to th e fMOn&nce escape probability. For a alowly varying captW'll crou llt'JCtion , it repreaente a conaidera bly better a ppronmation thaa t he Wigner appro:limatioD. An important cue of 1II0" ly varying capture is the cue E. (8)-1fl'E. U th e IlCattering crou .eet.ioo it corataDt. the int.egratioo can be ez plicitly carried out ; if EG- ..... the follo1ring e~oo rent" for the r-ooaace e-cape probability : p (8 ) _ ( l

+ +yf -Eo''l)-'' r ..

(7.3.2 1)

7.4. Beecnenee Integral8 7,·4.1, DefinJUon or tbe BetoDao~ Integral Let us ims.gine en infinite medium containing a bomogeneou. di.tribution of higb -energy aouroee whoeeIltrength it ODO M utron per om' and see. ut an a'-orher ' ubstance alao be bomogeneoualy diatributed with a density of N I.toma per em' . Let the I.t.orber have &0 isolated rMOD&nce .h~ widt h i,1DI&II compared to the coUition interval. Acconling to Sec. 7.3.2, the Dumber of Q8utroll8 a t.orbed per em' aDd IIllC ill (7.4.11

let 111. now ..mte ...

AI

•• - NI J E,.

17.• .•)

III

E, it the neutron flu (per unit lethargy) which would be pre.eot if there were no re.onanoe. Here

L;. -E•• +Na..... I

Op. oiL, P.13.

..

(7.".3)

.n .benE.... the~_leOtioQofthemodentor.odCJ'''''' atbe microeoopio WJ
- n~£\ Ih,-J- ::'{(.I)~.0'.. -J I.-JE • • fI, ••

•I

1+ 0',":(~Jl,. fl. {El

( I

1+ ", •

(/,

~•• • (7.....4)

In tho IIOOODd It.ep _ have in t.roduoed the following notation

Xo tI, - y

_c,. 11 +0'"".

(1.• .6&)

cr.. PYM UIe _ Ueriag ~ IeCl'tion (with tbe uOlIp t ion of the rMOnanOlllC&tt.ering) per abeorber atom. In addition,

~{.) _ ~_

(7.•.tsb)

it the ~o AIIODaD.OlI _ It.eriDg _ IeCtton, Le., the IC&Uering U'()8I IeCtion of the ..beorber ..tom mina. The defi.ni.ng Eq . (7.4.41 bold.t for .. .mpe ~0lI . The integration need. only extend a Te!' '-he rMOnanoe: but it can be oxtonded from 0 to 00 without introduaiDg &Dy error, liDoe the abeorpUon Ol'O&ll MCtion Tam.he. e't'eryw'hel'8 OQ~ 1he fMOOat'lO". the effective neotW108 integr&l for .. ablorber ~ baa m.&Il1 DIn'OW' J'eIOQ&QON it defined in t he _ .....y. prond.ed only U1a.t ~ bet_ *be .-onaaOM .... IUfficMatJy larp. In thia r Der&l. _ . " . ilnolonpr equAl ee N bot rat.ber it ginn by

0', ..

U,a.&1lr.

re..a

t..!lr.

,. _I_.-JfI

...,.1'.,

(7.• ,6)

Onl,. whon the expcmeot 11 amalJ. doel Eq. (7.( .2japply. 'The rtIIOII ~oe lDt.6gr&la defioed here for bomoBeoeout minur. are fl'eqoently ..ued ~ ~ c e int.egn.ll. H~ 1'MOQ&Z10l1 int.ogr'&bl, whioh we will Itu dy in another CIOD.D8Ction in Chapter 12. lin of gn&ter pc'acti.0&I Importance. The following , IOmewhat formal oonaideratiODll _rYe largely u prep-ratio n for th e discUMion in Chapter 12. Our soU here will be the O&Ioulation of the effective reeonanoe int.egral aod \h e reeonanOli _pe probability from the Bre it..Wigner puameten of t h e ~. Before _ peeeeed, ho_TW, _ must in troduce a few ad . ditional CODOlIpM. Aa Eq. (1••.• ) abo..... for a ginu ~0lI ablOrber, I . i8 largen wben 0'" _00, t.e., when the ablOrber i8 ~nt in the moderator in " infinite dilutiou"'. Tho.

1.-1._J 4M_J fI, (M)

fI. (EI 4:

.

(7.• .7)

UlJII&1Iy. l. i8 called tbe " rMOD&D.ce inlegr'al at in.fiDi&e dilution" or llimply &he " r-onanoe iDtegnJ" .1. caa euily be det.ermiDed nperimeu&l.1ly (el. Qla.p w 12). Ifill ~ with iDoreuing oonoeutration of &he .t.orber, i.e ., with decrM8ing fI" . Thi8 OOtnel about beoalZle the flUll: at the reeonanoe, whiob i8 proportioDaJ. to IJ(.r.+~, bu a depr.-.iOD which A. deeper t.be am.aIler r,. i8 oompand to E. (.tf-ehieldin,); " iDfini&e dili&tioo. \heN ill DO flUll: de~ becaIIIe E.< E,. I

~

N _0,

'PI" apPf'O'Mb--.

'33 '1 .... 2. Calculation 01 the B80nance Internl with Help of the BreJ&-WlgnerFormula For the purposes of simplicity, the interference between potential andreson&nOO 8C&ttering will be negleoted. We write, &II in Sea. 4.1.2 (7.4.S.)

with (7•••8b)

r" (") »

-., r

(Ea. ga)"

(7.4.8 e)

I+ ~

Then

(7.4.D)

Next we introduce

%=

E

n:

R . . . new variable of ilIt.egn.tion and u.e tbe ab-

breviation ' '''0'.10', ; if we notice that the integraUon oan be extClnded frorn ,"'" to + DO and l IB replaced by IJER bece.WJe of the aharpn_ of the ..-mance, we obtain

('1."'.10) The integral it tranaformoo by the aubstitution gI =,sI/(I+l) into

=

I eft

1 _ eft

no,ryl2KR

('1.'.11)

¥t(!+,)

IIf1tr.,12611 =

Yt+aJD'l

I ...

Yl+oJa"

1...="I1. r"'2EJI ... 2n'I~g

ffi

(7.4.12)

(7.4. 13)

ia tbe limiting value a t. " infini te dilution". Eq. (7.4.,12) mo," the effeot of .elf· thielding quite eler.rly. For very high abllorber oonoentrationJ, (1,/u.<1 and

letl~Yf1,/(fo 1"", . In the 0Me of V6Tf high absorber ooncentrt.tioni, the fennul.. developed hero are no longer immediately applicable . We urnmed thU the rMODIoDOllB were naJTOW, Le., that their width' ...."' amall compared to. oollition internJ.. AI ..

". uu.

rule, requiremeD ~ " fulfilled if the wid th being oonaidered i8 t be natura! width of the l'NOD&DOIII ; in other wordI, tho condition r « I -«)8.. i8 fulfilled if th e mod«&t.ion U b,light nuclei. However, \be relevant width i. not. th e natura l lridt.h of \be reeon&o08 but rather the riith 01 t hat on«gy internl in ..hich tbe rMOnanCle C~ eeotion - - ( ; : 8 -)" illargtlr thlln tbe OOnlt&ut _ t te riug en:-

1+ -

MOtion

'"

V;;" -, conditio n tbat is not alwaye fulfilled whOrl the

CI. , 'I'hia width ia of thern order of magnitudlt of r.,,1lW r

1_ _11 .

and we muat

require tha' roa be « 1-«)8 " . & ab.orber ooDoeDtration ia high. Thia. problem hal been ItuWed In detail by DauB" among othe.... We .baU l'elItrict ourwolt'CllI in what. follow. to the C&Ie in which fa alwa)'8 « I -Cl)8• . Unfortunately. Eq. (7.4.11) OODWna lItill another I6riOUS overaimpl.ific.tion : The Brei'-WlgOW formula in the form hitherto dilcUllled d-mbM tbe oroeo. I8etiorw for ~ readionI of neutrora with nuclei which are initially at l'elIt. In order to UM tbe Breil.Wigner formula in pnctioe we must modify it to I.&ko into acclOQQt t.be 1Jonlt:' tiled, whicb CC/IQM abotn beuuae of the the rmal moUon of lobe .t.c:wber n.uclei.

r.

1.4.1. TIle Doppler BroUenla( .r

BeeoIl&D.ef1

Let u bombud • group of aloonUo nuclei with

MlUnon.

UnN of velocity . . U tb e

awek nuclei are initially at I'M', t he fMOtio u rate it proportional ~ "O'{. ). We aaume, howe""", tb.r.l the nu clei ban .. Maxwell diatribution of nlocitiM

JI .\I

.li t'"

P (Y)dY = ( h i"1'~ , - u r, b

Y'dY .

(7 .4 .14)

The relatin velocity it then

""l -IV-~l

(7.4.16)

&Del the rMCtion rate i , then proportional to ~

f ".... 0' (11....)P(_...) dll... sa,II V - "1 0'(1 V-Ill) P ( V) d V

151

t>cJdf (" j .

(7.... 16)

In tbe Jut atep we have introduced a.n eJltditlt erou 1UIi.cm th .. t muat be used to deearibe t.he int.enction of btlutr0n8 with nuclei in thermal motion. We e.. n hnmed.i..tely eee from Eq. (7.• •16) tbat. ...hen O'{.,)_I/" , 0'(.,1= 0'. ,. (.,), .. 1&Clt. which it int.uitively cle&!'. On the other Mnd, if a it oona t&nt., O'IIft{. ) it by no DltI&Il8 &1••1' equ.el to 0', .. will be lhoWll in Sec. 10.1. H ere _ ere int.el'eeted in t be _ of r-.on.enc:e reactiona, in.hich 0'(. 1ill deecribed by the Bre it.Wigner form ula.. At one IlUgbt ~, in t.hia _ tbe na.tunJ line it bfOlldelltld, IinOll btlutrona in cident. oft r-:aDNlOll CQ. Rill "fiDd " nuclei for which the ~ condition it fnlfilIed. A mflMQJ'fl of lobe amea.ring out of tbe ene rgy g "' ,."'/2 of th e incident neutron ie given by the qnantity

-~ [( V + .)' t

( V .)'] _ :h,n V.,.

fo

P

_ 1V~6~ ~ .

('U .l 7)

'"

Here A a the Ina. number of the Itraclr. nucleul. 'Th1lll there "PPM" in pl_ of the natural line width ran effecti:no line width of the order of magnit ude of

l/--p

U ·RJ:T.

(.. long u

.

the latter bi Dot NIlan compared to F , r.e., .. JoDg .. t he Doppler effllOt i. not ..ltogether neg ligib le ). T he e&l.culatio n of croft (R ) with tbe help of Eqa . (7.4.14.1 and (7.'-16) &rid t he Breit.Wigner fonnula ill lltTaight forward bu t 1e.dA to vtlry complicated e xpf'e&llion.l. For th o important special ewe in whioh th e l'8Onance enl:lrgy Ep, ia > J:T. And th e interference term. are neglected, th e 0J'06lI .eet ioM in the neighborhood of .. rtlBOnaoce are glv"'D by

a,oa(B>-O', -;' ,, (8, % 11

(7.4.18)

a:::;(EI=o.4- ,,(8,z) . H ere

%

I'.

.

E- 8 a f··~ · un..ullrmore

agun ~ ;

8 =-~;

.1-

VU~tT.

(7.4.19)

(7 .( .20)

Tbe functio n ,,(B . a') no... pvee the line Ih&pe and. appearl in place of the nat ural line ..ba pe (1 + ~) -I. When TI _ O. Le., wben .1_0 and8 _CIQ, ,, (8. %1 approaeh. the fl&tnraJ. line ,hapo ("weak Doppler broadening " ). When 8 _ 0. Le., .... ben r<.1. then for z not '-00 large 1 I ,, (8 '%)= 2 y;i 8e- t • .. _ '2'

r - 1&'-·')'

y;i 'if II

«

>,

(7....21)

In thi .

ClaM of ".trang" Doppler bro&dening , th e line .hape i. a Oal18lian whc»e width i . th e Doppkr ~h Ll that WAI Int roduced above on heun.tic grou nd• .

When

~ > ;'" Y' (8,~) i3 a lwar- alIymptotic to the natwalline .hape, Le., ,(8 , ~) ..... (l + zI)-I+ bigher te rm, in (l+ zI )·I.

(7....22)

I n ot her worde, , ufficientJy fat: from th e line center th. natural line . bape alway. prevaill. DRZlBD ball given a detailed diAeUlllion of th e propertiee of th e ". funct ion. The funct ion hu been '-bulated by Ron u al. A detailed de ri ...tion of Eq. (7.... 20) can be found, for e DlD ple, in SoLBRla; a1lIo IIfJfI A»UR and NAUBOn. From now on we . ball drop the indez " eU" on t he ertl6lIlIOetion. So far, our CIOnIideratioDl have been bu-i on the _ umption that the ab.orbe r atom . have a Maxwell distribution of velocitNla. ThiI would be the cue if the a b80rptio n occuned in .. perfect gu. In practice, tb e abeorber II eit her a liquid or a lOlid, and the validit y of our formula ia in lOme dou bt . However. LA... ball . hown that if a IOlid body can be de.cribed either by t he Dun or the EI!fITKUf model and if the te mperature II nol-too Low. t he Doppler effect. can be ca.lcu1Med wit h Eq. (7... .1..I, aave t hat in place of the actual thermodynamic te mperat ure a .I ightly high er effective temperat ure aPJlNl-.

... 7.4.41. ~II 01 Ole a.nw.ee 1Die«nlTakIn, the Doppler Etreet laM A~llD' Sub.ti.tuUoo. of Eq. (7.•.18) into CU.f ) yield1

•

- i'? !•• •

-.

,, (6 .s) d:rr .

(1.<.23)

,, (6,.&')+'

It. iii DO$ po.i.bIe \0 enhaate t.bi. in\ep'U in elo.d form.. I>ua1f:n." _ u.. ADu:a .. .z.ohu CIleW&t.ed. it a umericaUy eed tabulated. it for UriOUA TaluM of8N1d ' ; -oDHI of~ l'Mlllt. .... lJhown in Fjg. 1.4.1. For very bigh dilution and for very higb oonoentht.ion of the abaorbor ' f---' ~ -- nuclei, the curve. approach the cutTe for 8 .... 00. Le., for T _O -K.; in between the Doppler eHect . Iwa,. inOl6MM the eeeo, nence integr al. and in faotineteaMll it more the . malIer 8 II, Le., the l~r the temper• ' ature b . Thill can be interpreted. in t he following ... ,. : At-very high dilutio n, t> l . .!. ;;cand , (8. z) oa.n be neglected oompared in the denominator 01 Eq. (7.4.23 ). Then'

·

"no-- ..---I• ,

~ '< ~:

.,,

' ''R'

to '

e--

•

•

,,

,

·,

.

•

.

r

--1II

t-nr

r

.....

_ a£j!1 _ 1

•

--' .

(7.4.24)

.

•

Eq. (7.4. 2-40 ) MY' that. a t. infinite dilution t he Doppler llftect de. not affect the int.egnted reeonanoe abeorption. In thi, cue, the energy dependence of t he nUl: D6U' t he rMOnaD08 i. Imall. and the rMOnanoe int.egraJ b lIilUply an in tegral oter

-

•

r (..,,·zT.rr'J

-....-.

. I

If1lJ - '~ f' IB,Zld.:t:

th e crou aection. which natorally ia not influenced by the Doppler eHed. At higher oonc::entrationa of tho abeorber' nuolei, the flu abo.. .. depreMioa MU the ~Cll. Owing \0 the Doppler effect, thw de~ it broader aDd &tier than. in the cue T. _O ; .. .. rwoI\, the .U' hieldiat: it amalJer t han in the _ T. _ Oand the rMODNlCll irltegr&l ia larger . rta. U .l .

n.~

. . . . . . III . . - ..

__

....no.

.

III

ca.

111

In tho cue of very high oonoentn.Lion thore 1.1 an extremely Wge au depreeaion. th.., the central put (If the l'MOn&rIOO doeI not oontribute .t all to tho TrJue of the reaonanoo int:egnJ. 1D the winga, _hiah then alone oontribute to the resonance integral, tho l'MOnanoo line hal tbe natural line . hape (lee Eq. (7.4..22)] ; for this reeeon, the reeona.noe integn.l 1.1 not affected by the Dopplv effed;, 10

Chapter 7: Retereneea Goo"'" Da U lIn. L. l Reeonuwe Ab.orption ill Nuolev RNooton. Odont·Loodon·N_ York·Pam: Persamon PreH 1960. Wmll'JlllloO. A. !II., and E. P. WIOII'D: The Ph)"lctJ Theory of Neutfon Chaln Ret.otota. Cbic.go : Chicago Univeni ty PreH 1968, Nped.J1y Cb..p. X : Energy Speort.rum During

-~

8peelaJl

KttnDe. H. : Nukleonik i . 33 (1963). } The A~ N1lDlber ol 00llW0na in J4J.acm1. W. C. 01: N\llll. Sot Eng . i . 338 (19611). Moden.tioa hl ElMtio Col1WoD. ' PuazH. G. ; Ph,.. Jtt,• •• t. ~ (1M6) 181owiD& Do.,. in Noo.-Abeorbiug Xedia). RoWUlfDe., 0. : J. N oel. Energy A 11. 160 (1960 ); A 11, 14. (1960) (SloW'ing Dow:o of Fia10n Noutrot:M). BIDlfDZ. R.: Nllel. Sol. Eng. 10,219 (11161) (E uct Solution of the 8lotriDg.Dow:a Eqution). CoOGOLD, N.: Proo. Ph,.. Boo. (London) ~ 20, 793 ( UI67). c.JC!Ulllti
I

WIOlClPl, E . P•• d at : J . Appt Ph".. !C,200 (111M) (lWoa.r.nce 1Iltegn4 and ~ Abeorption). ADLD,F. T••aadY.D. Nu.ooJ'J': J .Nuol.EnergyA&B ... 209{11l611. \ OonTSllL" G.: 0 - -. 1966 P/813 ; Vol. 14 p. "2 DoppIM' Rolli. H .• d oJ.: BNL.251 (1IIM) 1 WAPD·8R llOO (111M). DroAdenini of So Ulal:O. A. W.: Nuel. Sci. EllI. 10, un (11l81). R.eaon&Dce IJnN. _ Amer. J . Ph".. n, 251 (198 1). ADIoD,F.T.,1o.W.NoJU)lL&lJ(. aDd ~tion of the 1Woo&noe ID~ ~D$n 1968 P/I98l! ; Vol IS p, 1M. fCC' Doppler.BroadeDed ReMIDaD... Da ltl.... 10. : Nuel . Sci. Eng. I, 68 (1966).

G.W.HnoulC:I

8. The Spatial Distribution or Moderated Neutrons In thia chapter we shall calcwate the lpace and energy dUtribution of neutrollll during the slowing-down prooeall. Our gorJ will be to obtai n the flux 4)( ... B) or the Ilowing-down denaity q( ... E) ari.mg from given ~urcee. It will turn out that thi . genenJ problem i. coneiderably more difficult than the two lIpl'lcial eaaM of it previowJly dillCUlIIll'ld, m ., diffusion without moderation and moderation without diffUllion. We can eaaily _ the 1'lIMOM for the difficulty if we oonaider. for e:u.mple, the neutron field due to a point source of fait neutrollll in an inf"mite medium. At large dietanC611 from th e ecerce, the neutron flulI: ill predominantly due to neutrollll that have made no colliaions or at moo a few ImaIl.angle oollilionlJ that produce only a emall energy lou. Their distribution of directIonI IUangly anisotropio. and • deecription of the diffull,jon prooeea by FlCll's law il no longer

u.

I

ct. fooklote on p. 63.

138 ~ble .

For t hia ft&tOn. the treatment of " deep penetration" problema i. particularly d.iH.icult. They ue mai nly of interest in ahielding calowatioM, and ". wtll ClOIUid. r them no fwother hen' (cf . HOlll'a .. well .. Va M . and Wr oK). At amaller diaw.nDN - t bU limitation will be made more precile later - the dil. tribuLion of neu&l'on wreeti0n8 (with tbeuCllptWn of th. energy nngeimmediat.ely below the 1IOW"C'Je en&rgy) ~ only _kly anieotropie. and with oert&in lim itations Fla '. law i, nlid. The fiuJ: e&Il therefore be described approximately by an energy~ependont diffuaion equation. Even lhia equation ill not ~uble in general. and ""' ani fotud to ma ke furt.her approximation-. I n "h_vy" modera to... (Be, p phi"'l. a very aimple a pproximation called age theory I. JlO8'ible. a nd we ahallltudy it in dewl in Sec.8..2. For H,O und D.O . AI _II .. other proton. &nd deuteron.containing I Ub.'tanON, ~ theory givtlll very inaoeurate reeulta; and in Sec. 8.3 _ itball become .cquai nW with lho rudj menta of IIOm e better a pproxi mations. Final ly, in Sec. 8.• , we .hall caloulate the diJItributi on of th ermal neutron. due to .pecifJed. l ut noutron IOUroee in Mlveral l impl" CUM. HOWOl'er, before _ proceed with OUf . t udy of the vanoue th eoriM, _ . hal l introd uce . n important empirical quantity, the .a.called m&an . quued . Icwing-down dleeence.

8.1. Tbe Mean Squared Slowing-Down Distan ee 8.1.1. DtrtnltJoD Dr rI Let UI conNder a n i.atropic point IlOUfee of wt neutroN in an iufinite med ium . Let UI _UtH tha& _ know the tJowing.down denajt1 q(r . E>. which in tbiI _ i. -Pherical1y . ynl.lnetrio and only depend- on t he di.t&noe r from the ecuree . W. can tben c~ the . lowing-doWD ~ hy the quntity fl. tb e a",rage of the aquare of the MlW'oe d.Ytanoe at whkh t.he neutron- pu. th e energy I g . b it gi een by

i~'lr,E) fo lll ,ttlr

<1:• _ -",m-- - - Frequently, we ..t

(8.1.1)

,1,(r.Elfo1l .. tl r

(R.1.2)

where L. i. called t.he . lowiltg..dowrl /.e~A and T.ill called th eage ofthe neutron'. The reMOn for the latter nom.nclature will become clea:r later. A .aM.what dif ferent definition of tbe mean Iquared .lowing-down dittance it baaod on the flux lJ) (r, E ) rather t han on t be . Iowing-down dentity q(r. E ) :

j~ .( r,E) foll .. tlr

;:;'= ' . -- -- - -

(8.1.3)

! 41(r, E) fo ll ,t tlr

•

rp it the mean aqU&nl of the ditUnoe from tbe IOUl'Oe at which the neutrona ~ tile energy E. b it .amewhat greatef than~ . but in mOllt. CUM the di f. feren ce i. unimportant [d. Eq. (8.1.16)]. I I

Some empirieal dat.l on dellp penet.n.tion ('Aft btl follRd in Ch.p~ 16. i.•.• _ lIIond down put ttl. """"1Y It .

'30 Th e Ioignifieanoe 01 theM qQ&llt itie- i. lhat. they are .. dired meuure of the . prMdina out of th e M ut-rona during modere.lion. Th ey c.n Maily and ra t her aecura~ly be dew rmi.ned u perimentally (Cha peer 16) and un be comp&red directly wit h th e pndieti.ona of t he YNioUil appro ldmate tb eore t teaJ eelc u...tion-.

8.1.2. Elemt'Dtary CaleuW.IoDof 'I Let w oofl/lider u. infini te, non ..beorbing medium in whieh neutronlllow d own only by elutic ooIli.iona. Fi g. 8.1.1 . bo... .. typiea.1 p6t h of .. neutron du ri ng the F.

----- - ----- - ,-----~ moderation fro m the 1OUf'08 energy 8 0 to the energy B . Let,. be the free p" tb of .. Muwn after it. ... th colliaion. Th en (S.l ...) Th e l umm.tioD only extend, to a - I beeeuee on the ...th collillion the energy E i , pa-l. U we denote the angle bet _n r. and r " by D•.• • th en (8. 1.6)

The a verage ill to be ea.rried out 1. over tb e wmuthal eeattering angleef • • H of th e individual &C&t tering Pl"OOOll8efl i • 2. ever th e pa th length, ' . between the oollilio n.; 3. over the polar IC&ttering a.ugle8 fJ• •~ I i th i. i. equivalent to an average over ali E. ,

~.,

....

eince the energy 1068 in a coni. ion uniquely det.erminflB the BC&tterlng angle ; • . over All numben of oollisionl that can • lead from the IOUf'CIe energy E. to tbe energy E . Fit- 1. 1.1. Tbo .....1.... bol _ "-•..- • • We begin with t he avtlrage ovtlr the aU, ..-.. " . 04 10." muthal ICattering anglel . The conliderationl connected with this average will make it pceeible to tlIp nlll COlI 0•." in terml of COl 0•.•+1' COl 0. ..1-0+. ' etc., i.e ., in terml of the _ttering anglel for the individual oom.iOOI . Fo.r thU pu.rpoee. we tae a reeunion peoeedeee which can be m ade e1ev by ml>anll of Fig. 8.1.2. Let 111 I UPP'*' we al.reldy know COl 0•.,, _ 1 and let UI calcullte COl (J••" . U 9' il tbe aUruutbal_ttering angle of the ....th oolliliou. t hen aoeording to tbe COline theorem of epberieal trigonometry (8.1.8)

'40

w. mud avenge thia .~OQ Ofti' &l1 po SiDoe aU uimuthal &118_ likely. the term. oontaining dropand we obtain

eo."

-

n Ull, .. we

e&ll

"•.•- -

011'

8•.•_1 _

&r9 equally

".-1",..

(8.1.1. )

8. t1,..... . . . COl (}.. -1,.

(8.1.7 b)

eully ..,.,. 00- " . . .. _ COl "'. ,. tl cot

where for putpoeM of lim plicity we have dropped the overbar that lignifiee th e .."raging prooeu. 'The mMD diatanoe betwoen two colliaiona a ginn by

(8.1.8&)

i .. ,- r.~· ~,

~ -' _

1. - r.U'~ , 4,

- :r:~~.l =- f J:IE.) .

(8.U bl

•

Bere B. it the eDefIY of the neut.roQ after th e..,d ooUDioD.. Thu

~ -' I·fJ.: (EJ+·fJ., (g.)..._...· ·f1A. (B.)oe:-f,.•t , ... ooe8. _t••] . . -1 _ .

(8.1 .9)

SiDM the IO&ttrer'iq angle and the energy 1011 .,.. uniqlHlly connected by U.ela. . of eluLio oolliaiona. we oou1d now e~ - ' • •H in term. of B. and B,+t . W. would then etill have to oarry out tho averagO. over the diatribution of the B. and over the didributiOD of the oolliaion numbel"l. Es.0ll~ in t he cue of pure hydrogen , t.hie prooedure ia very diHiouit. to cr.rry out . 'Therefore, from now OD let W oorWder .. aimp1e appro ximate procedure which 11 n lid in heavy moderaton (..4> 1). In heary' modelaton , th e Ilcnring dOlFD I, due to many OOIli.liOM Neb of which prodUOlll only .. very amall energy 10lMI ; the energy dinribut.ion afte r the ,,~ oolliIion i, then rather aharply concentrated around ~he &~ TalUO E•. H . . DOWa~ on r the ICIottering anglee and notice chM _ 6•.•+1- - 6.+1,. +1 eto. -2f3.A . then . . obtain

~_2[t:~(E.)+~t:A.(R.)":~~IA.(R,.) (~ which . .

ll&Il

n

(8.1.I0a)

&lao write in the form

(8.U Gb) Hen . . hu. &pproDmated ~ a~ TalllM of A: and A. ovor the energy diatribvtion &her the N1 ooIliaion by A:(B.) &ad J.,,(E.). re-pectinly. For thi8 to be & good approsimat.ion, .A ahouJd be luge aDd 1.(8) Ihoold T&rf IIow11

with energy. U both oondiUOOl ue fulfilled, then _

can further

let;

'" (8.1.11)

and ol.t.ain

'1-

2

.-1

- -- j- 1; J:(lI.).

(8.1.12)

l - U"- ' Finally, beeecse of th e large num ber of collWons we C&Q replaoe ~he I u.m.mation by an integration. Th e probability that to colliaioo 0CCUl"I in the energy internJ (E, B+dEj P dEll E ; thus

"

'1- ( " ) !J: (E' )' : .

(8.1.13)

l I- IT •

Tho oont ribution to tha

eJ: ~on

of tho fin" night of tho neutron mun IJtill

be take n into .coount. Th o l urn in Eq . (8.1.12) ooot&ina .. term i~i~ which eGU1W wit hout making .. colliaion. Tho contri bution 01 th_ neutroJUI i. not Included in th e integration in Eq . (8.1.13). We muat therefore.dd .. " Iint.night oolT&Ction " ;doing 10 _ p I. i. du e to th oee neutrons that come directly from tbe

" J I('- IT)•

r),=2 A: IEgl+ - - '-,-

.t:IE') ~ .

(8.1.1.)

Notice here ttl" the factor l-~ h.. t-n omit ted in the oo1Teetioa. The nIUOn for tm. will become clear later (Sea. 8.2.&). Since th e ~ term u

;".01

amall in general and 213,,« 1, wheth er th e fact or I ia incl uded or oM plaY' no im port&nt role in the c.!culatioo of Simil&r ooMidentio Ql yield

'I . " ;P _ 2~ (EQ) + J 2 !J: <E' );- +2J:(E ) 1('- .. ).

(8.1.15)

for the quantit y;:P. Here .. "lMt.fiight. oorTeOUon " %;: (8) oocun .. _11. We uodentaDd tbe a p ~ of tJW, \erm if _ ot-Ye \hat in the caJoulati oa of ij: we tum.med only over the lint 11 - 1 oolliDora, ainee on tbe ..tl oolliJioD the eqergy puaed 8 i t.be energy it larger thaD 1l on the 11-1.. fligh t path. In 1.he oalcru1ation 'of if' we mlllt obYioully t um up \0 t.be 11 +1.. oolliaioa; f. ~n .. \e rm ~ oooun in all \h e tuma. We can euily oonvinoe olll'llelftlll \hat CUI

uu.

the correction te rm Ih ould DOt oontaio t he facto r l-~;r. The expr-ion ;P/6 - 'J} i. oooaaionally ealled the flw:: age. Eq. (8.l. UI) it important beo.UlIll in experimeot.&! determinatJOJUI of tbe 8lowing.down length 1t it ;:P that 1.1 Ulually meuund. (of. Chapter HI). The fonDuiu denloped bere hold nther aocurately in graphite and NpreMot a UlIllfa1 firA aPP"O:Wutioll. in other moderaton.

'" For certain future applic.tiona,

it will prove advantagooua to calculate the mean Iquared lIowing-doWD diBtanoe from a plane source in an infinite medium . Since in this cue the ll1owing-down density depends only on the distance x from

tho 8OUl'OO

.urface.

J• z"q(z. &') k ~ ='"",---­ /fJl:e, E ) Ilz

(8.1.16)

•

Since • plane source ('an be considered .. made up of point eoareee , it must be possible to 6xptell8 iJ; in terms of Some IIimple geometric considerations .bow that (S.Ll7.)

'I.

while in gtlDeral (with ,=1. 2, 3, ...) 'f=(2'+1)~ .

(8.1.17b)

8.1.3. FormulatioD 01 an EJ:~t CaleulatloD 01 ~

Next we wiD shew bow 'I CAn be oalculated by the eo-celled 1IlOrM1IU mdAod (originally propoeed by FaRMI). Thi. mothod can also be used to calculate the quantitiea ~. In Ute dieouMion of thia method, we shall.tudy the PH·approximation to the energy-dependent Uanlport equation. which when N =I i. the point of departure for age theory and oet't&in other appro:rimatioJUl. Let UI DORmer an infinite medium containing at %_0 an infinite plano IIOl1rtle that emit. zero-lethargy eeeecne. The one-dimenelonel traMport equation then reada X,
"'" j JE.tO'-+O,

...

1l'-+v)F(r, 0', 1"1 dO' dll'

+.,~ d(u) d(x) .

I

(8.1.18)

We bave aMumed the source to emit illOtropically, juat &II we did in Sec. 6.2. We proceed exactly u we did in Sec. 6.2.1, Le., we expand E, {O'-+n, u'-+u) in Legendre polynomials of tbe ICattclring angle (J,:

E,ln'-+n, v'-+u) _ ..I; ~ (2J+ljZ'.,(u'-+u)l}(ooa (}.)

•

E,,(u'-+u) - b !E,(O'-+D, u'-+u) l}(cot (}.) am (},Il{J. .

•

(8.1.19a)

(8.1.19b)

When the ICatterillg il ilOtropio in the oentel'.of.maBlly,tem, we can derive the u'-+u) from the relIulti of 8&0. 7.1: The probability that a neutron 01 lethargy u' hu a Ietbargy u after ODe colliaion i'

l'l1'OIIIf leC'tion E,(O' -+0,

.-.' _.-1_-")

,(u'-+u) _ _

for u-In{l/«)< ...'
,(v'-+u)_O

otherwiee.

'" It. lollowa from Eq•. ('1.1.6 ) fond (7.1.7) t hat th e ooeine afthe scat tering Mgle of .. oollillion in which the loth argy changN fro m .' to

"i,

ThUll for _- 10(1/«)<.'<.

.-.. -I ,- c

.-.-I)

E.• (Sl'-+ n. . '_v) _ I:• (v') J.... ! . _I_e 1- . -f·-1I') X

(

xd ooe(}.- A+l - - 1 - -A2-I ' - I

(8.1.20)

I

(8.1.21)

.

The factor If2:1t oceun beee uee .u azimuthal aeat teri ng anglee are eqU&ll)' likely ; th e cJ..fu nction t&kee aooount of the faot that only one val ue of 0.» B. COI'TMpondt to e&Ch letbarJ{y ehenge V-II', From Eq. (8.1.19b) it folIo"", that th e f in t eapeneion coofficiont i, giv en by , -C. -e')

E. , (u'-+ u) =l:', (u') ---y=;-

for

(8.1.2h)

. - 10(1/«)<1/;' <. ,

.r••

i.e., (u'- ul illl t ho Ct'Ool6 section for Ion _ ttaring p~ t hat tramport .. neutron fto m . ' to • . The eeccnd &1p6lWon coefficient I, given by ...

...d ll

.

• ,-Ie - II') ( .4+ 1 - (. ~ ~ )

_ v ) = L'. (t1 ) - I-=;-

- ,- '

A_ I

- - -,

c

. ~ • .))

(8.1.22 b)

for _ -lo(I, «) <.'
. /

E. , (v' -l> a)d v _E. (u'I ,

(8.1.23a)

" + Ia (l /al

f

..

E. a!. '-l>a jd u _E. (u') ooe8', _E, {u')

2 JA. '

(8.1.23 b)

.

I

U we , ubetitute Eq. (8.U 9..) in '-be U'amport &quation and peceeed e notly ... in Sec. 5.2.1, we find . 001 (J

"".'h' + E. (v) '(z, 0 , .) .. TI Is

J" c:~ (2l+I) E

. _1Ii(~1- '

X P,(_8)j' (z, 0 ' , . ') P'(0I»8' ) lin f ' 48' d.'

• Nen, the TeCtor flu

+ f~ 6 (.) 6 (%) •

(8.1.24)

a e.panded in Lepndre polJDOmiU I

' (z, n, v) = ... with

d(.'-l>v)X

•

yo (21+ I) Jj( z. ul P,(_ r-I

•

(J)

Ji (z•• 1-21f/I'(z, n,. )P'(- ' ) UD" d',

•

(8.1.2Sa)

18.1.24b)

.

,

If .... multiply \be traDIpOn Eq . (8.1.24) by }I(00.8) and integrate over D, then UlIing Eq.. (8.I.U), .... ob~ the follollring Bet of N + 1 integro-differenUa. equationl :

!.~~.•) + l; {-) F,(z .•)-

J•

E •• (v'_II) p. (z . 11') d .' + Q6 (1II)6 (*),

S-" U.J-)

J.±!.. 21+1

81j•• (z, _)

a.

+ _11+1 1_

2111_1(*. _' + r ,, ( ) h ... , so, 11

... J•

E.,(t1' - . )P, (%• • ') d.',

(8.1.261

s- lii( lIQ

N al" _ I(*' II) .,.. ( I SN+l h + ""1 ::11:• • -

•

f

1_1 ..• N- l r

('

.¥..'I d•.'

I' (

~I. 11 - . I JI

• - lii( lJa}

Eq•. (8.1.26) obrioUllly reprMent th e generalization of Eqa. (6.2.46) to th e energy-dependent CUll. They are the .t.a.rting point for variOUll approJimate methoda about which we thall leun in later sect.iona. We begin here with th e " momenta I t method. W. define the momenta purely formal.ly by

JI.,

---.

M.,Cv) .-

I;{%. w) tb .

(8.1.27)

JIt(a',.)" __ +/ 4) (%, . ) 4. i, the 'PM"'.lot.ept«l Dux.

Furthor.

--

81 28 ( .. al

-.f-

a-rl1 Mtfla) -

J-~

more. Jlltl_)_' ~4) (%. Il)d.. ; thua ~_

•

.l.r!1 _2 .1(.. (_1 S. M..ltl'

or generally (anoe M•• 1'a.nWlM for odd

"I

_'_=
z'.' . •

(8. ' , 28bl

U we multiply Eq•. (8.1.26) by rot"l aDd integrate over z from - 00 to 00, we obt&i.n & "litem of OOlIpled integnl equationa for the M.,{tI}. We oa.n integrate theM equatiOlll with tbe help of an e1eotronio oomputing machine aDd from th e momutl ao obtained dot.ennine the Matron flux diatributJoa lfI(r,.l (of. in t.hit conneotioa 00LDeTu1r or SJoDOU &Dd Fu-o); t.hi. procedure i8 one of the t:-t aDd moM. ae-al for the eallNl&tiou 01 ne utron attenuation in .bie1da. BO_Tel. ""' are priJlcipUl, iaw-t.ed ben in the CllLIculation of H"aDd M. .. «i.Doe from u.-momentl_e&nimtnedi&telyobtNntbemea.naqu.a.redalowina:-down d.iltanoe. The equaUOIUI ' or \he fint three moment. are 1; (. ).1(••(.) -

j

E••
(8.1.29 &)

. -IaCUoJ

l; CtI)Mulw.)-

J• I.d,,'-vIMu (.') dv' + IM..lvl.

(8.I.29 b)

.!1(w)JI.. (v)-

f• E..(.·.... ,,).M..(.·jd.·+M, 1(. ).

(8.1.290)

. -"01-)

. - la U/al

.

, +s

I n t be deri vation of theee equatione integraJa of th e type inw-grat.ed by parte : -

J; aJj~~!! cfz _re OIl

x_ ±"". F

The expreeejon -; - Fd z. U)i-'OIl vanishea beeeuee All

vani.hee luter

s

tb ll.n any power of z. Fu rthermore , th e fact that I Jj(z,.) 4z _0 for l+O .... a1lIO ueed . -"" Eq. (8.1.29 . ) for N .. (u) correepcnda to Eq. (6. 1.16) for the flu in an infinite medium witb homoge neoUB1y diatri buted aout
t:

We can nevertbel_ ltill dnw one impo rtMt conelUBion jut from the form of Eqe. (8.1.29). In the first plece, th_ equatima are e u.ct. SeooDdl, . we would have been led. to the same relationl if we had lte.rted from a ~ ...ppro xima.

,1

8)....

t ion for F(a'.u) (F(z. u ) _ 11 Jt(z. u ) + ..'; ~ (%. v ) ODe we caD eaaily eheck. It therefore folloWll that an y t heory of . Iowing down that b beeed on • ~ .• ppro nmation will give a COI'TeCt va lue of ~ . ... long ... no other appronm• • tio na have bee n made.

8.2. Age Theory 8.2.1. The Eneru·Dependeni DlthUoQ EquMion

an.

A,. Theol'J

In thb aeetion. we Ihall try t.o derive a li mple diffe~nt.iaJ eq u.tion for the energy· and I paclI-depelldent neutron flu in any medium. We IItart with a neutron balance for the neutroJUII in one em' : (8.2.1)

The left-hand lide ~preeentl the hIlutfORI roo per em' and ICC *hrough 110wiq down, .baorption. and. diffuaion: the righi-b and lide repnlMDti the ~. 'l'hi8 eq uation b eu.ct : it b • combination of our e&rlier b&lance Ecp. (7.2.10) _MttiW_ 11 _ _

"',...

10

".

The Spatial DUtribut ioa of Hoderated Neutrone

and (6.1.8 ). Aeomding to See.1.!.l, the slowing down density and the flux a re related by th e equation q(. ) -

,-(·-.I .

" f

')-

X, (,,' ) l'lt(.' )

11

II

du' .

(8. 2.2)

.-1Ii(1/1l)

We now . .ume that FIOK.'s law relat.ee the flux gradient and the cunent denllit y : J'(r, . ) = - D (. ) grad flI (r, u ).

(8 .2.3 &)

The diffusion oorat&nt i , given by 1 D (u ) = 3E,,(_, -

1 3L'. ('M)(1

2f3A) '

(8.2.3 b)

Eq. (8.2.3 &) u. certainly of very limited. validity ; th e oonditione of iu. applicability will be iDvelltigated in Sec. 8.2.2. If we eub6titure Eq . (8.2.3&) in Eq. (8.2.1), we obtai n (8.2.4)

Eq. (8.2.4) i' an mervy-tkpe~ dilfwitm equation that ie th e . tatting point lor .. variety of approximate method.. In order to eclve it, we mUllt simplify the oonnection betillt'een the flux and "owing~own denaity given in Eq . (8.2.2). In a " heavy " moderat.or like graphite th o maximum energy 1088 ia .mall. We can tberefont approximate the quantity I, (u')4' (u' ) by itA value at ,.,' =u t.nd take it before the integral j doing thi" we get f (W) =E. (w)lJ)(w)

• , -(II - .. ') _

f

1

II

cr

du' =EE,IJ)(u),

(8.2.6)

11- .(11.:1

i.e ., th e Ou" a nd elcwlng-dc wn dollllity are related e" actl y as in an infinite medium with hom ogeneou.ely diatributed IoOUI'068. With th e help of tru. rolation, we can write Eq. (8.2.4.) entirely in term. of either IJ) or f . U we &88UDle for aimplicity that ,r.... O and limit 01ll"lOIyee to a eouroe·free region. then e, (r , . )

D (,,)

8M - IE, (v)

(8.2.6)

V-g (r,u) .

U we introduoe the hBMl age T(W) _

.f

D(v' )

•

'f"

IE.lv,)dw ...

•

.l,J(B')

Sl(1

2p..tl

rdE'

(8.2 .7)

in plaoe of the energy nriable, th e celebrated cage equati<m reeulte :

: : ... Pig .

(8.2.8)

ThiI equLioD ia formally idenLicaJwith the differential eqUAtionof heat oonduction if we identify 9 with the temper&tQftl and T with the time. When E. ,*O and in the preeenoe of IOUI'OM that emit neut.rona. of SOlO letharxY, i.e., neutrorul with

'* -

T_O, the. equaLioa becotnfll

fig -

b~~:i-;f'('r)+ 8(r)~(TI·

(8.2.9 ..)

'"

U we introduoe /f' (r.1' ) =g (r,1')p IT) _here

(8.2.9 b)

the equati on for g. dOM not contain an ablorption W1rm ; thUll the abeorption I18Ual way by a I"MOnanoe MCape probability,. 10 the following , and in puticular in later applicationa, we .hall only consider the CUll I. _O. Of course, it ill clear that age theory i' only valid when I.
...

to

I

-Irary llfifs

t;;!41

•

•

~,

~,

-.

-1 -I .

,

z_

I

1

J

"..II.I.L 1'tIoI oIowlQr - " __ ll _ .,... _ " _ ~. 'i tJ .. 1lM_ . ... . .

•

J

.. ~

The .alation of Eq. (8.2.8 ) for .. plane IIOUr"Oe emitting Q r.ero-kltbargy M utr0D8 per em' and pel' -eo into NI. infinit.e medium M (d . Sec. 8.2.3) 11(#',1'1"

Q

,"If

tU=.:'

. »:

(8.2.10)

h

ThH IOla tio n i. dDplayed in Fig . 8.2.l ,.here we can eu.ily see how the . Iowingdown deraity, ....hich alwaya b.... O.118Ii&n . hap', Ipreada out with inCl'NAing '1". Tb e mean Iquared lS1owing-down distance i' given by

-

° --7._ '1__ -'of-w-=l'fJ":......,:;-O - .- "'iT u f~-~ ~

,, -

.

3

-

• •<

liz

h (B ),

(8.2.11&)

o

I.e.,

~ =h lK) - III

" •

JJ{SA)f A:(&')

~•

(8.2.11b)

&:cept for the fim-fligbt ClOI'NCltion t.biI ia identieeJ lI"ith our eu'Uer rwuJt, [Eq. 18.1.14)] . It ia Clharact.eriltio of th e &p approrimation that it dOtlnot in4 clude the oontrihut.iou of the tim mpt: .. lhallntumtotbUpolIl.iin 8eo.8.2.6. ConIuaion OM euily &rile in the UI of the t.erm ... aDd the lymbol1'. W. ahall denote .. Ule tJfIf. 1'. 0DHixth of \be; mOUl. equnrd Iklwing-down ellatanClt from .. poI.ni IOUJ"Otl [Eq. (8.1.1». 'l'h1I deflnlt.lOllia rDenJ and dON hOi depend

'0'

."

Ftnrt_ .

on . "hether age t.beor)' it valid or not . By T or 1'(81 we 0 &11mean th e variable in the age eq uation ; in .. medium in which age t heory it valid, T(E }"'T._ Finally, by Illf~ /Jf1C ,.1 we.ball mean ono.aixtb of t he mea n eqU&red t lowillJ-down dUit.anee determined from the flu di l t ribu t lon [Eq. (8.1.3»). I n the age approx imatio n then! i. no d iffoNl nce ""twOOft TI and T. , The name . imlicateol that _ are dealing with .. quantity t hat ino~ ... t he elowing-down prooolM ~ i later we .ball find that t ben! i. .. direct connection between t he Fermi ace and the lIowing-do wn time. . Before _ go into a ny appUcationa of age th eory. we lIhaU at ud)' t he condiUona of ita validity. 8.1!.!:. Ap Theol'J and th e Tran8 port E quatio n

In &0. 5.2.1, _ deri ved elementary diffuaion theory from .. ~ ... ppro][im. tioll 'OJ t he tI'anaIp "" equation. We TlQ W atto mpt .. Pa.. pp'·... im.ti"" "' . tho "''''r'fQ'" dependent V'anapoR Eq. (8.1.U ) ; th_ we IIlIt •

3

F (:r, n, 11) = ""ii" F. {Z', tt)+ "i;r- Falz , tt) e«t. 8 .

4)

M en E. _ O. th e foUowlng equatioRl for F. and

(8.2.12 )

Fr follow from Eq. (8.1.26):

-~~a~~ + L', F.(z ••)-

J• E,.(.·_ IlI~lz, Il')dll' + Q " I·)6 Iz).

(8.2.13 ..)

bF~;, II1 + E. J;, lz . Il) _

J

(8.2.l 3 b)

_-1111(1'" •

-}

E,l lv' _Il)J'd Z, 'lI') d" ' ,

_ _ 111(1100

In oontrut. to \.he energy .independent. e.... the second eqlUotion dON not. yield Flox', taw immediately• .moe \.he ril ht.-band m e cont.eJ.n. \.he te rm

JE n lll' -

Il)Ji (z. Il') d v' ,

Thill tbell:...pproIimation and diffueion \.heory &re no longer t.he 1I&1lUI in t he energy-dependent. caae. We now m..ke th e following additional UllUIIlptio n8 6)

E, I. ') F. (Il·) =L'. (Il) J'. (Il)+ (Il' - 'lI) " II [I , ('lI)F. (Il)] .

(8.2.1410)

~)

E. (Il·)J;, (v' )- E, (Il)F,,(v),

(8.2.14 h)

i.e.• we _woe that \.he oollPion deoait y ~ only .. lit tle in one colliBion interval.. In tu connection it i8 important to note tb ..t it i. COruUetent to appro ximate I .(to' )J;, (to' ) by finly fine term in it. Mrie. e xpe neicn, since for th e nJidity of the ~ ..ppro:l.i.mation we must . .wo e that J;, (z . 1l)
l'. I (Z• •

I

U~ lz.

) ~- 3t . tI -Zj3AI -

II) •.----

(8.2.16 10)

Uaing Eqe. (8.2:.14&) and (8.1.2210). we obtain U; ~z!..!l _ _ Iz

l ,[E.IIII ' .l z. IIn + 0 6(II)6(z) 8 11

(8.t .l lSb)

'"

Ago Thoory

from Eq. (8.2.13 a ). U we t hese eq uationll yieldll il q (:f, II)

_ . iII! . -

se~

q (:t, Y) =EE,(M) Po(:t, Y), a combination of bOth 1

bl q( :f, II )

3E.l.li;<)(f=2/3A) *-

iI~ ~ ~ + Q6(Y) 6(:t)

(8.2.16)

which i. etltl(lntially t ho MOle aa Eq. (8.2.6). In order t hat thi s eq uation be valid, the ap pro ximationll a, b, and c muet obvlouely hold . We now invelltigate how well fulfilled t hey are by etudying t he flux distribution near a plane Bouree; for thill purpose, we wrno Eq. (8.2.10 ) in t he fonn I Q - -~ E,(u)F.(u)= ·'· 11·_--- t 4 1(.) (8.2.17.) , 4n T (II)

Approximation a ill surely good if I;,(z, (8.2.17a), t hill condition beco mes

:1'<" tI( 1-

u) <~(:t,

u) ; using E qa. (8.2.16&) a nd

~/:IA)':"~(1I1 'f lu ) .

(~.~ .17 b)

Age th eory ill thus only valid at relatively emall distances from t he source. Thill condit ion becomee eomewhat elearer- if we conside r a medium in which E, dependa only wea kly on u. I n such a medi um , T= -3Hi - ~AI L'f "'" 3U-2;A)L't ' where 71 ill t he number of collislona neceuary for moderation to t he leth argy v . Eq. (8.2.17b) now becomes (8.2.17c) :t<2n.l, .

Thill limitation is connected with t he fact t hat at ~ diSt&nOO8 from t he IIOUroe t he neutron flux is predominantly determined by neutrons t hat ha ve undergone only flo few oollisioDII, whose energy and direction have not ebeaged much, and whose angullU' distribution ia very anillOtropic. Approximation b presupposee t hat E7>~ [2',(v)F,(:t, u)]< E,(u)F.(:t, u) . Using Eqs. (8.2.17a) a nd (8.2.17b), we ca n rewrite thi, condition ..

" < T.

E"1ill

(8.2.17d)

ThUll ma ny collisioDII must occur during moderatio n, t.e., the mod erating medi um must be heavy. Thi s condition is eertainly not fulfilled in water, and we expect that age theory will be a poor approximation t here. Approximation e can only hold if

E: .

[L'. (u)I;, (%, u)]
view of t he foregoing, this condition leads to

~ d~~")

< A.(u ),

(8.2.17f')

i.e., the CI'08i &ection can only chenge e, little in one collision interval. Furtber more the absorption muet be very ama ll (2'. <2',)i a1110 we mUll~ be lIufficiently far away (&everal A,) from any surfacee, eluce otherw i811 the angulv distribution will be strongly a nisot ropic a.nd cond ition a will no longer be fulfilled. I n spite of tbeae many restrictive conditiolUl, age tbeory is appli6l.ble to a wide variety of slowing-down problems; in graphite and berylli um , ~he conditione derived here are well fulfilled. Next we ahalllltudy some methods of solving the age equation.

... s.u. A"ueauoa 01 til. Are ~utioD Ie 81mple 81ow1n&'.Dowa PrehlemJ In thiI ~. "" .haD crJoW&te the alowinl;-d own denait1 for lOme limple arrangement. of point aDd plaDe eDW"OM in homogeneoua media. AA .. bonnduy oooditioD AI; free..m-, "" requm tbt q(rg +"d, '1") =0 foz &U 'f. where d _

,

-

r.(lo.1~'3A) • e uc\Jy .. in the monoe nergelio theory. Bere _ mut remember that the _tteri"l Croll teeUoo.nd thu the ertrapolatod endpoint depond on E . nd therefore on 'I ; taking thiI fact; into atrict acoount would lead to great mathe, matioaJ. oompliea.tion••nd t broughout th y sectio n we, han UIfI a .uitable average value of d t.ha.t it indClponden t 01 ene rgy. U we wim to t reat thia eff&ct m oro &oourate ly. we muat UIfI an other dMcriptlon of "he .1owing-doW'n ~, th e ao-o&Ued maltlgroup method . into whiob " aba1.I go further later. A IIimila r oolIlplieati on 000UI1I in the dMCriptio n of tb41 llo wing-down p~ in . medium oom~ of h.o or more ftIgIOfW th at con.t&I.a diUOftInt moderatol"l, In thia cue. \b. flux aDd CUlnlnt mut be oontm QOu at the interfac.t at awry lethargy. In _ h .. proble m it it DO longer of .dnnt&ge to \a& the ~e. l ince the relation W_a T aDd • Ji- by Eq . 18.2.7) b clifLnnt in difforent media; in other wo-n:U, .. the D&U\roD em:- an interf.oe, lUi age can mer- or deer - di• . oonti nuoualy. EnD if we ue Eq . (8.2.6), which it equivalent to t he age equation . to deec:ribe alowing down in aueh a heterogeneoUi med.iUDl, a .cllut ion it hardl y po-ible linoe a double i.nfinity 01 boundary oondi tlOl\ll mUit be ...tiatied at every int.ertao.. For IUllb probllDUl, mwtllfOup ml th od, are atronaly to be prefe rred . Pltu.. 8ou'tC tit alt Ilt/ittiU Madi"".. The ap eq uation tau. t he form h" - ,III, .. + Qd( z) d( T)

(8.2.I Sa )

with the bouDduy condjt.ioo , -0 fM z _ ± oo. Here .... caD till to ad nntage th. method of PawW:r 'ro "'/~, _hOll application to nrioUi probielDa i, cl.mbed in detail in t he book of SnDDOJI. Th e FourieI' VaDalonn of , (z, T) il

defined by

+ 00

-J.beoa_, -/Cw, Tl - f

9(z, T) , - I"lb.

(S.USb)

Th e invene t.nulafonnetion i, given by

1 q(z. 1'I_ 1 11

/(w ,T) e'· ·tku .

(8.2.18 <:)

h il ....y to ebeck that; vam.bea at infinity. the Fourier tnm.Iorm. of ia iwl and th e Fourier tftDltorm of ~" az' it -J/. Th Ui Fowier t.ran.. lorma&lon of Eq. (8.2.181.) 1Mda to

8,th

"«;; '1'1 +J/""Q6 (T)

(8.!.18d )

.-

(8.2.18e)

which hal the eoluuon

for

T

> O. Therefore

, (%,T I "' i~J ,-C.. . - I• •ldw .

-.

(S.2.I81)

...,...,. •

!~ •- "

q(z , T)=-

f

+~ (

-

..

I'

e - *,T_ .,. GW.

(8.2.18 i )

-~

WyT- ',~ _ Y, dw =dxlVi

After the euberlturicn out And giVM

2 r'l'

qlz ,T )-

tb e integral can be carried •

...

yQ_- t-"• .

(8.2.10)

Thi. lOlution i. ...ell know n in th e t heory of 00.\ conduction. h hu .tready been daplayed in Fig . 8.2:.1.

Poi"" &'ru

~,

WfI

fP, + i1'f ~,.- + bII + Q6 (z)6 (,)6(a:)6( T).

lII, ~r'-

ai Here

l "/i" itc Medi. "..

i ll Stl

(8.2.19.)

define th e Fourier tranaform +~

I (w, '1") -

Iff q(r. T). - h. ·.. iiI; fly Ga:,

(8.2.19b)

-~

+-

(t~l--fJf I lw,

--

91rt T) =

T).'_ ·r

d~ dCIJ dw.,.

(8.Ug e)

Since the .lowiDg-doWD density "aniIhee at infinity, Fourier tra.na!o rmation of Eq . (8.2.19&) lead. to

,.

81("'. 1")

+ w'/ (w, T) _Q6 (T)

18,2.l 9d)

which baa th e ectuucn

for T> O. The mn r'16 ~onuation ia carri ed out euetly .. in the cue of .. planCllOurco .nd when we note that w' = w~ + m: + w: and. "' -~ +r' +*,.)'ie 1dll

o

- ~

q(r .T) _ - - , e ••

- •--•-.

Th• •

(8.2.191)

14:u )

tI=oJ~ -(h -T), 411:'" dr =6T(J') while in general

-

'" J .--'.r _ •

(1,+ 1)1 'I'(E). ,... - - _ . 4,.rlli'r _ --. ----

•

(4 11f)1

,1

(8.2.198 )

(8.2.1 Db)

". Poi1tl &1uu ~ &M Ct*r 01 11 be

S~ .

l..et the effective radiu. of the . pbere

R. Thel1 f iB. T) _ O and' (8.2 .20 &)

o.

+~ 6 (,)6('1" ) . f, JI'

.!(' t ) = DI(' f} a~ IT

(8.2.20b)

Next we deftlop 'f in a Fourier lleriell (S.2.20c)

Fourier t r&naform ation of Eq . (8.2.20 b) t hen yield.

-a. + M.(TI

( ' . )'

R

QI

(8.2.20d )

A,(T) = l Ri d ('r ) .

Thie equation haa t be solution

.

,

A, (T) _ _QI «- ('")' It ,

(8.2.2Oe)

forT >O. Su betituti ng Eq . (8.2.20e) into Eq. (S.2.20e) we obtAin

Y . '-'R'"i!='t'mn

IJII'

Q

11'

- I'·f· " •

18.2.2Of)

.wn

The Wrt T i., the hot ter th e on the right.-hand aide convergN; when T>(n{R)' I only the I _I term contri butoM appreci&bJy to the neutron di.tributoion. When R_ DO t.he .alution given in Eq . (8.2.201) thoukl red uce to that. p Ten in Eq . (8.2.19 0 , In tb.ia limit we e..n replace th e . ummation by an int.egration by mean . of the . ubnitution. ",_ b IR, dlll = nIR ; Eq. (8.2.201) then become. ~

limoo 11(', '1") = a_

-,Q a .·-f , ain(wr) e- '" w dw .

I., •-j. ,

After' t.be n bd.iw tion .mcur _ e euily and yield. Eq . (8.2.191).

-,f

. the int.egration

(8.2.20g)

C&D.

be carried out

Poirtl 80tuu i" 11" 1_/i_tldy Ltmg PiU . Let the eHect iV(l lengtb, of the pile', . ide- be G and b. and let the ooordinate ' )'IIte m be orien ted as lIhown in Fig . 6.2.3.

U the

a611ftll1

ia Ioc:&ted at

~,

Y., z., then (8.2.2 1&)

." gain . . deV(llop q in q (#',

&

Fourier Ilene. ~

.

y.s, T) '" "'" A,. (.I,T) aJ.n

".-1

hu: .

mn ,

G' - ,m - -6~ '

(8.2.2 1 b)

with tbe coeaicienta Ai.(t, T)o;

:~

..

JJ••

q(ot , y, 1', T) un

':Z

ein f d:l= dy .

Fouri er tl'anaform.tion of Eq. (8.2.21.) th en giYN

....(,,) +••(!,+ .'J aT III b" AI. ("' .. •

"'I.

I I

.. Q1
(8.2.21c)

I

(8.2.21d)

"' ~ + 1Ib

'I'M ltOlution of thiII equation, which c.n be obt&ined by tM meth od deeeeibed in connection with tbe plane ~ , i.

• (t , T)"'I.

. + '," . j • "Q .- . ' (V-."..",.. . o.n -' ' Zs.- ,m II

Finall y,

ql%'. "

I',

-~ - '-

"'''' YI

-lib

T)= -·i!=

f. (e-.. (£

"'~} h 1' 1,.-1

X (e

•

~gain th e Ilum convergee ra pidly for large T; ..hen grftln by , ( a:'IF.. ' I', l'}_e-

-rr-

I ..

Ito )X (. - ..r . bz . ""11) ,. am - , 81n ··-

+ :, ) . lJin

I"z. ain -

(1 - ..)'

(8.2.21f)

~ ( ll + -~-)1' ;;;: I.gi'

.-..,. eln ~ ,lin.!!1.. -i'II

~

(8.2.21e)

~

llimply

.

Po;nJ Source ill II ,,_. PiU. LIlt. the effecti ve Iengtha of the pile'll , WeII be ii, and e. The coordinate a J,:(lII are oriented u . boWD in Fig. 6.2.3, and th e source ill located at zt . ,. ' 1',. By denloping g in a Fourier llfIri(lll io all three coordinate., _ c.n obt&in th e following IOlut ion II,

" (-'" '•-.- + ... ,.. + .,• am . -'liZ" llm . -"'''Y'81n . ~ 1I11.r.) , - -'Q LJ e e" - X lI ~e l," . _ l '" ~ e . l1Iz . ."11 . X( 81n - .- o n - ,- .

-

I"" "J

aa. -

'''III')

I

(8.2.22)

8.%.4. The Phy.ieal Sipmeao~e of th e Fourier TraDslorm of th e 8!owing. DoWII Denallf

In Sec, 8.2.3, we used the meth od of Fourier tranlfortution 118. mathematical too l to aoln the age equation. No.. _ ehall _ that the Fourier tranaf01'1ll of th e elowing-downdenait y d ue to a point .auroe in an infinite medium bu an important pbyaieal lignifican ce. Thia Fourier t.ranaform ..... introdlloed io Eq. (8.2.19 hl, which we Ilhall tint write in a more convenient fonn. Since in an infinite medium tbe Illowing-down denllity dopend a only on t he dietence r from t he eouece, we can uee polar coordi nate. in the integration :

.

... I • •

!(w, 1') =

,f ,f f

g(r,l'),-i"·-·"'4rd" IIin 8d(} .

18..2..23)

..

,

The inlejp'at.W1UI over f

and,

are tririal; when they are c.rri ed out we obtai n ~

/ (OJ,T ) =

f• qCr,

_cu r

T) ---.,- "n ,.adr .

(8 .2 .24

Thia de finitiOllof the Fowiertruvdonnalioo il identical with thu of Eq. (B.U 9 h) : t b_ _ wrote / (w, T). but it turned out I..te r that / depended. only on w'= w' When __ lYe poll r OOOrdi MtN In r.tHip..oe, Eq . (8.2.190) giVOll the foUowin., tI:lpreelion for the inVflMle t ,.naformation

f lew, ~

1 11(', T) _ (h )'

•

, in cu r Tl ...,- 4nwldw

(8.2.26 )

[d. aJ.o Eq . (8.2.201 )]. For tbe lake of .i.mplicity. lot u. oonaidor .. non...beorbiftlt medium and We the ~ to be of unit Itrength. Then l ew, '1")_ ,-'" rN ul"" from Eq . (8.U ge). We can apparently write tb e gene~ aolution of the age equation in .. finit.~ ' medium containing IU'bitrary aoW"llN .. folIo_ :

l: 8• . R.(r ) • e - ~ · . • Bere ~ are the Iligeanl_ aDd R. the eigenfunct iona of t he equ.tion

(8.2.26)

V- H (r) +

(8.2.27 )

9(1', T) =

,81 R (,,) _ O

with th e bounduy condition R =O on th e effective l unace of th e medium. The S.. &I'll the ooeffioillntAI in tb e IlIpanlion of tbe eol1nle distribution in th_ eigen, fun otiona:

f Ir. T _ 0).. 8( ,.) ... l:8..R. {r).

•

(8.2.28)

Our prerioul l'MUlt. for the finite pile [Eq. (8.2.2211 and \be . phere (Eq . (8.2.201)] are apecial ca-. of u-e geoera! equat.iOOl. ThUl, ""' eua d.eribe the dowing. d01lt'D p~ in a finite, bomogeDeClU.' medium in a purely formal . ay .. foUow. : The ~W'Ce dUltri bution i. cornpoaed of individual component. 8...R..(r). Duri ng tbe .10wing..d01m ~. t heM individ u.J component. (" modes") di miniah eiDce th e neutroRi can leak out of th e medium. The deereaae of a component ft during alowina:do1lt'D ill ginn by tbe factor .-r,,-""' / (w _B•• 'l'). The Fourier tnDIlorm of the . towing-dOW1l. denait.y due to .. point. -auroe in an infinite medium / (B. , 1') ill th118 the probahilit.y tb.t. a Deuf,ron of oompoDCInt. • dON DOt. eecape from the finite medium daring moderation to Fenni age 'l' (non~pe probability ). It oan now be mown th at to l'NuJt il muoh more general than t be above coraid erati ol"ll indicate (of. tbe proof of the M OORd fundamental t heorem 01 reactor theory in WIblr.sao and Wrona ): Let UI ooneider an arbitrary bomorneou IlCat te ring medium. Le., let 118 drop the ...umptiODl 01 age th eory. I n t.biII geoe ~ _ , the Fourier tn.Ddorm 01 the atolVing -down density due to a point IOQl'Oe in \he infinite medium ill defined by ~

,f

_in .. ,

/ (w.B)= q(',i') . ,

4 " rl d r,

(8.2.2h)

lnol~ioD

of \he Fint. Flight in Age Thoory

.,

t.e., Tis replaced by the energy variable. Th en , f (w, E) is ag&in the non-escape

probability, Le., in the finite medium with an a.rbitrary SOUJ'Ce distribution S Ir)

18.2.29) holds; heee RII and BII anl eigenfunctions and eigenvaluee of Eq. (8.2.27) with the boundary condition R=O on the extrapolated. IUrface of the medium. For Eq . (8.2.29) to hold rigoroUily theee mUlt be a lingle tlJ(trapolated. surface on which th e slowing-down delUlity vanilhee It all energitlfl. Beceuee of the energ y dependence of th e scattering Ct088 lleCtion, thil condition is generally not fulfilled in practice, and calculatioMlU'O CMried ont with a suitahle averuge of the extra. poleted endpoint. Eq . (8.2.29) ill then only valid at pointe more than IIeveral mean free paths from thelurlaoe ; if the dimensionl of th e medium anl only a few mean Ieee paths, it il not eppliceble at all. Eqs. (8.2.2-h ) and (8.2.29) are of great pnetical importance since they permit the ralculation of the Ilowing..down density in any finite homogeotlOUll medium without UIIe of a lpecifio Ilowing-down model. To tw. end, the Ilolring..down density due to a point SOUl'Otl in the infinite medium ill determined experimeot&lly (ct. Chapter 16) and the Fourier tranaform according to Eq . (8.2.2410) ill obtained by graphical or numerical integration. Mter normalization, the nlIIult.ing DOnescape probabilities are inserted into Eq. (8.2.29). which immediately yields q(r. E). A. one or.n eaaily IIhOlll', this procedure worke even if the eC6ttering medium is absorbing. There is a simple connection, which we lhaU derive in th e interest of completeneee, between the Fourier tr&nIIform of the slowing-down density and th e momenta ~ . If we develop the term sin w r in Eq . (8.2.2410) in a TAYLOS'eeeriee around r =O, we obtain oo

.-~

f(w .E)=

( - I )"

~

"'"Ii..

;}

(2,+1)1 ,..-w = 1- ~li - w·+ ·I20w' _" '.

(8.2.30)

Here we have made UIIe of th e fact that the source is of unit etrength and E. = 0. Thie expreeeion illend, t.e., it does not depend on the a&8umptionll of 110gB tbeory. In age theory, on the other hand

f (w,T) =e - ·· r =

~

.'=1

<-:;)' T'w

t'.

(8.2.31)

By comparing ooefficienta we find that (2'~ 1 ) 1 T"(E)

(8;2.32)

a fefJult tbQt we derivod directly earlier [Eq. (8.2.19 blJ.

8.2.6. InelUSIOD 01tbe FIrst FlIght 10 A~ Theory

In comparing the mean squared elowing-dowu distancea calculated. directly [Eq. (8.1.1.)] and with the help of age theory [Eq. (8.Ulb)]. WI saw that age theory did DOt include the eo-calIed .. fint.f!igM oorreotion". This omieaioD etema from the fa.ot that age theory doea not deecribe the energy lpectrum of the neutrons properly. Let WI ooneider the spaoe-integnted neutron field in an infinite.

·..

non.absorbing medium containing .. unit source at u =O. The eollielon density i. given by (8.2.33 )

[cf. Eq. (7.3.9alJ. Tb 1I01ution of thi. equation for u > 0 i. the Placz ek function dilCU88fJd in Sec. 1.3, whcee uymptotio value i. I /~ . At # =0, the collision deM ity hu .. d·function Bingulauity lince all neutrons undergo their fint collision at u = 0. In the age appronmation the equation for tbe collision density is

I k' ·

(8.2 .34 )

= 6 (u)

[el. Eqa . (8.2.14&) and (8.2.13&» . The IOlution of this equation ill l/~ for all # >0. Thul not only are the" Placzek wiggles" not described, but also the aingularity a.t u =O. In oth er \Vordl, age theory does not include th e "virgin " nflutrona that have mad e no collision, -.nd for thia r&lI8On the firat.flight correction is not included. in Eq. (8.2.11b) . According to F'LOooJ:, 1lg6 theory can be improved by limply taking into account tb e spatial di.tribution of the virgin neutrons. If r.(E,,) ill the IlCattering CI'O&ll eectien of neutron! that have the ecueee energy, then the probability that they make their lim collimon at a distance r from the aoW'Ofl i.I W(r ) dr-Er IE,,)

'.r'

~-.l'p,l'

4nr'dr.

(8.2.36)

In dtll:lCribing the slowing-d o'IIVD deWlity with the age equation, we now use 8 (r) = Q. W(]r-r,1) .. the eouroe density inetead of the original point ecurce at roo For a point eouree at the origin of an infinite medium there then reaulta

f

Q

q(r, T)= - - -, E.(E,,>

c-.l',~,lr'

(01 111")

Integration

I~

(r - r j'

- -

'r" e II

-

4>

dr' .

(8.2.36)

to the formula (8.2.37)

,

wbere erf(t) =

;7i !e-r"dy. •

U we calculate the mean squared slowiT18-down diatance using Eq . (8.2.37 ),

we obtain just Eq. (8.1.14) Iwithout tbe factor

T'=-~'3..t

in th e lint.flight cor-

""""'oj. We might auppoee that the variant of age theory jut deecribed ill suitable at Leut in a rougb lint approximation to deecribe the alowing down of nentrollll at lu-ge ditltanoee from the eource , but that problem ill actually more complicated, .. we aball _ in Chapter 16.

Other Appro:dmate Method- of Cal\l\llating the Slooring.Dolm DMIlity

161

8.3. Other Appro:dmate Methods of Calculating the Slowing-Down Density 8.3.1. Th e Selengut-Goertzel Method A serious limitation of age thoory ia th at it ca n only be applied to heavy moderators , for it pre8UpJlOfl6& that the eolliaicn deIlllity can be described over a eingle collision in ter val by onl y two term", in ita Taylor series [d. Eq. (8.2.141')]. If a moderator eontaina hydrogen , a neutron can jump o ver a n s.rbitrarily large let harg y inte r val in a single collision , and the IIimple Taylor series approximation is sure ly ina pplica ble. In t he ceee of a hydrogenous mixt ure , S CLJ:N OUT and GO I:RTZE L recommend the following approximation : Let us assume, all we did in Sec . 8.2.1, that FlOK'S law IItiJI connects the current and the nUll: gradient, 80 that we lIhali be ab le to . tart from all energy-depende nt diffullion eq uation :

(8.2.') (Use of Froa's lew means, all we ha ve seen in Sec. 8.2.2 , not onl y a ~ .approxima . t ion but also neglect of the correlation between tb e &Cat torin g angle and tb e energy change in a ec lllsion.] The slowing -down denlli.ty ie now deecmpceed into a part qll (r, u ) that acoounta for th e neutrons th a t have made their l... t coUi&ion with a hydrogen nucleus and a part q...(r,. ) that ecccu nta for the neutrons that have made their lut coUillion with a heavy nucleus :

H.re (8.3.1)

Le., we use th e age approximation for moderation by tbe heavy nuclei {cf. Eq . (8.2.6». T he moderation by hydrogen, on the other hand, i8 treated e xactl y (cf. See. 7.2.2): qs (t", .) =

f•I. lf 4) (t", . ') e- (M -M·) d.' .

•

(8.3.2)

Eq. (8.2.4), together with Eq8 . (8 .3.1) and (8.3.2), forms t.he balli. of the Selengut. Ooertzel method.. In the quite 8ptl(li&l caae that we can neglect t he oontribution to the slowing-down deDllity from ooUmone with heavy nuclei as well &II the ab o IOrption of neutrons, we can oombine Eqs. (8.2.4) and (8.3.2) into a single simple di Uerentie:1 equation for q 1:

"j ell

-"-- ""' ---D (n ( q+ "- - + 8 (t" e ) ,

e"

r.s

'

(8 .3.3)

U we ....ume the neutrona are produoed by a monoenergetio point ecuree in an infinite medium. th e Fourier transform of the slowing-down deDJIity aati.8fi81 the • We can ahnYI do th• • but In l enetal the rMulUng eq\l6Lion U Yery oompJica,ted. We d~t~ Eq. (8.3.2) with rMpeot to II, IMllYing for '
obtaiD Eq. (U .3) by

..

,

!"··" __~(I IW 0' +'''.' ' ')+610' \ 0'

"'.

....

'

8.

11 /: -)(1+ ~~) - -!: f(w,lll)+6(M} .

The aolutioll. of thia equation ia

",."

•

J

- .. ~ftr•• ""

/ (w, v ) _

,

• • -'---0-,,= --

(8.3 .6)

Ill. onleI' too obta.i.D the mean Itquared -Jowing-doWD diRance froQl thi. u pn-ion, . . IZumi eaJcaI&te (l!fIfla ~..._o [01. Bee. 8':1.4, Eq . (8 .2 .30 )] ;

D) -'Ie- - - ("'I) -~ -- +/'010' -1;81 - - -' , .... __• - (-r.••-. •

U .. oow remember tJw D=lf3[E.... ll -2f3Al+ I E•• l

...

,

do . then find that

~ - r._ ~.[.r..+s.z:.~tl-27jAil.-.:~· + j -E;itr.. +J~..II- ii£4If · •

(8 .3 .8 )

(8.3.7)

'nM tim term ill. kind 01 fanl.-Oiaht coneoUon ; it iI e&rt&inly too l&tp ain oe in the 'PP"ODmation bei.Dg 00DAicL!lred. here only fint oollWona wiUa protoQl &nI YokeD into A(I(lOQDt.. Eq . (8 .3.7) neTel'the_ ia .. n ry uefu.l approrimation 1M in

T.

....waDd other bydrogenou oompotmda. While age theory ",~DtI .. limple approzifu.tion for .J0lring do wn by bea'1' moderaton (A> 1) &Del the Belengut-Goertzel method .. aimple appro xima. tion for IIlowmg d01nl by hydrogenoa moderaton, neit.het of th _ methodl i, putico..larly Rood in the region in between, e.g., for modention by deute rium (A _ 21. In the latter cue, one can prooeed by the method. of OoUTZ:lL ..nd OUULmO, which we .tudied in Sea. 7.3 in oonnection with apace.independent tlowing-down poblem. (d. M.i.~ and ZW1UftL). 803.1. The Jla1t1poa, HethCHI In all the metboda of aa10uJating lbe , ptotial diIItribution of moderated neutronl h_tofore ~, the energy or the lethargy .,.. \l'N.kId .. a oontin uOUl Taliahle aDd _ tried to find analJdcl lOlutio na. A different kind of method &hat ill al-o qmte fruitful ill to diride the eMlJY or lethargy range into lnterrall or group aDd ~ lha* t.h. neuWolY ill each of theM group diffue lib monoeDer'J'Iti o ontroua : moderation by elMtio or iDelMtio -ueriDg ill then a ~ thM t.nMporta ne utrou from ODe POUP to &DOther. In thil method. we mu t 101.. the difhWon poblom in each J"OUp Mpu»e!7: howeYW. _ 0Nl ap P 7 all the appo:Dmate methocb for the Vea&ment of _ group wan.pon problelU bithmo &l'nJoped, in panieaJa.r the P,- aDd 8,-metboda. 'The muJticroap method it the onl7 ODe with whieh we OM ~ , Iowina: down in iDhomopD8OUll media, a problem that h.. pond intnctable .,hen ut.aobd

with the comparatively limple foie theory. Aleo. a practical. deacription of the neutron field it poeaible only with the multigroup method in aituatione wbere the Ilowing down it predominantly due to inelamo acattering. In the following, we ahalll'e6trict onneJ.Ve8 to the discu8&ion of the eo-celled multigroup diffuaion theory, i.e.• that theory in whioh the energy-dependent diCfullion equation ia eclved by lubdiviaion of the energy range into groupe j ....e Fball furthermore reetrict oUl'fl&lvea to a medium ....hich slowe neutrons down only by elastio scattering. We begin with the diffuaion equation

8q~~ uj + I . I1>(r , v)- Df'lI1>(r, v)= B (r, v)

(8.2.4)

and divide the entire lethargy rango (0......) into ,. lethargy intervale (groupe). By integration of Eq. (8.2.4-) over eaoh of th_ groupe. we obtain the following ayetem of coupled. differential equatiora: ~

q(r,

Vy) -

~

~

q(r . 1'._1) + f I.I1> (r. v) dv- f D VWtl1(r, u) dv = f B(r. v) dv .

...., -

With the abbreviations

.....

-.

...-.

18.3.8)

~

4>.(r) =

f tI1(r. u) d1/;

B.(r) =

f B(r. v ) d1/;.

I

~

~-. ~

.. _.. . _.

/ E. ~('" uj,hI

1: "'" _. ; - - --

(8.3.9)

/<,1)(l',u)d"

~

J DVt<,1)(r, to)d"

D -

-ee-~.c----­ ~ J V"1fl(r, uj d"

and we caD. write thia lyatem of differential equations aa - D1V'l l1>l + 2'. 111>1""' 81- ql'

- D.V-tI1. + 2',.. 41.= 8. +q._I- q.

)'=2,3, .. .•

(8.3.10) fl. }

In order to continue _ must know the oonneotion between the 9. and the til•.

,-.•

In hydrogen thiI oonneotion obviouaty hal the form. 9.-1:0,..4>,. since neutrons from all the lethargy groupe lying belo.... VycaD contribute to the slowing-down densi. tyat u, . Ho_ver.if we 11M only .. few.rather wide groupe (e.g., LI 1/; =3). theprobability that a neutron will jump over a group aa the fMolt of a oolliaion with a proton iI very lmall . For all othel' nuclei. this prob6bility it zero. and we o-.n Il6t 9.- 2'. . lIP......here I .. fa a ...Iowing~own" 01"0Il6 l6etion. In order to determine tho group OOn8tants E•• tond D. IloOOOrding to Eq. (8.3.9). we mUlt know the vuiation of tP(r, v) with lethargy in each of tho individual

'80

The S pe.t ilol Dialtribution of Moo..rated Ne ut rCllUI

Since .. a rule we d o not know this vari ation , we muat make eome M8WD.ption concerning it ; thia &lI8umption introduoea a certain &rbitrarineaa into the method. This problem is particularly troublesome when the medium cootaiM re&OnlUloc ..becrbere ~at C6\ll1O strong local flux depressions. We must t hen introduce eHeot.ive oroBlJ lIllOtiona like t he reeonsnce integrala dealt with in Chapter 7. We limit OW'lMllvM here to the aimplest _umption, viz ., that (E)_I/E . Then groUp6.

~

f

D (1I) d "

D =!!::~-- -. •

11. -11' _ 1 ~

f E.(uj 4.. I.•• =

(8.a.II a )

.!>-~'::-:-v,.-V. -I ~

f E.t..) elM E•• =

~. ".

The number of

1

in group" per em l and

OOllisiOM

mU8t make N =

. ...

"'-;-=1. colliaioDa before

lI&C iB

I •. 4'J•• Since a neutron

it is modera ted from _. _1 to II,.

fJ. =I•• tPJN and

E.. = t-'- :·- llJC'

(8.3.11b)

U.mg Eq8 . (8.3.11), we can now &Olve the group EqIJ. (8.3.10) . Let us now eonstdee th e epecial CaM of a monoenergetio unit point source at "0 =0 in an infinite

...

medium. If we aBIlume I .,=O and eet ~ = .;!!.., eben

::;~ ,

!til (r ) =

f

!ti (r)=

•

and in gtlneral ~

I

I-f

, r -

.

~

I

II'

.... t. "'. (p.), - . -., ... d"

'"D.I,,-p,1

(8.3.12)

1

rh_,~,_,(p,_,) t - I. -,,-,'JL,.

,,,D,lp-p,_11

I d

",-I"

Here it ill again advantag60ul to Introduee the Fourier transform ; in analogy with former definitions of /(w, T) and ! (w, E) we write here

OUl'

•

!, (w) =

. iD 1»,

f Eh!ti.(r), lU'r'"rld,.

•

In particular,

to (,)

• r..

f •

', (w) =-

I, (w) -

, -.I£, l in 1», I - D,- --, - - ., - rl dr = ' l+ ~fLl '

l+~Lf I+~Lf'l

.-,n •

I.(w) =

(8.3 .13)

1

l+""'r.:'

(8.3.1'a)

(8.3. l n )

~

Approsimflte Ket.bocb of C&louJating the Slowing.no- DetWty

U'I

Eq. (8.3.14b) can MAily be proveD. lrith the help of the coDvolu*ion theorem lor Fourier t ransfOl'Da (d . W&:1lCUBO and NOD.ua). Now we ean oaloolatoo tbe neutron flux in each group by inverting tbe tranafonna in Eq. (8.3.14bl ; ~

LI-u

._l . n+ ILl

E..
j

,-riLe

t t) ..

(8.3.16)

b Ll"

In view of the fact that ~6 _ _ (at/. (w)/awt}.~. the mer.n .quared 1I0ll'ingdo wn dietanoe i.t given by :I (8.3 .16)

t - Lf+I.4+ ... +L:.

It i. ouy to _ that in t he limit "-'N,_1-0,,.- 00. ~f6 approache. th o Fermi age. Th lLl age theory i8 the limit of multigroup diffuDon theory ... the number of groupe become. infinite. The tn.nai.tion to t lUl limit ill natunUy nol Ta1id when the medium containl h~u, for then the origin.aJ _umptiOll that no Mutron could jump OTer a group ill riolated. In pnctioe one uuaIly ~ by (i) experiment&lly determining the .I owing-down denaity n6U' .. point IOUI'Oll in aD lnfurite medium . (li) Fourier trawd'orming it. and (ill) trying to fit the experimental .....Iue of / (01, E j th Ui

determined with .. product

If. I+;"q

containing u low faclora ... po8lIible.

Th o inverted transform ill frequently celled the " .ynth otio" . Iowing-down densi ty. In maDy CU88. it turns out that a good approJimation ill .chiem lri th relatively few groupe. For Pl&ny pr.otica! ~ . a limple two-group theory. in which all the fan neutrona are lumped.together in ODe group (the thermal neutrofll are the .econd group). ill a .uf6cientJy &OllIlBte fin* appronmat.ion. In thiI theory. we let L.= l~ (with tho energy 8in the neighborhood oftberma1 energy), i.e.• . . demand tha* the . yn thetio . Iowing-down denUty oorTeOtJy reproduoe the meuured value of~ . In \h ill cue. the tl'eatment at heterogeneoul media compoNd of diHerent materiall ill puticu1arl.y limple, .moe . . eeed only require continuity of flu and eWTent in each of the two groupe at each interfaoo. 8.3.3. The BrApproIimatlOD We th all now deaeribe another method for the trea.tment of I10wing-do1nl prohleQl.l that ~mb1e. ~he ~.method but yieldl muob more aoourate relw tl for an equivalent n penditure of computational labor. Le~ UI oonaider, ... in the derivation of tbe momentl method in 800. 8.1.3. an infini te medium with a plaoe .aur(le of I ... t neuuora at :11' _0. lnetMd of the differential flu!: F (%. o. .), let UI conaider itl Fourier lrt.nIIorm. which iI defined by

.-

--

1(01. n ,. )-

f F(r . n. . )e-'·· dz.

(8.3.17)

I

After mllltiplioation by , - ' •• and integration over all %, t hekUllport Eq . (8 . 1 .2~)

become.

_

(iwOOfl "+L', (tI») I(w. n, tI)= •

.-

1. f

J n',.') • Jf_"'" X

--"wIN.

i

/ (w.

(21+ 1)Ed(·' _tl) P'(oo. 6') X

1/01 1='1 P' (oot 6" ) lin 6" dB' d tl'

(8.a . i 8)

+ ,~ 4("1 . II

,,, NO'III',uin the trMtmen~ of the oniin&ry tran.pon equ.t.Uon ./ lw. in Legendre poI1'1OaUaJ. and the ex.pllnaion kuncated alter the 1 N f(w. n,.) - ". -jI '5'" (2t + I )/,(w , u ) l} (cos 8 ), ~

n. til itexp&nded

N." tenn :

(8.3.l!l a )

~

'.(w, 11) - f ,,(x , _le- l,u dz.

(8.3.19 b)

-~

Here the qunutiN Jj (z, Il) are the expansion ooefficientll introduoed in DOW mul~y Eq . (8.3.18) by P' (ooetI ) (1-0, I• . . . • N) and int.egnte 0"" IOlid -.ngIe. we oMain N + 1 cou pled d.iffe~ntia.l equatioTUI t bat are identical. with the Eq . (8.1.26) .XOIlpl t hat /,( w••) repl&Olll Jj (%, til I!Iverywbo"" In other wonU, wtI have obt&ined .. P.rapproximation for t he Fourier tBnd'onn of th e diHerential DUll: th at II in no .I.y diHerent from the Pr&pproximation for the differentiaJ. fllI lI: iteeJf. The _noe of the B,.method it tbat we fint divide the tnDapon equation by " +~ (. l and only then multiply by P' (COII8) and intep'&M. w. then obtain tbe foUowln,: I,..tem of equatJoM :

Eq. (8.1.25). U _

,w _

N

.~. ~ (21+ 1)

f, (cu. _ ) _

,

-ll-Ij -

'-I

J Ed(II'_ Ill/,(W, •

._'-1'"

v' )A" d. ' + Q6(; IA-t

f"~(-') t_') . D>D ,"'e-'+ (Il) IIln II •

(8.3.20 a)

(8.3.20 b)

•

Th_ coupled int.egraJ equationa can be ecleed, for example. by th e multi group method i once we bave determined I,(w, tI) _ can find Jij(z. til and thlUl F (z. n, til by innrling the Fourier tranafonn +~

Jij(z. tI) =-

,I,. /,,(OJ,tI)ei·· dw .

(8.3.211

-~

f.(w. tI) yield. the flu:r. 4'(z. tI) and aU the tQomeulol ~ . The aignificance 01 ebe B•.appro:r.imation procedure can be made clear A8 follow-. : Let U8 ueume that all t he I " lu' _ u) vaniah e:zoept E.. (u'_u). Le., th at the ~teriDg it. iaotropio in t he laboratory l)'lItem . In thia _ , the angular diatribution of l (w. n.tI) illlimply ronat../(i wCOI 8+ E,( u) and only one of the Ecp. (8.3.20a) - namely. that with ; -0 - iI needed to calculate f.(w. tI) and thQl to 101,.. the problem completely'. On the othe r band, in a normal p.approJ:imatioa many equation! mQlt. be 1I01'nld for an &OCUrIte caIeulation of the angle-dependent Oux rep.t'dJ- of ..be~ t.he _ ttering it. iaotropic or not . TbIUl in the B.-appro:r.im.ation. we l tal" from the angular dilItribution that the Deutronl would have in the _ of ilOtropio BC&ttering and calculate th e deviationl from thil distribution. It ia olear that thill procedure convergel much lu ter t bln the Pr metbod . A detailed decription and numeroul appli catioM 01 th e BTappro:r.imation. can be found. in BftR. tl al. ._. I 111 Lbo _ 01 Mowopie -u.inI. _ 4f• •'" hu ...... round. aU the ot.ber I,fw. '"

- -- --- -

CUI

be ob&UIed " ' - the reIalioft

fJI..

"j- •' i

E..

- ~ lJoI

("'- _}f.(_._1A ., .....+'Q6 1_' A. , •

The Dinnllutkm of Thennal Neuwon.

163

8.4. The Distribution of Thermal Neutrons Arising 'rom a Speelfled Fast Neutron Source Distribution When a neutron reacbl'lll thermtJ energy at the end of the 8lowing.down process, it diffU8el1 &ll a thermal neutron until it i8 captured. or ee.e&petJ through the 8udaoe of the medium. Thia tranllition from slowing down to thermal diffullion prceeede continuously; we shall di&cuss it and tbe general problem of "thermalization" in detail in Chapter 10. In fint appro:rimation, however , we can eseume a diacontinuous transition, Le., lI'itb the meebode developed in the lut few eeeuone we can calculate the Ilowing-down dellllity 9'f,h (I', E) at an energy E just ebove the tbermal range and use tbill elowmg-down dellllity &8 the 801lUe denaity in describing the diHu.e.ion of thermal neutroIl8 with elementary diHu.e.ion theory or eome other better approJimation. We shall illustrate this peccedure in tbi3 eeeucn by working through several e:om.plee.. In doing 80 we ,hall contradict in two plecee &ll8umptionl we pnlriously made, viz ., (i) that the elowing-down density could be calculated .. through it aMll8from the eluUc oolliaion of !leutrona "frith free atoml 1nitia1ly at I'Mt (which 11 no longer true at energiM below about 1 ev) , and (it) that the IOUl'Oe of thermal neutrone emitted neutron. already in equilibrium with th e thermal diatribution (el. Chapter 8). The erron oocaaioned by tbOIO inOOIlliltenciM are UDall, .. a rukl, and may often be oompenaated by simple eorrectione, all we shall see later. A4 an elementary limiting cue, let us consider firllt an infinite medium in which tbeee ill a homogeneously diatributed source of faet neutroae. Let the lob. sorption during the efowing-dewu proce8Il both here and in aU8ub8equentexampies _ be negligibly small . The Ilowing-down dene.ity q il then indepeDCIent of both 8p&OO and energy ; 6COOrdingly, the thermal OUll: tPf,h is allO space-independent and is related to 9' by

4>u. = -!--. <.

(8.4. 1)

Remembering that the epithermal Oux per unit lethargy tP~ =ql'X. ill OOIl8tant between 1 ev and several kev beeeuee of the constancy of ,X" we _ that (8.4.2)

With a given epithermal flux , the thermal flux ill larger the luger the moderating ratio [ef, Sec. 7.1.3). This relation i, very uaeful for many pllf'JXlflfl8. 8.4.1. The Mean Squared SlowinS·Down Dlstanee and the Migration Area Let us again consider a point eoueee ct fast neutrons in an infinite medium. Let tPlII (r) be the thermal nux it prcducea. We now Mk for the mean sql1&l'6d diatanee

from the source at which a thermal neutron is ebeorbed. It ia ~

I ,.tE. ~(rl"lt,.tdr

-,.a= --.. , - ---- --

(8.4.3)

,I E.4lu.(rlhr"dr u'

." In the oue of .. purely thermal aouroe ;i _6L1, .. we have already seen in Beo. 6.1.3. For .. fMt -ouroe we oalCiult.te;J from t he following relation : rl»' -~ + ("-"I)I +2rl ' (r

iJ _il _ (r, + (r

" .)

(8.4,4)

1t'bere "I ia the point u which tho alOW'inl40Wll prooeu CODles to an end and tbe neutron rea.ohM tbenna1 energy. However, Ff -~ ... t1rIllI. the mean .quared lIlowirla;40W1l lencth to t.hermal energy; furthermore, (r - r 1)1"" 6 £1 is th e mea n aquared SO a beorptiou from .. thflmW point. 1OW"Ce. The mind t.erm. in Eq . (8.•.• ) ~ upon .. ~ eteee there is DO ClC/I'ftllatioo. bet ..~n the directioaa of ". and " - rl • Finally then.

mn.ooe

(8."',6) Th e Q1WUl8t1 Je- -£1+1'lIl is .ued the _ tgrutiolt i ocoui.onaJly M _ ~iott Je-vc4. Both are meMUfW of the mean .eparalion. from .. point IIO'Ilf'l:le that .. neuWuq allhieftJII in tbe eoune of ita life bmory .

VV + T... ia 0&Ued the

8.4,!. PolDt Bouree In lUI Inflnlte Medium

We Iball eaklu1&te cfIu.(r) for the following t wo form. of the denaity :

alo-.ins~own

............ '

(8.4,6a)

(8.•.6 b)

In both OMOI ft""' ChIlo • We e&Il caIoula.te t he t hermaJ.fiu J:euilr with the belpof tbe

~1Mll rl"Jw.­

.--l:.mMI that ... introduaed in Beo. 6 .%.%. '11le number of neutron. th at become lb._I.I in tb.

~heric..l.hoU

bet ween r'.nd " + 4,' iI 4nr' l q&ll (r' ) rlr'. The

thermal neutron DUI: is thon given by tJlYl(r) -

f•' lIl (r' )O.tr'. r ) -hi r'1 dr'

(8.' .7)

•

with O. gi nn by Eq . (8.2.10). A ge TMor)' : c1'1II(r ) -

Q ::: {e-tl.ll+erf(l~

-

Yf)1

-~£ I'- '''(-'s yT; + t"')JI L 1 when Nf(*) i , the error inte,ral [01. Eq . (8.2.37)).

o-,T..,." QU

I .-"1.

.•-rfJiiii I


I

(U.8a)

(8.•.8b)

The Diltnlmtion of Thenu1 NeutroM

16'

n is iJatructive to conBider the behavior of the f1w: at lM'Be di stanON from the eouree . Aooording to Eq . (8.4.8b), the flux at large diatanON i. given by Q£I .-rIL 4>tII (r) = bD(£1 TIb)'

(V > TCb)'

(8.4;9a)

Q£I .-rlplii "1>u.(r) = -."'-D(TUo £I) r

(TUI > V) .

(8.• .9b)

ThUl when Y > Tu., the ratio of the thennal flux to th e epithel'lD&l flux inCrea&e8 with inoroasing distance from tho eouree: at very large distanOO8 from the SOUl't» we have purely thermal neutl'Oll8. This is the _ in graphite and heavy water, where the phenomenon is used in oonstruoting "thermal columns" for reaetbn. On the other hand, when 0 < TtlI the behavior of the th ermal neutrons even at large diatanOO8 from the eouroe it determined by the Ilowing-down denaity. and the ratio of thennal to epithermal flu:lt .ppro&ehllll th e oonatant value __~ ... jEt _ _'-_ 4lfpl 1"; l-£IjTUo'

(8.4.9c)

nul il wually the C611l1 in ordinary water if the source neutrom have lufficiently high energi.lIlI (a few Mev) j clearly purely thermal neutrone cannot be produced. With the help of age theory, we obtain Q .-rIL "1>Uo (r ) ="inD .'10 _,-

(8.4.10)

at lArge diltan06l from the source whether T> Oor < V . 10 other worda, in age theory, the thermal neutl'onl alway. predominate at large di8tanoell from th e /lOUI"(lO. This oonclusion il inoonect ainoe age theory i. not valid at large w..tanON from the source. 8....3. Varion. Arrangemenla of Either PolDt or PlaDe 8oUI'UI We limit ourselvlllIhe", to giving the th ermal flux in several typical aituatione. The e1owing-down density will be gotten from the /lOlutkm. to the lie equation o btained in Sec. 8.2.3. The thermal flux ie calculated with the method of dif· fusion kernell, using the 8OIutioneof the diffwion equation oht.ai.ned in Seoa. 6.2 and 6.3. In fmite media, a Jingle effective surface that ie the same for both thermal and non -thermal neutron. i. Mlumed; such an a.aaumption ie only a very rough approz.imation. Pla~ 8C'Uru '" a1l- l11-1'1I-tk Med,um :

(8.4.11)

18.U8)

...

The Spatial D;.tribumD of Modented Neutrons

Poi'" I10twtA at

z.> "0

~ {%.y.zI =(J;j) Qs- I£' ~ ""Ill z:

a"""."

( .

1.l'I~

lUI

.

b'fi"~l1 Long Pm :

""Ir. - " l~ + ~) ')

lIUl -.-~ -.- ·t·

X

1,_-1

X X

L,.(.-!,:~ [1 +erf (~-:-V~ - [0}+/L~ [l -erf(rV~ +[0]) x .

J"z .

.m . -am

(8.4.13)

..11,

,, ~ - .

Chapter 8: References GeDerai

_peoia11,

AluLDI. E. : 100. ci~ . f 71-80 D.t.Vleolf. B.: Neutron '1'hnlIport. Theory. Odonl. : Clarendon Pr-. 1967, ... ~iaUl Part IV : SJowing·Down Problema. 00t.Dn&nf. H. : ~t&l.Mpeon. of Re.ctor Sbiekl.iq , R.dirI(: Addlloon·W.ley, 1e411, .pooi&1ly Chapt« 8 : c.IwlaUMa of F&fi Neutron PeMtnltion. flld.81UK. R . E .: Tbe 8lo'lt'inl DolPn of Neutrona, Rev. Mod . Phya. It. 186 (1Df,7). 81111.11ooM. J . N. : FOIlriel' TranIIIonna. New York.Toronto -Lcmdon: McOraw-Hill Co•• 1961, NpeolaUy Chapter VI : 8lowill( Down of Nouttolll in Matter. WUIl'UIlO. A. M., &nd L. C. Nop. . . .: Theory of Neutron Cha in Re&otiona. AECD-3671 ( 1i.5 11• • pooially Chapter IU : Slowir!i.Do1m of NeutroM. Wanuo, A. M., and E. P. Wrona: The PhyaiOlloI Theory of Neutron Chain ~. Chioago : The Uniftlalty of Cbie&gO Pr- 1958, .peciAUy ChaptM Xl: DiHuaion end Th~Ib&tion

of Fut Neuwlll.

8peetal l BLUlCIldJ), C. H.: Nuol. Soi. Ez1K. .. 161 ( I ~). FuIn. E. : Rx-o.. Soientifiol. 7, 13 (1936 ). . OoJ.Dn'EIlf, H ., aDd J . CIlItTU lfK: Nuel. Sci. Eng. 10, 16 {196 l). Hoaw.&y, G. : Phya. Re't' . 10, Il97 (1936).

j

Calculation of the Me.n Squand Slowing. Down Length .

PL.i.a:u, G.: Phya. Re't'.It, 0123 (19oI6). VOLUS, H.C.: J . AppJ . Phya. 1:1,121 (19M)• . FDUU, E. : (Ed. J. G. DacJ(D.Lnj, AECI).26601 ( 19SI) , eapeclally Chapt.er VI : ) The SIo1rmc.J>o,ro of Neutron.. Age FLiloo~ S.: Phya. Z. U ,oIU (11M3). Theory. W.&LLAc•• P. R., aDd J . uC.mcK' AECL.336 ( l lM3). H u.wrn, H .• and P . F . ZWKIJ'IIL: J . AppJ. Ph)'• . 1:1, 923 (10M). } Se lengut,.Goertr.e1 SIIION. A.: ORNL.2098 (1966). ApproJ:imation . LavINa, M. M.. lit &1.: Nuel. Sci. Eng. 7, 101 (1960 ). } Goertr.eI.Greuling M.&c •• R . J ., and P. r . ZWI.I,IIL : Nuel. ScI. Eng . 7, I .... (1960). Approll:ill1&tlon. BIlTRa, H . A., L. TOMU, and H. Huwns: Ph,.. Re't'. SO. II ( 1960) (BN·Met hod). EDuCH, R.. aDd H . Hu.'I'TrS: Nueleoniolll:, No 2, 23 (19M). } Hultigroup Method. H.U;t)L, Y. E.• &nd J . HowLa'T'J': Geoe... 19M P/oI3O, Vol.C;, p . 0133. GoLDSTUN, R., P. F. Z.... J:IJ'n., and D.G. F08TO' GeM'" 19M Pf2376'j Vol. 16, p. Varioull Hethoda HVIi,"", H .• and R. EH1U.ICH ' Prop-.NllcJ. Enlll'iY' Ser. I. Vol. I, p . 343 of C&leu1&ting ( 1956 ). . S lowing Down WlL&DIS, J . E .. R. L. RllLLalfS, MIl. P . F. Z....J m'"U. , Geneva 19M P/fIn, in R.O. VoI.C;, p . 62. HOLTI, G.: Arki't' Fpik 1:.623 (1961); .. 209 (1961); 8.IM (19M}' j Neutron Dilttibueion BraNCa&, L. V., aDd U. rno : Phya. RoY. III. 0164 (lalll ). at Lt.tge DLltanClel Valla, .... and G. C. Wroa : Phya. Rev. 11, tl62 (1M'). from the Souroe.

m.

_..... _--I l'f.

-

f~

on Po Q .

16'

9. Time Dependence of the Slowing-Down and Diffusion Processes In thi. chapter, 'IV" Ilh.n at OOy the behavio r of neutron fields with non· titationary aoUf'CN. In doing 10 we thall round out ou r clieouMion of t he diffulion and . Iowlng-down ~ .nd in puUcular P~ptore ounoI. V'N too undentand tbole impon...nt method- of m. -unl ment that employ non.mtionary llQUl'CIN (t.b_ method. will be ~ in Qlapklf 18). ~ we -hall oonaider tbe time-dependent. Ilowing-down proo8M in the at-noe of diffllllion. the n the ti medependent diUueion prooeN in the abMnoe of do'tJina: down, and fin&1J.y t.he I~. time diatribuUon during moderatioD, th ough only in the age approJimation . We . banalm08t . Iwlra _wne .. Tery than pul_ of neutr'Ofa" the M>Uroe (S (' )-d II» . In practice, tbile i. t he m~t important _ . Alto, with ,uitable no rmaliza. tion the rMUlting tfI(B. II 0Nl be oonllidered u tb e probability that .. neutron produced It ti me aero b... the I nergy g at time ,. Thul the life hiatory of an " a vera ge " neutron ca n be read directly from the IKllution ~(B. '). If we integrate tb e t ime.dependent eclutlc n over all t. we mUllt again find th e reeult we obtained earlier for a . ta tiona ry 101lrOfJ . In Sea. 9.4, we . haU oonaider, in addition, th e ('I lle in which t he neutron IOUf'OIIII Ire periodi~ in t ime. !

9.1. Slowing Do"" In InDnIIo Modla For . implicit y, let 011 "WIle thl. t the medium ill non..beorbing. The Ou 4»(11, I ) th en A tilfiee th e equation

~ ~ "'i~ 'l = - E, l1l+

J•E,<1' (~') r~l~~'" dll' + d(Il)6('J.

(9.1.1)

. _ 1Io 11a

Here we have ueumed a monoenergetic, ' patiall y homogeneo1ll unit l1011roe aDd have su pposed th.t the . lowing down i. dee to e1aati~ ICIttering th at is illOtropie in the C..ystem. A simple mathemat ical a rtifi ce that permita UI to ccntlnue our t reatment of this equation is to Laplou lnJulorm it wit h reepect to t The La.place transform of <1' (a , I ) is defined by ~ (11, .) _

w

f

4) (11, I ) t -tl d l.

o

(9.1.2a)

'The inverse t r.nalorm is given by O+~

<1' (1I, IJ- l"k

J

4i(1I, . )e- d• •

(9.1.2 b)

.-~

U we La.pl&oe trandorm Eq. (9.1.1), it become-. (9.1.3)

FormaUy, t hil i. the equation for the . t&tiona ry neutron n Ull: 41(11) in a medium with t he m.cf'0600pic abeorption 01'0II aection , /", We oaR therefore apply tbe relult8 of Chapte r 7 to the further trutment of Eq . (9.1.3 ).

9.1.1. 8Icnt1nc Do_ ill B,Uopa AecordiDg to Sec.7.2.2...ben there i. a unit Dux 4>(11) in hydrogen i. gi1'en by

IOUI'ftI

at 11 -0 th e neutron

(9.U a)

• r ,+ -!!iL "Itol E +--

41(...)

(2:.+ -,' )"

..

(9.I. ...b )

E, + -

"

Hen ..e ban _ ed that E, i. eonnant in erdee- to be able to e&rry out th e inMgral in the expone nt in Eq. (9.1.""). "Il ie the TeJocity of the IOW'ClI ne utronl (to _ O). aod .. iI the " Iocity of lobe _trOD.t with lethargy 11. Th1ll " -"O, - -n . U we noW' innn. the tl'andorm with the help otEq. (9.1.2b), wefind (cf.M4MRAIt) (9.1.6)

Here lbe ~ term cIe-:lribM the decay of the aDool1idfld Dux. In oede e to better 1lDCI.ent&Dd the fint t.erm. let III ooneider the .. ..ymptot.i<: " reg ion of . ma11 _rpM, ~"o ' H ere E."ql>l 10 that ..hen ..e introduce 4>(".') = -:. l;1)(tl. ' ) we find

4>~' ( '" ' )_ (E.",)' ,- Zo.C"'" ~' - ' .

(0.1.6)

'I'be uympt.otic On depend. only on th e dimeo.si onleee quantity Z_E." l. We oaa r..d direetJy from Eq. (9.1.8) either the "Iocity distribut.ion 4> (..) at an arbitzary tlme' or t he tim e diltribution 4>(') at an ar bitrary "locit, e. Fig . 9.1.1 illutratM tbe latter dietribution for lOme velooitiee. Here E. oorreeponcb to the proton denaity of the

IOW"OeI

litO.

The validity of the Mympt.otiOIOlution PJ'"uppo-e. that

b...,. enorgiea tha' are very Inneh larger than t he energy at which th e

IIowirII: down ia beina ooneiderod. F'urtberm ora. in order th at Eq• . (9.1.•l and _ ha' followa it be ,..lid . the _Ue ring 0,... MICtion mUllt be intlcponJent of the eneflO' . 'I'bi. u. true iQ. hydrogen bet_een I and about 20,000 . 1'. It i. of intereet to form an a....rap from Eq. (9.1.6):

J-4>... (. , ' I i, ,,", -i;;.

(9 .1.7a)

•

Thie equet.ioo hal th e following: interpretation. In th e ate t.ionary ltate. the Dux

froln a unit

IOQrC)CI

i. 4> (B)- b (cf. Sec. 7.2.2) or

'1D ....Jit,• •(•• tj mlWt ha Eq.(U.e) iI otmo./r ~ witboll' \be db d "M _

-..n,

l;1)(.) _tfl (E') ~ -

i...

em·'. w_ _ the rigb \-habd. _ide of 'J'hM IlOIINlI abou1. t>ec.u. _ b..... _J-ified the ~

~~. . -.

.

,

When lntegreted over a.ll time. the non-stationary solution giVeti the same value tho 8tationary IIOlution with an equivalent S01U'C(l. Thie is to be expected. Tho average time;'; for moderation to an energy E (v61ooity tI) i, then given by

&8

~

f

t
I) fll

r.=-l!.../ -lJI.------ = 1.•. ., (• • ' ) d l ,

(O.Ubl

Since we have used t.he MymptotiO 1I01ution in this oaJou1&tion of 1.>it doH not depend on the source energy. U we had calculated from Eq. (9.1.15), we would ha ve obtained for Il< "'0

r.

(9.1.7c)

- --

.._.

•,

--- - - --r> II

" ,.", "

/

•

' ,I """'" (IX/tI)

J

- -

,-

-j

'/

'.

-

..... 0.1.1. The

u_

---

-rr-r-

1---

1\/

4-1lll ~\.r.

I.JlI~"

/ \

V V-,

X \

\

1\ -

• ,- •

d"l.rlkl.lo~

- - --

("V)

'" tho ....._

fbi. bl _

\

• •-,

at. ~ ........ doo

When u< I1g. the difference between l~ and ~ b unimportant. Aocording to Eq. (9.1.7 b) . ~ for moderation to 1 lIV in water i. 1.6jLMO : to }OOIIlV, O.161l1eC: and to 10 kev. 0.016 loL8eo.

It. iI. ,,-1110 of interNt to form t.he quantity

(9.1.7dl

,

The quantity

.1' =~-l1- = (E,tI)1

(9.1.70)

i. a me&8UJ'Cl of the di.persion in time of the neutron flux at • fixed energy. In

terms of ;; I LI ie

-I!- ... ~~,3 . Knowledge of LI i. important for jadging the ~Itp

reeolving power of neutron .pectrometere that employ pubed aoUrt:el and bydrogeneoua moderator (d . Sec. 2.6.41.

&

Time Dependenoe of the BIo.-ing-Downand DiffUlion ~

170

In order to calculate the time variation of thtl aVtlrage Vtlloeity, we muat ltart from _(tI) =(I)(e)/tI. Now 1

fA(",,)tI" "'"f ~

.ince the

~

•

•

lIOUl"O(I

'1>~,~", II d ,,=l

(9.1.8a)

produCN eu.etly one neutron per eml • Furtherm ore, ~

-tI(') - f •

iiI(,) =

aDd

4)~. (tI,')

d,,_:t;i'

~

f

t!

•

4)~. (", ') dtl=

(E~')I

(9.1.8b)

~

VI(,) _

f •

120

"'4>~, (",')d" = (E.1i"

(9.1.8c)

(9.1.8d) Clearly the entlrgy .peetrum during moderation in hydrogen ie alwaya very breed. Thi. COroM about becaue a ne utron can 10M! an .rbitrarily large fraction of itll el'ltlrgy in a .ingle eollleion. 9.1.2. Slowing Down In Heavy Moderato" (A+l) To deacribe moderation hy htl&vy moderatonl, we can again etart wit h Eq. (9.1.3); but we immediately encounter the difficulty that th ere ie no elmple IOlution of the .tationary' alowing-down problem when E.. o. We muat therefore uee approximation., . uch I I thOfl6 with which we became toequainted in Sec. 7.3. Uling the Goertzel-Greuling method, KOl'1'Jr:L baa recently obtained a very elegant appro:l.imate 8OIution to th e time-dependent elowing down problem to which we lball unfortunately be unable to devote any .pa.co here. Rathtlr we eball limit OW'IeIVN to obtaining hy the method of &UBaa..u: lOme useful averag e valuee [tIOlTHponding to thOM! of Eql. (9.1.7) and (9.1.8) for hydrogen) from which we .hall be able to obtain all the important phyaioal information we need. For the following conaideratio ra it i . bMt to . tart from the time-dependent elowing-down equation for the VtIlooity-dependent deMity ta( ", f) :

+

all(", ')

h

f./,. . ,,.

- a,- -- =-- - oE." (tI" )+ I _1l E. _(tI,') -.; -'

(9.1.9)

• I Hwe. .... .-1Il ahould.mt.epa'- 00.11 to "0 aDd. abould. 111& t he oomplele IOlutiora ratber 1>(E."O)-· we ean 111& the uymptotlc

thazI jlM& the u ym P'Otio IOlu b. H _ , ...hen aolatlon and int.epale k! Wlnlt,.

171

We have omitted \be eoueee term here ainoe we are _ king Iln uymptot.to aolu tio n that ie .,.&lid for velociti ee much lm.rJler than the Ylllocitie. of lb. IOUMe neutron-. We ean in troduoe z = _E.' .... new variable in Eq. (9.1.9); th e lat te r ca n then be weneee (9 . 1.10 )

Now we define t he momen tll of ,. by ~

M,-f z'1t (z) d z .

• U we multiply Eq. (9.1.10) by ,} _I and integrate, we obtain

(9. 1.11)

t he following re o

eunion formu la for t he momenta :

M,=M,_I [

!:f-_!]'

I- I 1- .. t 1- 1+1 1_.

(9.1.12)

!

A. DormaliDtioD we take M . _1. U ....e try tc ca lculate J{I fro m M. by mN ra of thi s recunion formula, we en oounter .. ,light difficulty, viz ., t hat on th e right . ha nd elde of Eq. (9.1.12) there lltaoo. an ind eterminate expreuion. However, if we let 1=1 a nd tAke th e limit .. ,_0, we find

+,

,

,

(9.1.13.)

M' ''''-fM, = "I and thut for It:: 2

' M,_,2.-1 II -

-1 - -. -- ..C.

(D.1.I 3 b)

2 1-.- .-_ I._a . _-. +1

,-- -

In uae A;> I , we ca n a pproximat.e th e reeereice form ula (9.1.12) by' M/ _

A +2 2 - H, _I ' 1+ (3-1) 3(.4 + #

I

Thull if we take M. -l a nd neglect te rm . of higher order, _ find that M, ~A + 2J3

[d 2/1; d. Eq.17.1.19))

M'.=A (A + 2)

(9.1.16)

M1 _AI (A + f )

M, = AI (A + 20/3). We .haJJ nezt . how the conneetion of th _ e quatiOlll wit h Eq•. (9.1.7) and (9.1.8) for hyd rogen. Let ue oonaider the time de pend ence of the ne utron n Ull at a filled va lue of II. Since

•

_(z ) d:r ==.(II. II T dl fA

- +1.

-r_ (1- ------po)'

I Eq. (1. 1.14) ON be dttrind by ..placing. konn. in it. Tay lor _ _

_ +1

(A + r

by the firR LhNt

",eobt&in Mt

=-; "" .I.!• •

11(., t ) • tit

'l'hit equation M analOi oua to Eq. (9.1.710) and N p that wben int.egn.t.ed over aU time the l pectrum il the Nme . . in the equivalent l tati Onuy ewe. Furthermore 1 JI. r.. ...-;r -JI •

(Q.1.I6 b)

• •

(9. 1.16c) (9.1.16d)

With the belp of Eqt. (9.1.16) _ find for A> I

1: = A + 4i? .... . A_. • E, . E•• '

(v. l .n.)

(Q.U7 b) %

,I

2A.J3

.11 = li - . - ~ .

(9J .l 7c)

Thlll.1 I~ ....2f3.d.

In other worda, therel.tive di.peraion in the time of moder. tion to a p-.rtJcula:f e'*'BY ~ with u.e-..anc M_ Dumber. Tbi8 beh. rior originat.M in \he fad that for la.rp m.- num.ben the D18utronI an alowed down by WarlY coJ1WonI, MoCh of which rMU1" in only a &ID&11 energy 10M. Owing to the large Dumber of oo1lWonl, the l tati.ticaJ. flu c\o.atiOOI in the e06tgy 10llll per collision and th e fligh t path betweeD IUooeuive COlliaiOIll cancel out to a large e:deDt ; in firat appro:liw atioo therefore we Table U .I . "A ~~ " BIowi. ,. ClUI dNcribe t be alowing-down proc- &II • Do- 2". _ 101_ continuoua decreue in a re1&iiTely Ib&rpIy defined neutron eDerBY• Be (l .78 p · )

C

u.e

Wem"l

Pb( U .U ~·)

.

...."',

10.18

--, s.e

....

' .7

of I. and .1 "'" EqI . (9J.16 b and d) lex ""nnJ bea,.,- moderaton. Table 0.1.1 give-

vaJUM


Th e fird few momenu of the velocity diltribution at a fiud time &nl given by 11,

1'(1) _ E~' -

01(' ) -

JI .

-a:;'J"

_ J/. .. (I) .. (E, 'l' .

2

EE., •

(D.1.I8a)

(D.l.lS b) (U .l Be)

'"

When ..4:> 1, t.be mean energy at .. nxed time ' ia given by

iii (' ) = 2'"

J/, • .04' (E. f)I ~ 2(E.,), -

At

1E,1)' O.622fl~ cm.

...

ev

(9.1.lDa)

and it. w...penion is ginn by

(?:t- E';.r

IIIl

3~ - '

(9.l .19 b)

Eq. (9.l .ISb) aho,," that t he energy lpeetnun d uring moder.tion in heavy llubl!tNlcea i• .JwaY' ve ry eha rp . This tharpn_ ilia result of th e nearl y conti nuoUl nature 01 the alowing-down pfOOelll. For lead (A "", 207). for eumple, y~ _ 11.• % . The ec-eetled 8lowing-
,. - EE.. ~,

II.

(O.I ...20a)

lr,

-11'- - - ',

(9.1.20b)

tI .

. ~-('!' •-...!_J".U:, .. _ -'-) .

It t hen foUo. . by integn.t.ion t.h&l - I E,

•

~

til

,

., (Il - IE,"

(9.1.20c)

T he de rivation of thia equation prNuppoeee that t he M u tl'ona hu e a d efinite energy .t every time. but .. we h.ve _ n thit ueumption i. only nJid lor nry lar~ A. For amaUlIr A. Eq. (9.1.2Oc) i, only oorrect when " rep~nw th e . V1Ir. age velocity. Eq . (9.1.200) Y ofUln UfMId to calculate the t1owing-doorn tl mo. but .. wo O&n _ l rom oompa.riaon with Eq. (9.1.17.) or (9.1.l6 b). tbi. prooedure i. only oorreot lor large A (for A _ 1 th e error a M%).

9.2. The DltrQ.81on of MonoenergelJe Neutrol18 (Pulsed Sonrte8) 9.2.1. The Tlme--DepeadeDt DlffuIoD E4IIl&UoD The M lltron b.l.n0ll 'equaUoa. in • t.ime-dependen~ mouoeoergetic M Utron field Il&II. be wriuen . 1 a "'( f'. f)
. --,,- =-

11 we _ume that the CUl'I'eut a.nd the den-ity gradient lU'e related bl Fral:'. I. .... Eq. (9.2.1) immediate1l yieldll the time-dependent diffUIion equatioa. : 1 U~ (f'. f) • - , -, -." D VS t1' (r,')- E.
(a.2.2)

AnI ~

validi~y

of tha equation ... we re boundary lurf.t.oeII ~d localiaed IOUl'ON mna\. hew away and E. must he < E,. In Sec.9.2.2. _ IhalJ. _ U1at there AnI .t.i.lI other nlIItrict.ift oooditiorw. In l pite of ~beee nWDerouilimitationl. Eq . (9.2 .2) ia of greu practical importance. In ~cular, i~ fonn. the buia for the anaIym of the non-atationary diffusion e:a:perimenta die . euseed in Chapter 18. We ...&11 non di.scu. .ome of t.h_ IOlutiODB for a pulsed IOW'Oe (8(r, ' I_ ~(')S(r) . We migh~ S W'Dl.ille t hat the res ulta for monoenergetic ne lltroDB would also hold for IMI"IJIal neu trons. In fiht approximation. thi s oonchwon it oorrect , hut we ,hall_ in Sec. 10.3 that cert&in oorrectiolll muet be a pplied wben the reeulta for m.OIIOfIMJ'l(8tiO neutrons are usedfor a pulaed th erm al field. When _ AnI dea.ling with a put.ed .-oun-e.... ean rem ove the ablorptkm term with the l ubAitution (9 .2.3)

The

AIDe

oondit.ionl

for th e

heC.'IMI&l)' in Cbaptrer 6 for \be nJidi~y of the ~..pprosimation.

m.,

Th•• (9.2.4) We OUI. euily _vinoe CM.ln8l"'-that in the ill:lpoc1allt apecial cue E._III/the lbeorption be taken into aoooant by the aub.titution 9.!.3 even for 8RBl'lY.dependent problema. In p&rt.ieu_••e ooukl b..ve taalN'l .. J/I/·.."-orption Into -:lOunt in t hi. ...y in Sec. 9.1. 1'hie pl*1bility of .p1itting off th' ..bIorption aI...,. 8lIi.t. in puI-'. neutron probleme and dcMe aot c:Ie~ OIl tha lYe of diffu-ion lb-,.. In orde r to adnnoe our treatment of Eq . (9.2 .4) . we may note with advan tage that it it formall y IIimilar to the. equation. whieh W&I di~ in Soc. 8.2. U _ make the l ubst.it ntiona and q _tPj" . Eq . (9.2.4) beoomee formally identical with equ.tion, and '"' can U8fI all the reaul... obtained in See .S.2.•. OUI.

the.

T_D,,'

Some important e..- AnI : PkUM 8oW'U ilt 411 l lt/i lttu Jlrdi.",1

tP(z,I) = T

;<--.,- e

," :lID !>1

Point /JQuru i n

43

- ~ _.._- - . r,,1

q.

'D.'

l lt/in tu M lldillm

(9 .2.6 )

-

.-TD.'-·r",

q. i I..""D.I} 1\ follo_ &om. Eq. (9.2.8) tb&& &be mMD aqund diatanoe at lime , ia tP(r,I)-

;I _aD _a.

(9.2.6)

(9.2.11

P

for thermal neutrona after' 10 _ ia 46 em in a.O and 110 CIIIl in graphite. , iltiU Media.. UDde!' the _mption lha~ the n- dimenaiolll of the medium are large oompNed to the mean frM path, the Lime-dependent On M ginn by t/} (r, ' I -E S." a. (r)e-{·Z.+D.~I. Here ~ a.re t be eige nnluM and

R" (r) tbe eigenfnnctiorul of t he equation Vltlt +,BItP _ O

I 10 the

(9 .2.S..)

1oUo...mc M)\uLioN lbe -roe .tnngtb q dcMe DOl h.•• the

.me. q .. the tot&IlIlUIlbw oI_ttona _tained ill • palM.

(9.2.8 b) d~

MC-1

'" with the boundary oondition IJ) -O on the effective . urfaoe of th e medium. Tho B. are the ooefficiente in the esp6lWon of the lOuroe fun ction in theeo eigen_ functions. The neutron nux iI eeeepceed of an infinite num ber of " mode. " that dee&y with th e rela:u.tion limN \&1" mode predominat.ee: '

r.+ID

Bl' For ftry long time. t.he"fund&mon. , (9.2.8(1)

Hore

(9.2.8 d ) and R, i, t he lo_t. e~nfunction and 1JI the 10_1. eigen nlue. Thia lowNt e~nvalue i. frequently e&lIed the lnldlillf' 01' , more preei_ly, the pwmdrW b1dli"9. The eigenfun et iorw and eigenval_ for .. rectulgUJa.r pualleJepipetl wit.h t he effectin edgeI 0 , b. and e are gh tln by

I I

' r... z 1WI-.. . ", I1Il-,. - 1WI. p . " .1)' B:••- x' (Qi+ V + e--

RI•• (z, Y.*I

. :116

In particular R ( e, y• .1) _

~ ..

. ""II nz an . lUll

T:PI,

. c,-

IUD 1U

(9.2.91.)

(9.2.Db)

The contribution of diff\I&ion w the decay COQIItant iI obviOUAly greater Lbo amaller the lin... dimenaiOni of the medium ; t he eharacteriltic unit. of length in the medium. iI th e diHuaion length. Aooording to Eq. (9..2.8dl. wben th e Iineat' di moruliofUI 01 t he modium are 1&rge oompared \0 the diffusio n length, Ol.... . I.: bewesee, thia rMwt h.. no prac1.ical. ti gnificance beeeuee under the cireu.m.t&n C!f.ll being COIlllidered, .. very long time mWlt olapee before th e fundamental mode alone i. pr'eHnt. The eigenfunCtJOlll and eia;envaluee of • oylinde r of effecti ve radilll ,.. a nd height ,. &l'e

,R,. (,..,)- J.(tIt :.).mT Bt. - (-;'-)' + (¥ )'

}

(9.2.10)

if for amplicity we _ume radial. . ymmetryl. tit if; the I-A root of the B -1 funot.ion J. (cz,, - 2.406. m,, -G.620, cz,, -S.6M, ~ _1l .192) . 8J~D baa . um · mariI.ed. IIOme edditional eola Uonl. of Eq. (1I.2.Sb ). 9.itt. The Valllllty of the TIme-Dependent DlffuJloD EqU.UOD We can _immediately from Eq. (9.2.6) th.t the deecriptJon of time-dependent neutron fielda by the diHuaion Eq. (9.2.4:) cannot be entirely eorreer : For ' ... 0, the neutron fiux vam.bN eTefJ"ll"here except the point ,.=0; to neult ill correct. llf ~ _

AnI

radialIy'ymllMltrie, Oftly radi&Dy .ymmetric: _ _ AnI ueited.

For 1-6' >0. OD ~ other h&od. (r.') • .mabeIllOwhere , <hough it is Imall for aU r +O. In reeJ.ity, OO_1'W, (r, I) mUit yanish ouWde u.. Iphere r _ " 6' beea1lM in th e time 6, no neutron can tl"'&vene a path 10lll"r th an" 6'. That the t.ime4epeodent diffulion equation ~ not delcribe this eHect oonectly comee abou t beca1lM FJ.ox ', law, which underliee Eq. (9.2.2). doee not take into account the finite COllWOIl time. In order to _ t bD clearl y, let UI remember th e elementary derivation of Fl olt'slaw ginn at the beginnl.njj: of Cbapwr 6 : There..., tlX~ the curre nt in tlrma: of tbe flu l[ ... follow. eo ' a •

s

sr«: :~ E.J J

J ooe(J (r)e - r"d rd'l'tiD.(Jdfl .

•••

(9.2.1l a)

ElI:pNUIion of In a TaylOl'..ria Jed Immedi.tlly to Flox 'alaw : .~

1

J _ _ 3E; h

'

(9.2.11 b l

This prooedUl'l ia corT'eCt in a ltation&ry neutron field only if { . ~~ < <1>. In a noa-ltationa:ry field, Eq• . (9.%.11) obfloual.y DO longer connect the flu aDd CW'TNI.t at the aame time ainoe between the Jut coIliaion of .. neutron at tbe point r and it. anival at the aurfaoo element d, a time ' =r/" elapIN. U we . .WIle in fint approlimaUon thal all ~ DlUb'ou make their laat; oolliaion at a diltanC'l of one mean free path from 4', the " ret&td&tKlD" time ia I,.,E, and Flox'alaw

......

J

(

%.

,+

U we denlop J in a po J (J:.

I ) •• •

8. (1, '1 . - I..- -3..

8.

(9.2.12a )

in 1and break it off after \be eeeced tml:I., I

,

I

'.j..,)

I)+ .t; "ij J(J:, 1)- - SE, - .- .-

or in tha general three-dim.naional

, ,. J+ , r, Bi _ _

WI find

(9.U2 b)

OUI

Dgrad4> .

(9.2.120)

U the change in J in one OOIliaiOll time can be neglected, .... can drop the aeoond term 011 the left..haDd aide and obta.in Fum ', law in it.. WUN fonn' . We 0&11 now combine the balance Eq,. (9.2.1) and (9.2.120) into a lingle equatioo by dJfferentiating the fInt with .... peot to time and the ItIOOnd with ....peat. to poeiUon . In a l1OIl.al»orbinB. lKlUf'CI·free medium th e relult ia

iI'. ". ".

7ji" + n i l f - T V' .

(9.2.13)

ThiI IqtiaUon hal t.he form. of the well-mown telegrapher', eqaWon of electrodynamiw. .\ ia a familiu' propen1 of the telegrapher ', equation \hat it predi cta t.he e:Detenoe of .. _ n front whieh mo... with the group velooity " 0 for the ~ oI11:J7 dilturbaDoe ; ouWde thiI_ve front the dimJrbance Taniahee

oompletoJlly.

.........

I Eq. (t.l.IJI_1l.o be _ _ bJ . 1:;--PfI'Ori"'I' ioQ .. t M ~ ,"-PO"

171

In our _ , ...; = 1,01/3, Le., the velocity of propAgation b tI! y!. We migM have expected th at t ho velocit y of p ropagation would be 0, and In f lU'll. the factor Vl ia .. ecneequenee of using the Pa.approximation, ,..hi llh dON not deecribe the angul&r dYt.ributi on of the advan cing aouroe neutrone 'Very well. WKDfUaG eed NODUJ:Rha n obtained a complete 8OIuUOD of Eq. (9.2••3) for an in.finite mediUDI, but we . h&ll not reproduce it here. hu t he wave,like form. montioDed abon . i.e., it ia u ro for r > o'/Y3 &ad for r
n


'1_E S. R. (r)T. (I ).

(9.2.101'

Aooording to Eq. (9.2.13), Tlltt) Ja given by

~~. + 315 ~~- - ~ ~T... If we eel. T..(, ) equal to t:..... tho foUQwlng ebuacteriatio equation for

,t -

(9.2.11ia) II

reeult. :

W• ' +-" ,BI- =0,

'I. 1- lJ~ ± ,~

yl

(9.2.I15 b)

12J)1 ~ .

(9.2.150)

Thus two upc:»l(lntWa occur in the ~ d_ r of NCb mode. In ge ueraJ, .. PI' approJimation a-u to N +1 -.J._ of II . For Iarp I. howe. .. 0011' &he • .m&lJe-t val ue of II ia Iignifio&nt. Eq. (9 .2.16 0) giftl (U :.l 6d) for t.hilII emalleet. value. Th u in ti n.t. appro:dmation we again ob tain the re.ult. of elementary diHu.Rion th eory; howenr, oorTeCtion terme., which are Iarrr the larger ill, do ooeur . We ehall return to theee traoaport oorreotiODll in another

n:

connec tion In Sec. 10.3.3. For ~> ~ _ Eq. (9.2.16d1yielda Imaginary valuM of • . Tha mig ht indi cate th at. for . noh lID.a1l ' p tem. no upol18ntial time

f Z1r.

mUlt be aware that. B:> ~ of our IlimpJe diffuei on theory - i.e.• p".ap proldmation - 11not permitted. Much more laboriou eoIuti one of the t.ran.tport. equ.t.ionI have to be found in order to deeoribe the oeutton decaf in IHD&ll e)"ltem. (d . BoWDD. Kuoxu, DMrQR and E"OOLO) but. we aba1l ~ ~ them bere.

deca y of th e neu t ron flux e:D-t.e. Bowenr, _

holda only In very . mall eY'tema. where t he _

9.3. AJfI ThflOry and the TIme-Dependent DlttuJlon Equation In orde r to deeoribe the time and ' pa.oe dietribution of t he oeutronl during the alowing-down~, we mUit. ,t&rt. with the bU&nee lfCluat.ion

..1 Boook_IWItU. ....

, . (..... f)

a,

-""*'"

• 'f _ _ .L'.tJ)_divJ_ iS-+ 8 (r,tl,')'

(9.3.1) 12

n- Dependenoeof lhe SlowiD.g·Down. &Dd DiffUlliorl ~

178

u

we now introduce F'IOK:'S law in iw elementary form J "",-Dgrad
r. f+

"f

f E, _ ....". - - I E.

D

f E,

.!!.

J71,- ". +8 (" ._, r).

(9.3.2)

We reRrict oonelVel now in the interNt of llimpliaity to a DOD-abeorbing medi wn aDd a monoenergetio.auroe 8 (" • • • ' ) _ 8 (")6 (. )/(' ). U we trandorm Lbe tede-

'*

pende nt nriab1e from lethargy to age T=

f•~~: l ,,',

we obtain

I .,l " + " - J7I, +' S (,,)6 (T)/(' l .

D•

(9.3..3)

'l'hia equation iI c1early a pnenJ.i:ution of the age equation to the cue of nonltationary toIlroM . It can euily be verified by . ubmtution in Eq. (9.3.3) th ..t th e time dependence can be separated in the fann

' (",',1')_,.(",

T)J(e-j ~~)

(9.3.b)

• where 9. ('. 1') iI the .alution of the time-inde pentlent age equation ~~- - V'9. + 8 (")6 (1').

(9.3.f b)

We can euily lOCI what thit equation meant if we . .ume a pu1.eed

toUl'Ce

j~), Le., , and are not independent • variablea: to each t.ime ' after \he beginning of tho tlo wing-down ~ th ere ~odt a uni que nine of given by ,

/ 1')=6 (' ). Then , I', ,.1')=9, (,.1')6('-

'I'

.

'I'

' ''''f~~ ;

T=fDtl IU .

(9.3.5)

• • nlt i. to be ezpected on the baai , of what we learned. in See. 9.1.2. Th ere

'l'hia we that lor ..4>1 - and it ia only in thi.. ease th at age theory i. valid at all - the DeutIonI formed a D&JTQW ene rgy gro up d uring mode ration. Th ua we ND..peek of a RD&le aharpl.:r de£med energy E (' ) or l,tharsy a t' ). To each time therefon then ~pondlI a uni que leth argy and thua a unique Fermi age :

.l~tl.· =J,1~ ~~~ tlI'-JD.dt' , .

T=J

(9..3.6)

• • • [CI. EcfI- (8.2.7) and (9.I .20a.)) It UI thiI direct oonnectioo. between T and, that ~

&be DOmertclature "age". It UI DOW aodentandable why the tiJQ.Kependeat diftuaion ~tion diAouued i n Sec. 9.2.1 la nry IIimil&r to t.be age equ&tHMt : the tim e-depe ndent diffUlion equation caa immediatel, be trwwformed into lbe age equation by changing variable. frora. ti.me to age. In order to deMribe Ule 'J*'l' and time dimibutlon of the neu trone during moderation in a pW-t medium ... thUl need DOt _k any neweolutiOlUl but can refer to the .alutionl of lbe ltatiOnary age equation d eveloped in Sec . 8.2.

Diffu.kln of Monocnclllot~ Neuttonll from 10 Harmonkal ly Modulloted &uroe

178

9.4. DWuslon of Monoenergetic Neutrons from a Harmonleally Modalated Scuree We conaide r lut t he 1p&C4l and time d istribution of th e neutrolUl in a medium in wh ich there i, a ainuaoid&lly modulated neutron ecueee . Such a ecuree can be produced. for e:r&mple, hy periodic Interruption of the ne utron beam. from a reactor. In the neighborhood of .uch a source "neutron ...TN" &l'(l prod uced from the nat ure of whoee propagation ... ('.&Q dra.. c:oncluaiora a bout \.he properties of tb e medium. We , hall retum to \.he Rudy of Iouch ne utron Yaftll in a..pter 18. The ti me -dependent diffusion equation for a point IKlUrce Q.+6Q,'·' I in an infinite medium takM the form

.!.• ~ _ D (~ +.!" aGl) - E• "'+ Q'+'Q~:!. 6CI '" 8~ ar . ,,~ r . Ita solut ion is

(/J(r, , ) ... (/J. (r)

with

Q.

(9.4.1)

+lJ (/J. (r )el • ,

(9.4 .2&)

e- rlL

(D.4.2b)

IP. (r) _ " :I'D - , - . 'Q e - ' /lw lJ IP.(r) - 7iD- - ,-

L. i. a

(9.4.2c )

" comple:r" diUU&ion leogth given by I

L::'"

x.+ iI.fe D

1

- V

..

+ .D· ·

(9.4.h)

Whe n w/.,E.> I. Le., .... hen t he period of oeeillation is Iomall 00IDpuN. too the mean lifetime against at.orption,

).; .. (I+ 2: ) 1

Thus

V·2t= +.(1- ~:) Vlt~~'

_ 1I(w )+ ik (w ) .

(9.4.3

b)

(9.4 .4)

The modula ted part of t.he ee utece penet ration

t

f1U ll

ie a d amped wave with the depth of

' D

z ; I+....... ..VIOl ' X. II"

the wavelength

'

..

-..

(9.4.6&)

fII

-D'-!~ ~ " b V!£! I I_ .'X. •

~

(9.4.6 bl

••

a nd the pbue ve locity

., . '" Y2D-;.c!. .. 1{2 D 1'w . I _ !~ J' I

. .... .. - -dOll1uat a1.I0~ be ::s; f4tiz1Oll fob.

•• _.i.nlzIgth

0I0D De U!'

(9.4.60) be JlfIlotiu.

IZ'

180 The

rtIpO'IloN

it given by

16"'.(' 11 ",(r)

_!i ~~._ q. ,-,/£ .

... 5d (9.. )

We can eMily _ Ulat. at high frequenciee t he depth of penetration of t he wave ulJll&1l. CAuaina lobe t'Npoi'M to fall off rapidl y wit h in~ r. At e nremely low &eq~... CD approach. aero. " aPJl"O&C'b_ IlL. a.nd I: ap pro.eb. U fO, i.e ., the lO1ulJoa beoomM qUMi..tationuy. The 1Pe of diffueion th eory fo. the deeer:iptioa of Deutnxl ........ i, oorTeCt. ulong &8 w< e E " i.e.. .. Ioog.. the period of oeeillation it long compued too \.M time betwee n l uCOleNin eoUWona.

Chapter 9: Referenu8 General Aau.D1, Eo : Lac. dl, _peeiaJly t 70 aDd t tlf,. VOfl D.t.UKL. G. F. : The In.~ 01 Neut:oM with Matter Studied with .. Pu'-l.· NeutroD Souroe, TraM. Roy. InA. Teclmol. 8tookholm " . leM. )L.1I3JU,..I;. R. E.: The Siowinc»0_ oJ Neut.rona. Rev. Mod. Pbys. 11,18.5 (19·1.1). WanDO, A. K., aDd L. C. NODDD: Theory of Neutron. Cbin Ret.etionll, AECD-U11 (I SSt). .-peei&l.l,. p. 1---8S: The Tt-OeperldeDt DiffuaiorI EquaUon.

81*1&1 DTA»'lPI.0.. . ad Eo P. B.t.T.ulfJ. : A&oalDay& G ( IM t).

o..8TUlf. 1.. a, and O. EoUlILI:n

_

EDeI'Ji,.. I',

I

1Ph,.. .. 478 (1m).

~O"U'K'V' """" ~ l865..1 XonIlL, J . U. ; Noal. :Sci. i'.rlI." lIiT liMO ). EamMoIf. Ko.K : .Art.iTPJlIik It. 1 (.I~). 8T.urntOlM, N.: n--.
Time Depelldeo06 01 ta. 1no......-I:Iow. P I ill H~

1"UDe ~~ ~ Lbe SIo'lrina:-DoW1l Pro- IA Ked» ...LIlA + I .

WA.L.LU., I. : 0 -•• 1868 P/11I3 Vol. Ul p.4.'IO. s"OsrLt..:I1D, N. G. : Nukleollik I , Be (18M ) j Arki. Fyai.k II,

I

m

(1i6ll ) (8oIu~ of ~"' +B'''' O). &un.u, V., Md J . Bo-o'W'l'n: : CoQl.pt. Rend. us,lm (19M) (Neutron W. . .). Bowo_. R. L.: TID·188M (1M3). D~ P. B., and D. B. ElIOOLU: Hucl. Sol. Eng . 17. 212 (1962). Tran.port Thea )U.ADQIl. fL. : NlI~'. 14.7 (1M4). &lUi r"f ~D. N. G.: Arid. ~ II, 10 (1" 9). Pul.ed Xoderaton. Wpo. G. M.: .b Iatlvd~ \0 ~ 1beoryj H.... YOI"k: John Willy -J 80M 1Do.. 1M3.

...

10. Thermalization or Neutrons Our tint ~ of thermal neutl'On fielda in 8&0. IU wu "--I on ~be jdea1iringMIWDpUora tha~ tM DlluVONiwere in a true ltatoe of thermal equilibrium and tIlu had aMaxw.1liaIloeneruclil&ribu~ with the toemperawre of the medium.. FOI' _ _ a1rMd1 men'"-'d tbotn, the ...wnption of I d an equilibrium. nat41 el nnoto _ rilOroul1 true. I n thy ehaptel' _ IhaII nady the energy cJJ-tributJon 1M neuttun ene,..1M bolo.. abou& 1 ev In \lotan. In doing 110, ... llilall to 00IlD8Ci the theory of &he \hennaliza&ion ~ wi\h \he t.heol'y of &he alowin&-down prooeeI t.ha~ .... previoUily de"loped for energiea greater than abou~ 1 ev on &M bMia of oolliaiOlll with free, .tatiolW7 nuoiei.

_tn."

18' The thermaliutioD problem ia _peci&lly diffieult. In \b e tint p1aoe. tM elementuy prooeee. m., tho 1C&t.t.ering of yery Ilow neutronl by IOlid or liquid modera.ton, ia complex and hu not yet beee fully e xplored. Seoondly, eYen when th. _ttering law ia known. th e oaJoul..t.ion of atrongly .~pendent neutron . pectra (the 111011&1 cue)1eada to difficult mathem atical problem-_ In thY chapt.er, ,... ahall re.trict OW'IMIIYelI to n pl:aining th e maR important pb.)'8ioal phenomea. and to jnd,iuting t he direction towardl an end tl'e&tment of neut ron .poetra. Th e l ubject iI.t ill very much in .. _tate of flu:! and the preeentat.ion in thia chapter mUlt be ecneldered preliminary. In &0. 10.1, the problem of inelutio _ u..ring of slow nflUUoIlll ia oolWdered . Sec. l0.2 ia ooncerned wit h the actu&!. thermaliza.tioD pro blem, i.e., wit h t he determination of the lpootrum 4>(&') or lJ) (r. E) in .. medium with given IOUroee. I n 8&0. 10.3 lOme propertiM of tbenJl&lized neutron fielda , i. e., neUtron fieW. that ani uymptotical.ly eetabliahed far from th e IIOW'C& (or in the cue of pulled 101ll'QN, long after the pu!ee ). ani explained. '!'be OOtWden.t.iOlll of thilleCtion ani imponant for the inte~..tion of lb. diffaaion experi ment. on thermal neutron fielch 100 be diIeu-.d in Diapten, 17 and 18. Finally ill Sec. 10.4, the queltion of how an alrNdy Jarrly therm&liz.ed l pectrum. approaehel the uymptoti o diatributioD i. dj~.

In the Ml'lied period of therma.liuLion phrUc-, it ...... cutom&r}' to appro:li . mate the t.bennal neutron apeetrum b1 a MaI1t'ell di8tribv.Uoo. 1t'it.b an effeetj"e toemperatore T : the oWn wit ...... \hen to cakulate the deYiation of t.hi8 te mpera.ture from the temperature T. of the moderator. There u no /J prV.wi phflli.caJ buit for lIuob an approximation. and it t una out that in lll&n1 CUM the neutron temperatun CODOI!Ipt le&lhto quanLit.atil'ely falee rMw ta. The toemperature concept U MYerthe'- ueefu.1 for qualitatiYe purpoe8lI and _ nail use it repea~y (&c-. 10.2.1. 10.2.3, 10.3.1. and 10.4.1) to explain the ..riOla pheooomena. of neutron thermaliz.ation.

10.1. The

Seat~g

of Slow Neutrons

Th e -eattoring pt'OO8M u oharaot.erized by the diHerentJal -eatte ring cro. lI6Ction a,(E'_ B, O'_O). In an iaotropio IIUbetanoe fI,Cr _It. 0 ' _0)... ,.1_. fI,CE_E, coo 'II) where '. u t he IItI&tterin8

angle. Frequentl y for LbermaIiu-

• problemt knowledge of (1,(8'_E)_ J'fI,(8'_ 8, ooe'e> d 00lI '.i. lIufficiont. Lion Wbereu for energiN greater' than . ... w . caloulate fI,(E_ E) fro m th e toW

-.

e&Q

_ tte ring en:- eection 1t'ith the help of the ..... of e1utio oolliaion (el. 800.7.1). in the thermal energy tall80 we mUllt take tho th ermal motion and tho chemical binding of the ~tering at.orna into aooount. We ahall treat thie problem in two .te~ : Fint we &b&ll neglect chemical biDding but take into aooount the therm&! motio n of tbo a\Omll, i.e ., we .han O&Ioulate tho _ ttering from a hJPOlhoLifI&J 1......'lCu with t ho dcHwity and temrontu", of the ~ thcnn"I&i~ hlNham. Nut we IIhall indieatoe 00. the eHecta of chomia.! bindi.na: in rea! media can be taken into .coount. Finall1 we aball cliacut T .nODl uperimental metbodl for

Itudying the IO&tterina of .row nout.rona.

.82 10.1.1. CakaJaUoa of o. f C_E ) for aD 14M&. Mon.a&omle 0.. The following derin.' ion ia due w Wlons and WIUI1flt. Let the moden.ting I" COIlSin of atom8 of 01_ M _ A "'" and let t.bNe atom.ll ha ve the energylndopondent IC&ttering croee 8eCtion Fur\bermore. let the . .. atom' ha ve a Maxwell velocity m.tribuli on :

a."

1/

P (Y)dY _ ( z... ..... _,

)'

_ .M'~


Uf'· ·h: V' dY .

(10. 1.1 )

We fint ClOfdider oolliaiona betlreen neutro"" of veloc it y,,' .nd atorna with a particular ",Iocit,. Y. SuPS-:- t bu before tbe colliuon tbe direcu orw of the neutron and. the au atolD make an angle c. (00II • "" p ). Then the relati ve velocity i.

v....=yv·· +V·

h ' Y}'.

U the atomic deDility of lobe g.. ito Natoma/em', there are N P ( Y) d Y gu atoDUI with .. ..Iocit y bK_n Y and Y + d Y per em'. Sinoe all direc:tionA are equally likel y fOf' both tbe ntlutron and t he I" atam . the proh&bility of .. oollWon angle between I and .+,1, ia d".rJ. 'I'hut. t he number of .uch COtliaiOM per em' .nd

...

;,

'-

d,, -tl..t ·a« :N . P( Vj dY · . %

(10.1.210)

•

We can aJ.o write dp in the form d., - ';daa,,·N

(lO.l.2b)

dtt.-",.

....bere u. tbe micro.oopie croee fleCtion for .. neutron of velocit y,,' to collide with a gu atom whe- velocity V ia inclined at an angle . to ~be direction of tb e neutron. Obrioully . ,.. . fI./ Il,. da.·r,.= -- ..- - - P ( V) d V-2 ~' (1 0.1.3) We now wiJIh to find tb e probab ility ll'("' _ . )d " th.t t be neutron f&1lII into the velocity inte rn J (., .+d.) ~r lucb . ooUiaion. We determine it by con8idering

the oolliaion in the neutn:m-s ..·.tom center -of.mallll l)'ltem . In tbis l)'IItem, the nentron b.. the velocity 'A~f 0... before th e colliaion. Thil velocity only changes I.. direc tion in • oollillion; tbu tbe velocity in the laboratory l yatom after th e ooI1w on • given by

.-V~ + ( A~I

r

w',.. + h

.

A~I- v", co- ".

Here v. '- the nlooity of the center of m_ and " ill tb e lCattering angle in the oenter-of-mus lyatem. "- iI given by v. =

". '1+.04" "1+ 2.41""It A+ l

I

A.uming tbat the _ t""ring in tbe oenter-of·mua Iptem iI i.mropic, we find

'ha'

g("' _v) =O .

.

= ~ -":"I '

- 0.

.<....

11•••< 11< ,,__

·>"_1'

( 10.1.4)

183

H ere ,,__ and 11_ are respect.i.....ly the largNt and am&lleet velocit. neutron can h.....e ..her tbe oolliAioo. m ., A

v-...=". + .A +I

tbat the

A

II.....

Ai-"' "''''''

"..... -= V. -

II we now eombine F.qll. (10.1.3) and (10.1.4) and integrate over ..11 d irootiorUl and velociti8ll V. we obtain th e IIC&tt.ering (If'OM lI&Ction for Pl'OO\'Jll8N in which .. neutron of velocity tl ill ecettered into tho velocity inte rval (v. 1I +dtll : a , (o' _ vI dl' =

J'lI. 14.1.1.

U we

n.

ac......... '"

...... _

IWIq _ .... oIea.-1..~

A +I

MIt

~. . ~ .

'I- i VA

., -,J" tI., J dV

ilIA

I,., e' ,

PC r l g (,,' -II) /lv.

_0' . , 'fT e.

_

of

J'lI. 10. 1.1.

'.A

"'- srA ' tbe

(lO.U)

- :"::

OJ '" u fll' no. ona-...... _ _

01

_~

• _

A -I

and

II Nl

lioii u.or-I

0.,..-..

"",ult of earryil18 out Lhe integ rat ion

an be written 0',

("'_ to) - '11

~~ ,,{ ed

IV;;;.. e"II± erflVo!':;.; ('1" -

('1" +

e.·)J+

+ up I 2~ (II"- ~) } . (e~~iii.~ ('1p' - ell>j =F Of . rl

IV;:;.; I.,'HolD).

Tho upper li gn hold. for lI'> tI, t ho lower for "',(8 '_ 8j _

(10.1.6)

tI '

r·l:f·er lU'{'-.'U·'erf['l V:;:

< II,

-r

a.( E' _ E) is tbe n given by

e y. t~J+

+ ,-~ll2'. erfI'1V-~.· - e V-&;l-I'-·'·"·ed f'1r~~ +e V-~;I_.-m.•rll.( ':; +. (-&-11},

(10.1.7)

A. an illu\ratWnof Eq. (10.1.1), <'. IE'_E). rCI-_! for Tarioua Tal_of r

"

liT,

i. plotted in Fig. 10.1.1 for A =<1 and in Fig . 10.1.2 for A =16. U i. d ear that u E'/1T,appro&eh. infinity, <,.(Jr-B) ap~hee the beharior found in Sec. 7.1 ; in thiI ClUe the tbennaJ velociU. of the _ttering nudei ean be negltJded eceapued to the velocity of ~ neutron. At .tuUer neatron energiM the thermal motion 01. the _tt.ering atom. beoom. no\iOMble ; in fa.ot, colliaiOQl in whim the neutron gainl tlnergy become po-ible, and the probability for ooUiliORl with an tlMrgy Ie. become. . maHer.

"< 'Tho I6ltJl ~"9

(TON

-

,

Modioli u.(E') =-!a, (B '_E)dB

e&Il.

be obtained

I

directly from Eq. (10 .1.3 ) by integrating over V and p . It it

"',(8")_ -

IInI;:':'," tilt

#" Vii "CP)

wbore tr,;jF. .. ill \be re latJ.n n locit1 . 1'enged over

(10 .1.8 )

Y and p./l'-dr/MI,. &Qd (10.l.9)

The function y (ft). known in the kinetic lbeory 01 g__ • b.. been tabulated by JUl'II. Fig . 10.1.3 .ho,," a. (E')!,,_/ .... function of~ . 'The at lmaU P. I i.e .,at energ:iee E' < 1TJA, oomell about becaU8e a very Ilow neutron iI fre. quently Itruck during its flight by the more quickly moving IU atom. (tbe I eo-ealled bllDlping effect). F or A>I,

mcre&lMl

•

,

1-- -

\

O'. (E ' )"",O'" -.inee the inCl'Nolle of O', (E' )

I

lint beginu tenergiee below g ' '"" kTJ.A . We caD alto caJeulate the o"Plo' -t J diMrihlioa of t he Pflut.ronl _ u.ered - lr. by .. SU of atom. . 11 we continu e to .... tG.U..n . _ I , I I ~_ _wne iaotropio _ ttering in the een___ It .. .,. It lMnMI ...... toer-of.1D&M lyatem. t.h. avenr COline of \be _tt.ering qle in the labonto"1 Iptem i8 ginn by

,

I

, V·Jr' -

•

_ I. - ix erf~)+~ {erfcP>-

y;. {J.-,.}.

(I O.1.l0 )

When E'> l1"JA, t.be angula;r dilWibution i8 the Mm, M for the IIO&ttering of neutrons on .tationary .toDll. Howonl, .. 1:' falla, eoe '6.decTMIIN monotonicaUy, and for 8'< i TJA . (l(MI 6.... 0. Le., t he Kattering iI aI.o i60tropio in th e laboratory

.ymm. We Ibal1 now invettigate the diHerenti&l crou MOtion a, (E"_E'I 80mewbat more cN.ely. We begin by noling that the nprelllion in the braoel in Eq. (10.1.7) "maine ~ wben E' and E' are permuted. Thu

r

.-'M. a,CE' _&I _Ee-u:roa,(E'_E').

(10.1.11)

In order \0 make the phpictJ ~C&IlOe of Eq. (10.1.11) oIear,le t u remem ber that in a 8tate of \nul t.hermodyaam.io equilibrium. (whiob would emt in an infinite. DOO....beorbiJII medium). the neutron Dux bu • Maxwell diatribuli on 4"(E)..... g.-.IH'. of eoergiM. Eq.llO.l.lI) th en "p that in equilibrium .. many neutroM make kaluQtiOllol from the llnergy B to the energy E' u make tranaitiODII from t be energy r to the eoergy & . ThiI _rtion it the generaUy nJid priJtdpk 01 ddGil«l llGlollU. _11 known fro m natI.tioa! mechanic.. Eq.

- I-JILf.-. -- -Sa- .1..... The KiDetio Tbeory of 0 -. c.mbridrl Ullil'eni ty

~

IHi.

'86 (10.1.11) bolda for the .c-.t.tering oroM .eot.ion of M.'y arbitrary _tWlrer aDd in particular for the cbemioally bound . ptena to be diacUllled later . We tbaJJ. _ later that the ke&tment of m08t thermaliu.tion problem. ia ItronPy inlIoenced by tim nlation. Panioularly uaelul for IM)( H appli _ tiou are u.. moment..

-•

0'.(4 Et -1 a, (E-E)(&' - at dE

(10.1.12)

which cont6in. information about the energy 10M per OOlliaiOD. We C&D calculate the a, tJR)' by integration uaing Eq. (10.1.7), but the fNUlt.ing fonnulu are compUcated (d. VON DdDaL) . However. wben .d>1 aod E'>iTJ.d, thMe

formWu give very aimple

e ~OIW

fen tho fint, few momen t.. :

•

f a, IE_EHE'-E) dE _ ~ O'., (E' -UT, ). • ~ - f a. IE' _EHE'-E)'dE _ ~ O'oIE' .iT, _ ~-

-

•

The higher momentll are of higber order in I/A . Eq. (10.1.13 .. ) for the average energy 10M in a eolliaion i. puticularly inatl'Uctivtl : In .. ooUiaion wiUl .. . t&-

heavy nucleus, n .... 2 g '/A (eI.Sea. 7.1). When E '> i T• •Eq.( l0.1.13a) lead.l to the Ifl,me nMIult. AA tbo neutron

tionary

energy ~. 110 dcee the ooergy 10M per OOlliaiOD; it nnlabe. when aDd when E'<2J:T, the DeUUOD gaiM '''''Il1. BMide. ~ and ~. the quantit y

Jl._(J,). jjJl(g,) ••

X

OJ

) P,{Af......'£. . . .

.,

/

r _2iT,.

'" -

I

'""

(10.1.141

Here M (8) Ui

&

(IO.I.13b)

" ~ ""Eq." ~Yr

""n. _""

I/A _

..... lO.U. eqdI

~

ar - . . I - . -

X (8"- .I')I (I', (&,,_ B) dB' dB

wal.o important.

(10.1.13&)

Mulrell diatribution

,

"

oq.....s_kalJl .... _ _

~ t o r. _

_ B_"7 _

W

_

d"_

,_a,'u'. normal.

lzod to one neuuon pel' om' per eeeced , JI, i. a meuure of the mean Iquand

energy eJ:cha.nged by oollilion in one aeoond het_n a thermal neutron and the ICIottering'"atoDll I, For arbitrvy A , ...1

M I

-JJ ..

an. miP'

-~.eaa

~11

80'.

.-8 -(-' -+-1)" (1.,= -" (~l"+';j~)"1

(10.1.15)

Wy too chanet.wiu !.be --V flI:ll~ hI !.be Iu-.- qIaMIUt.l' M, _

M (r}(E"__ 6)0'. (E"_E)4r 4.r. He"nr,· .,

be; ptond u.1ng Eq. (I O.I .II).

M,,,~,,.playllilla1ll'''';.oo

... and for .4.>1 , (10.1.16) Fig. 10.1.• IIhOW1l curves of M J a..... function of I tA eeteuleted from Eq8. (IO.I.Hi) and (10. 1.16). . -, 10.1.2. Tbtl General Natare of Neutron 8eaUering on Chemleally Bound Aioms

The model developed in &C. 10.1.1 ill UJU'O&listic (though nonethele&!l ueeful, .. we lhall _ later). At the denaitieB a t wWeb th ey occur in eolida or liquids, tbe binding between tbe _Uering atom. can no longflf be neglected, &nd. pa.rtic. ut... ly not at neutron energiM that are emall compared to the binding energy. If tbe binding were completely rigid ••10w neutrons could not eIchlUlg8 any energy by colliai.ona eince tbe atom. would have an infinite effective maa8. In other wordll• .cattering would be elaatic in the laboratory IYJllt&m . Thill bowever i. not th e cue ; in,ltu.d the neutroltl eJ:chlU\ieeneflY with the" internAl " dogreeI of freedom of tho _ttel'l:lf. Th_ degreee of freedom are lattice vihra.tioM in the cue 01 80lids and moltlcul&r rotation. and vibration8, as w611 All some more or Iesa hindered tunal.lione, in tbe case of molecular liquids like H.O. In tbelle latter cues, the diHerentiaJ scattering erose eecuon ean only be calculated correctly UlIing quantum machanica and then only when the partici. pating state. of the 8Uttering lubBtanOll8 are known . In the following. let us consider _ttering by a lubetanoe tbat consists of only one type of atom. Let th e IC&ttering be pwdy inrohtrtnl, i.e., let there be no interference eHoot. at all. Th e Born appronmation then yields the following approximation for th e dif· ferential oroea ~ion I :

a.(J.,

Here K' and K are respectively tbe wavo numbel'1l of tbe incident and outgoing neutroDlli K = pfli => V2",E/A. eec. ia tbe (total) SCAttering Cf'Olla eecncn of the

(1

rigidly bound atome. vit.• + ~r (c!. Sea. 1.4). x gives the chr.nge in wave number in a lingle oolliaion. x =H' -H. Aocording to tbe law of eoetaee (IO.US)

II'

The matrix element I... giv8lI th e probability that in a oolli8ion in which th e neutron ",avtl number changee by Ie, i.e.•in which a momentum lile ia taken up , the scattering ' )'litem goee from th e l tate 0 to the ,tate b. It is aummed over all initial and final state.. and tbe population of the initilJ atate i, weighted aooordins to BoL'TUU.Nlf·S ft.ctor P.(TI) =e-..,t"·/~I -"'U'· . The 6-function

•

guaran'- that only IItatee b will oontribute to the .um for which the oondition 8.- 8. _ g ' - g I. fulfilled. Finally, the oVCIrbtor lndioatell that an average ia to be taken Ovtlfall orientaOOni of the acattering l.beWiIle with respect to the inl Ct., ..... E. AJlAWI. 100. en., p. 601lf.

IS'

eiden\ neuUon beam.. Aher thU a~ , the _UDring Cl'OM Mletion can only depend on th e magnit ude but. not on th e direction of x . If we replace the ~-funetion in Eq. (10 .1.17) by ite ftlpreeente.tion .. a Fouril:l1' integral

6(Ea-E. +E-E') and further lIet

t~,

J••e'1.. - " U - " w/, cU

-.

(10 .1.19)

where H

u. the Hamiltonian of the Ipt.elll. Eq. (10.1.17) beooma 1

••

0'. (E'_E,0088.)= ::.

with

I C" , t ) _

V; f

-.

(10.1.20&)

e'1" -r>4!' I (x, l)d'

r p. ( T.) <".1' _1.._" ;Z RiJl eo· ·..c l BI,11, . ) .

( 1O. I. 20 bl

•

If we identify ' with th e time, we can eonaider tb e quantity ,jBII. &Ii

e-....-;Bill =- " . ...

( 10.1.2 1)

(1)

a Heisenberg operator. Setting F _ c;· ·" . we can finaJly write 1 (It, ' ) -

r p. (T.) (, .IF- (OIF(' ) Iyo. ),

(IOJ .Wc )

•

Eq. (IO.l.20c). which _aa derived by WICK (d . 61110 Zr;xACtl and Ouoan), i. the starting point of variOut a pproximate calc ulatioM of t he inelast ic IC&t te ri ng Cf'088 eecuon. A . urvey cf the vari ous met hoda of m..king lo eh calcu lationa, .. _ II &Ii numelOUi cit&tion.l from the Litera t W'&. een be found in NJ::Lltllf . We .hall reIIItri~ ounelvee here too q uoting EWalt. for 8flvera! important .pecial CNN. For

&II ~ Aa~ie ~ .

with the frequency w (10 .1.22)

.-. OIl

_

•• ( 'J

1:.~ 12';

_ -i" -

1

1

g t he average oeeup&tion num ber of the oecillator and M = .4 """ ie the m_ of t he lle&t t.ering a tom . Suhetitution of Eq . (10.1.2%) into (10. 1.2Oa ) then giVtll t he d ifferential IC&t te ring en- 1fICtion. Unfortunately the time integratk>n cannot I

In deririna Eq. (10.1.20). _

ball beea made of the okMure propeny•

..,iL.

fill

I:<... !'lI...)(¥tlti l... )-<... la18I... ) 1 h_ . ... , _Iu• • •

• . 1.... ...

_f B I

Il _~·""

..

heQ the -"-in( &toaI Yo bound ill a fWod barmoakl-m.lOr pot.BQtiaI. 1'1l._ _ ia pnaticUl, rMIiud ia aD EiDMeiIlIOlid or in moIeouJ. ia 1rbich _~ aa-. .... bound to a .....,.. alicbtIJ ~ nml P.

*

Th~ioa

'88

of NeukoWl

be canied out in eloeed form, but .. lerie. oxpr.naion of %(Ii', '). which leads to a aerie. expanaion of a. is poeaible. Setting

~ coth

X(Ii',tl =exp{- 2-;'

_~_I .' 1/ - 8

itT.

we have

~A AC;- (!I + ~)l . •}expIl"J(w .inh - -

2";;

'''',

I

(10.1.23&)

The &00000 f.etor in Eq. (10.1.23&) ha.a the form of the generating function of the Beseel funetiOlUl. 10 that

Z(H,II -'''{X

~ ~ooth-:0-;}X

~

z: e-1"•• ,,--co

....!.

n:l', 1.

(

....

A 2MQlliDh -~-

)

'

'''',

Subltitution in Eq. (10.1.20&) and term-by-term integration gives

a,(E'_E, 00I0.)=

~:-

V:

-+- -

X

.'l:._e

(lXP {- .;;,-

t oath -ti;.-;}X

_ o -. ( nT. t, 2M... .: : __

)

A~ _ ~(E' -E +dw) .

(1O.1.23b)

I

( 10.1.24)

'''',

We _ immodiately that. the individual lummand! on th e right-hend eide cor. reepced to pl'OCM8N in which 11 "phcncne" are either transferred from the nentron to the oecillator or from the oeciUator to the neutron l • The term with n=O dflllCribee elutio ~ttering. Le., lC&ttoring in which the quantum 8t1.t.e of the oecillator dcee not change. The exponential fWlotion before the IWD playw. the role played by tho Dobyo-W.uor f.m.or in the theory of X_ray IIC8ottering. To _ this let UI oonaider elMti c IIC8ottering, for which E' ...E. K· ... K. and W' =

2K.(1-00lI~,)=rn7~r(Where J. =2n/K is the de

Broglie wavelength),

and note tbat the "average amplitude of oscillation " of a quantum mechanical oecillator of temperature T, is .\"

1

Aw

VI _ - --- ~ . - - roth -·· lUI Aw U1".

Then - wban we di.uegard the factor

be Mt

(10.1.26)

1. (--"'~-A-) ' which can frequ ently 2J1w.inh -"'-

·::~D~~:.~E:E~:::_;~'"\-3~:"~~)'j. (10120)

Tho chanctemtic -.nguIar distribution described by Eq. (10 .1.26) r.rilIOII beeeuse the neutrona are not IIC80ttered from a point ecattering center but rather from a " tbennal cloud" that is Bmoarod out over a region of radiUll til . In t.he limiting CUfI g'>Aw and iT.> lw. tho Cl'OlllI eection should approach the acattering 01'081 fIO(ltiOD of the monatomic gall derived in Sec. 10.1.1. However, I rOl'~

_ . the.qJ&Mion ia c.lIed the "phonon"

e~.

we e&nIlO~ Ib ow tb..~ t.bia ia th e C&M UIing Eq. (10.1.24) llli.nce ma.ny termtl then contribute to the Cr'OIlJ JeCtion, t.e., th e phon on e xpanaion oonn fIM poorly. We or.n anin ..t .. more oonnnient reprMent.l.tlon of th e Ol'OM JeCtion by denloping the function X(x, ' ) in Eq. (10.1.22) in powen of lte e:a:poDen~. Tb il procedure lead. to X(x, ' ) = ~d

i: ~I [;~:'. {<'H IHc'· t- l)+_ ('-':· '- II}r

•••

•

•.. li '" iT• f+..... VrB "'"'

O'. (E"_ B , Cle- lI'.l -"iii"

Ir

-rl..

\

X

.-. -. X[ 2~"w {(ll+ l)(i · ' - l) +_(' - ' · ' - I)}r

(10.1.21) . )

IU.

(IO.l.2G b)

The individua l term. on the rigbt.ba.nd side &nI of the order of magnit ude of I/A- . In our diaouuion of .oattering by. mOMotomlo gall. we t&w tb ..t in the 0alIfl A>I ma.ny e~oDll aimplifi ed conside rably. If we rettricl 0W'8ll1Vllll to bIMvy n uclei, we C&D. truncat.e the e xpanaion of X(x. ' ) after the eeeced term (" bIMTY cryatal ..pproJimation " ) a.nd obtain

••W- E.... 6.J- ·:-

v;:!(.- .".:t ~(' H'J)6(E-E'J+

\

+ -tj{ ~ {Qi+ l) 6(B·- g- Aw)+J1 6(E"- E+ Aw)}l ·

(10.1.27 .)

Th. n

If .

+l

0', (&,-8): [tI,( E' _ 8 .coe 6. l dcoe 6. - tl..

)( 6 (8-8') +

B'+E'1

.

2A1...inh _

V;' [(1- ::;; coth 'I~;;';) ( (10.1.27b)

)(

..

"T -. 01-

x {,tw,-6(E' -E-Aw)+.11£.6(E" -E+Aw)}J ~

[(1 - j~ootb -ii¥.-)+

--

tI. (E' ) _ itl, (E'_E) dB ""'tI..

"---.:J''''

...- _

(10.1.270)

.

-00 ....

When E' > AIU. Eq. (10.1.270) beoomet tI, (B' ) _a..

(I_ ~ ) .

(IO.1.27d)

Thia oorrecponds to tbe retult for the hIMvy gall on page 184. There we f ound

a, (E')-a""but a",,,,. (

e~

• ""'tI..

1+. )

conti8tentJy neglect here.

(I-

~ )+higherterm.ein ~ ,whiCh"'CI.I1

,0> In analogy with what we did in Sea. 10 .1.1, we canD01t'ca1culatethe.~ (10 .1.28 &)

U,"ng Eq . (IO. I .27 h) , __ find that

",

j:.(E'_E)(E'-E)dE=a.. •

f2A .ifth(~ IUJ2n'J X

{

1~ . s-. X y l- -r- 12E'- Aw)t 't2'. -

tf:'""':AV1 + -r - (2E' + Aw)t - _U....,.•_j]•

I

(I O.l.28 b)

When .&"' > l w. it foUowa that

~ - {- aot{E - Awooth'2~;-;) '

(IQ. U8c )

} O'ot lE'- l W} ("A-2 a.. [E'- U T.l

110.1.28d )

fcJ,LJ11 -

lW> kT.\ "",
The lut rMuh &gain corTNponW 10 .. rM uic. for tbe heavy gu, Eq. (10 .1.13 &,. We can lee clearly how t he chemical binding graduaJly becomCII ineffoetive lot high energiM. Fi~y let lUI ealeul.t.e ..vI :

N ,=

c~Ji

--

JJ

J{ (E"HE' - 8 )l a . (&'_ 8 ) dE dE"

•• (l ...)I K. (.,l ';.J

(10.1.29)

=17,..-- - -- - - -i l - . 2 A (M"J ".inh t ;;"

•

(I n t.hit. ClOf\fteCtion, d . PvW HlT). Here K. it the modified. Beeeel function of the IeOOnd kind eed eeeond order. For Aw< kT• • .r

'..2\ ,,_. •

r

K. (';;;d - 8 ( ~rand

1'\

-

I'-• ,

. _.

M . - 8a,./A j tbi. re o . ult it analogoU8 to Eq . (10.1.16). Fig . IO.U 9 - A. , howl M . A/8a.. &8 .. function of 7:' ''' }t." (8 ill th e Eiruotoin t.emperature l. T hi. fiKUrtl

•

•

. how. partioularly c1euly t ho effect of chem ica l binding on t he thermalizing po wer of .. mcd er r . , ._ ~ D .. ting n t.t.ance. When T. > 8 . biniling effClCt.A rc....u . n._-*~ ... are barely palpable. Wh en T. fal r. below 9. M. (nIMl. . d ~ rapidly. 'I.'hW OOIlHlll .. blu' becaUM on the one hand energy ao. ~ are im· probable .moe only .. few ne utronli in the M...• _II dWributioa have energiel in e ll.... of 19. and beeaWI& OIl the otheI' hand - U pin P"l'" are alIo improbable owiQg to \be weak &herm.al u aitation 01 tM m oderator. Nen _ IhaD ltUdy -eatterinc by .. ~lid body. In the .unple caae t.h.U the latUDe Yibm.ic::Jm are harmoaic: aod. t!W &here ia only one aIom in each unit

_-

• .__ .. ....-._"7........ __..,. ,. .

. .. _""_......,

The General N&lw'e of :Neutrorl &.u.erini on Chemically Bou nd Aloma

l ei

eell, the function %(x. ' ) i. gi n n by

Z(x. ' I- u p

{2~ j';(W )~ [OH IHcf"'•

ll+ Ji{e- ' '''- I)] dW }'

(10.1.30)

Thill e xprMeion diHera hom that of Eq. (10.1.22) only in t hat th e exponent i. averaged over the normalized diltribution of lattice vibratione . According to t he frequently ueed Debye model, thia diatribution ill p (w ) -3

A)' wl, (-fBI;

"'D-

O .s; w ~ -"

(10.l.31)

Here 8 D ill the De bye temperature. We can aIao upand E q . ( 10.1.30) in a phonon expansion. but ... ah&I..I be h-.mpored by the MIne difficult~ .. in the C&8O of the EinItoin crpt.al (d . s"OLt..lfD U in \hU oonneotion). It ill nonetbel_ i.neiruetive to calculate tho ~ _tion for prooeeseI in whieh a ~rtg~ phonon ill at-orbod or emitted. We fiDdl

(J+dB'-B,

_II.)-!:- ~, _,w .;'...

:rt. j'l Atu._E_E',

P (w.> [eoth

8 > E' i

G'_d8' _E, 008 11, )= 0:

Here

,-'W... u p { _

;;"

v;:,-''' .:w.

-J

•

p (w,) {ooth ; ; ;

E <E':

P(W)~eoth};;';dtu}

- l

llo.l.32a)

+1]:1(10. 1.32b)

Awt -S' -E . '

ill th e Debye.WaUer factor.

Eq. IIO.I .32al aho. . that for TNy 10.. incident enorg~ the . peetrum. of the _ttored neutronl givee a picture of t he .pecirum of the lattice 't'ibratione (albeit di. torted by the temperature flo01oOr). I neofar ..... can keep t he contrib ution of multiphonon ~ . mall• ..., can determine the Ilped rum of th e lattice vibrations hom me&lJllnlmeDY on the inelastically ecattered neutroDl. For th e following let .. poet ulate th e Debye model. Expanaion of Eq. ( lO.l .30) in powers of th e exponent of Z(x, ' ) end truncati on of th e lIllriee afte r the second term (" heavy cry.tal approxim ation " ) lead . to

I

G', (E'_E,

001 D.l=

!.; y:::: lll- 2;~~:)"! tu (U+ l) dtul X\ 18.,t_

,.... I

X d(B-E')+ tM{i9D) "

for !E - E'I< .l: 9 D •

I:'-N

l ':'.--II - 61IU'.

I

(10.1.33.)

G'. (E'_E.cOld'.)=O otherwile

holD whieh il follo .... that a.( E'_ E)= G'.. {I 1_

~~ d)(7'w'9 D )! d(E-E')+ j

' U ' T=. -(r-=~Itl' " - ' :l + T• lVIE 7'- (16"D,'for ---;-:;:---c,-----

(I0.1.33b)

IB' - EI:5i .l:9D i O'. IE'_ E ) _ O otherwiM.

1'Ibe phoooo u p&Dlioo wD&rriId _, In lbt foUo..inJ"'1 : lint &be Deby..Wal!er faotor .. fad
'"

_11I'1'.

The function

tP('l'JB») ={ +

-lJ; f ;.~- hu been te.bulated by BU.OIUUNN

I

•

•

We can _groin calow.toe a.(E'). ~ . eto. from Eq. (IO.l.33b), but the l'Multing ezpreeaiOWl are very complicated. Finally, Bhown in Fig. 10.1.6, iI

M. =a..

f..'"

w' XI

U' U(.I;6D)I(i'TJ"

.

(2'iT~ W)dm

(IO.I.33c)

( •• )

1mb - -

o 2M', .... function of 8D1'1'. [of. alao Eql. (IO.1.20) and (10.1.31)). For 8»1'1',<1, the "floor. of the binding ill &gain h&rdly notloeable. u. while for Imaller temperatU1'el the lOud gt'&du. /.' &111 "freeull up", --M ~ We Cl'Ul &leo use the quantum mechanical form..wm developed here for .. very mort derintion of the aea.ttering CI'OlI8 section for a a. <, monatomic gu, Eq. (10.1.7). In thiJJ CaIlO , the wave fun ctioM appouing in Eq.(IO.l .20b) I de.cribe plane wevee, and the Hamiltonian ill that of froe p6l'ticIM . Then (el . ZJ:;lU.CS and na.00lIIG.1." n. OL.l.UBJ:R) .. _' -

K:

_. .-

-

,

, , •

•

~Il...-w.u.

__ ,.. . . ..., 0.,," Ir7*l ~ ">

... -..,. ... -

)

{'" (

%(x,t)=np 2M

"- Tt'

"')}

(10.1.341.)

I

and 'lrith the holp of Eq. (10,1.20&)

a(E'-+K. ece 0,) ,,.. I n- l { h i r { - T' YF.'IT,.exp -

yr

,iT..... Jl

[

....

(E'-B)- U,

,.j

(IO.I.Mb)

.

At thiI point. we need merely to integrate OVt'll' alIlO&ttering angl81 {}.to obtain Eq . (10.1.1). It iI inltruotive to oompa.re Eql. (10.1.34.1) and (10.1.22). U we C&ITJ' out in tho latter an expazWOD of ,,'.' and ,,-'.' in powm1J of the time t, we

obt&in I(JI, t) _exp{ :~ {.,-

{1I+: I AW,I+higher poweR of

,n.

(IO.I .36a)

In the term linear in I, both exponent.. agree. In the Fourier trandormation that 10adI to the 0l0llI eootJ.on, unalI' OOmlllpondi to large energy tranafen E' - E. ThiI meaDl, however, that the diHerentiaJ crolIlI 1eoti0Dl for both bound and Iree atoml t.end to the lIaIDe vrJue in tho limit of large enorgy tranden. 'I'hillimiting valueil

,.(J"_B.""'6~_~",r}[ _'_f+:rp{"I~" + E-8'I}dI 2 Yr 2:11l_... UI" - 2-;'

V;

6(B-8'+

~

I

(IO.I.36b)

[8'+E-2 VEE' OOfIDtl)

which beeoeaee O'"A (III/ (.d+ll' O'.(8'_E) _ 4r- - . .- ----uI

...

0'0{

(I ..)8'

. for

"

",8 < 8< 8

(IO.I.36c)

IL ~ . " Tbe 8pecifio Ilea, oI801Xi1o". IlaDdbuoh dR Ph~. VIlli, 377 (IN7).

&ft.er int.egration Onl' aII l1e&ttering &ngIee'. Here Gl hu the aame meaninlu in Sec . 1.1.1 ; Eq. (lO.l.35e) i8 then the reeuk given there for _ttoring from free, 8tationary nuclei , We _ now that the term in Eq. (10.I.34.a) quadratic in f e~ modification8 of the lIe&ttering CNlU II8Ction oaued by the thermal motion of the _tterinj: atoml. AA 1'._0, thU M>ooalled Doppler term. vaniBhM. A Dop pler term aJ.o appears in the eue of t he OlIcillator, bu.t inatMd of the thermal energy iT. of th e free particle., the mean t hermal enm'IY of the oeeillaton a ppean. The quadratic term (M well .. the higher tenu) i8 ch&neteriltica1ly de termined by the chemical binding. We K denote .. th e Doppler approximation that approldmation in which only th e term 8linea.r and qUAdrAtio in , (the latter, however, reU6Cting the nature of the chemical binding) ani used in the calcula tion of 11. (In thi8 h , IO.L1. If.. connection, d. PtrKoBIT and Rui.ooP.AL.) 'The nlidity of th e prirtcipe 01dd4il.etl bob,," for the _ ttering 1 )'8tem1 000aidend here . tiD rem6ina to be inV'Mtig a.ted. In order that thia prinaiple hold [d. Eq . (l O.l. II)}, we mUllte1el.rl.y ha ve

_-..-..,,,.rIPI__

_ ~_

I

t1'.

+-

j lr - ~

!X CI',f)'

tI,_.

,

•

_ ...

0'

+-

~ (r - ~

--

fI l",')' - .-

Eq. 1I0.I.38b) will be true if

I

_~

H",

C (.

. ...

- r>

I

!X (",,). -.--

af

(lO.I.36a)

tI,.

(lO.I.36b)

-<>CO

+-

j

(. - r) (

a' =fx (I' ,f )' - . -

-...

:1-.')-:(-. - (.- i;;)).

r-

II) w;

(10.\.3<10)

We can ea&ily conrinoeolU"llfllvee uaing Eqa . (l o.t .22), (10.1.30), and (10.l .Ua) that thi8 condi tion i8 actually fulfilled by X('" '). Eq. (10.1.360) bolM for every X(I',') defined by Eq. (IO.l .20b). Finally, let WI oonaider neutron aoattering by ... free rotator. Thia problem baa been atudied by ZDU.CB and GL.ltJ]lU .. _11 .. by VOL&Of ; it. eun f;re&f;ment Leada to ftlrycomplic&led e~ aod alIo'" the cUculati Oil of the oro. -::t.loo only in.peciaI CUOI. Bowner, al.uf6ciently high neulroa eDllll'giel and t.emperat1lnll 1 it i8 ~bIe to UI8 • aimple lMII.ioJa.io.I model. Let 118 000. cider the oolliaioo., depioted in Fig. 10.1.7, of a neutron witb an atomiolluoleUll of m&8I M tha.-i.l 00IlD80ted by • rigid link to a find point in web a _y that it can rotate about thi8 point in an arbitrary faabion (" rigid, free rotator "). Le~ the momentum. of the neutron oh~ from p ' to P in the OOW't18 of tile oolliaion, i.e ., lel the rotator take up the recoil momentum. p ' - p _ Ax. Furthermore, lel t he rotator be a t rest before th e colliaion. In the 00Ql'I8 of the collilio n, the 1O'-'or will reoein an amowIl of enel'l1' .dB ; bowenr, only componentll of Ix per . l

11'. .. ..an .. r aboWd be IafatI fIDIIlpwed. to \be - V of u.. ..... rot&&iouJ Ienl. e.a., $hit..p.. "'elliot ... abou t 0.0001 IlO '""'" Ib. ~ it tulfiIW __ ,.

h U. O Y6p0C', wide,.,... 0I

....-rw-. .-_ ".,..

.'t'

11

.

,

Tbennalj..tioo of Neutnllle

peodieu1ar too the linlr. joining the rotator to

oe1ent.in8 it.

~

fixed point are effoctin in ac·

Thu

If tho nuoJeua 'fVeftl oompletely free,.dE would equal AlxI/2M. In view of the relation between the energy and tbe momentum uanaferred, the rotator beha.vea .. if it wen a free nucleUi WhOM "eHoctin tat. . .. depew on the angle 9 between ,. aDd the uia of the link. If ""' .."rap over &1.1 ana:1eI we obtlUn

.. .

1l1 H ' l f

n - -i.v" ..nth M. _3MI2.

-

....

1in1 9 d00l9 - U /ea

(IO.1.37 b)

Averaged over all directJODlI of x, the rotator behave. like ..

free Dueletul of m..". M. If , For polyatomio moleculee, the calculation of tb o ef·

fective IDa. a of

OOUl'1!otl

more complicated. To calculate the

CJ'08Il

eectlon, we ca n

Eqt. (IO.l .Ualff.• replacing M everywbere by Molf' Thi. simple approximate procedure originat.ed in work of S.l0H8 and TU.UR and of BROWN and Sr. Jon ; its validity baa been investigated in lOme detr.il by KRIson and now

U88

NUJUN, among othen. However. etnee free rotation doee not occur for the moderating 8Ubetancee importAnt in pnotiOl'l. we Ih.n Dot go into this problem

&I'Iy m ore deeply here, 10.1.3. NelilVon 8eattertDA' in Water, Beryllium, and Graphite Now with the help of th e formalilm developed in t.he 1&lIt. IMIct.ion, we shall t.ry to det.ermine what. can be Mid about. the ecat.t.ering of n6utrone by nal moderators. We mUllt. expect right. from the at&rt. tbat. the t-t we . ball be able to ach ieve in tbiB .ay is a very rough delllCription of t.he ecattering pl"OCftl8 eince t.he act.ual motiOlll of the atoms in aolid. and liquid. are muoh more oomplicaW than those in the models developed in 800.10.1.2. Wow . The free water moleoule can eDCIut.e oeoiJIatiolUwith quantum energi6ll of 0.198, 0.• 7', and O.4.88ev. It.. rotat.ional .peotnllu iI nry complicated and bu atatea lritb quantum energiM do,," to 10.....,.. When the water molecule iI DOt free but interact. with othel"l in a ...mple of liquid water, the vibrational energiM remain _ntially the Mme. On the other hand, tbe rotation of the water moleoul. iI It;rongJy hindeftld by their strong dipole interaction with their neigbbol"l. Indead of rotation, a band of tonional osoillltotion appean, whoee quantum .nergiel (aooording to mtlMurementi of Raman spectra) are centered. roughly at 60 me,.. Thu for g'~60 me,. a neutron ean tt6U8for energy to the tonional a.aillatiOlll, aad for g' > 0.2 'IV it can tnnafer energy to the vibra. tional motioQl . The inverwe prooeuM are rve since at room temperature both the vibrational deer- of &-10m and the tonIional 08cillationa are only slightly excited. For thit f'tIoMOn, .hen B' < 60 me,. the neutron ean only exchang. energy with the translatory motion of the water molecule u a whole. These latter motiona are aJao hindered... _ know. for e:u.m.p1e, from motJlurementi of the apecifio hoat: but in \be O&1oulation of the IC&tterin,g Cl'OII eectI.onthey oan be OOIlmdered free with an aoouracy adequate for thennaliution ea!owationl. Swtiq with tbMe ideu, Nat.IUM carried out a caloulation of the _ttering

'" for ...&ter. He let (10.1.38 &)

Ir ('" 'J- up {:-: (,, - ~. p)}

llO.l.38bl

hi the tran&lat.ory pan with M _ 18 "'Jo . ZIt d~b08 t he \.o~onal OlIcillation : it i. treated .. an ordinary a.oillatjon with an effective 10. . . ",.. _ 2.32 "'Jo

(KaJ»o Ba end NaLKO'). Then

b (lI• •)_exp{2 ~:QI [cJH I) (.' · ' _ 1) + 'i(e-f. '-l))}

(IO.J .38e)

with Aw = O.06Oev . Finally, the "ribrational p&R ia give n byl

%,. (11.

Il-UP[::,. {3~ (e''''- I) + " ~ (e'-.t- 1I}l

(IO.I.38d)

where "CUJ. = 0.20 ev, and the '11'0 "ribntional .w,fa at 0.47. and 0.488 elYhave been ecmbleed into .. lingle .tate at lCI.tJ "",O.481 fI .... The m&8ll ..... hi det.erminl:d. by the following eonaid6!'atioM. For very large energy transfers, the IUttering tl'Olll Boe<:tion should go OYe r into that for free p rotons , Le.,

Iimr lx,'J- UP {,"Ill, '" it}. ,...

(10.1.38e)

U we exp&nd the argument. of theeJ:pOnentiaJl in Eql. (10.1.38 0 and d) in po_n of ' and combine EqIJ. (IO.l.38 h. C. I.nd d) . _ obtain

1"' 1'1/.- + iii;, + -;;;' I·" +...}.

l (Je.tl = u p \ T

Thul

II"" mtat equallJM + 1/",_+ 1/",.,. from which it follo"

(10.1.381)

that "',. = 1.96 "" .

The Fonriel' V'andormation tha* Iead8 from Eq. ( IO.I.38 a ) too *he _Uering croae eeeuee eanno* be oanied out explicitl y in general j ho_ yer. in particular range. of inoidlln* neut ron energy the expreuiona for Z(x. ' ) can be .till further

amplified.. Thu for incid ent -J'Iiee under abou t 0.20 lIY. only el..tio inter. actjOI\l wi~ the Yihrational de~ of freedom need be taken intoo aooount. and Iv ('" t ) redUON too a Dobye.WalIer faotor . U we introduoe a phonon upaneion in tJu. region, th en we obt&in

17,(E"-+B.tQI'J -!:

« t; wh...

~V~,~.~J~,~.;',.;. e:lp{_l: p}:%.:-i-:;; xl

... 1. )up {- m J(.l,.. (E- r (t .... aln.b2'iT."'" ,

,'.. '}

- lll w+ -ty)

1' .......1

•• 'I'~ + "'"' I+ OOU,llT;

p "'" 3,"y

1JI. · ti

{Cf. &p. ( 10. 1. ~) and (IO.l.M b).] For 8 <0.1 ev , t.e., in the true thermal energy range , only the \enD, with ll -O and .=±1 are important . In other wonU,only I

sa:- lT.,< l-. Uld. ..... i_ O _...-.4. II'

I" prootlI8M need be taken into aooount in which (lither no Cln6rgy ..t.1I is eJ:ohanged. wit h the tonional o-cillatioJll or only one qU&l1twn i . emitted or abtorbed . Above O.2ev and in particular .OOvo O.t8 ov. we mUllt take into accou nt the 10811 of energ y too t he vibrational modM j however, XT(IC, t) may t hon be aimplified. by a 10 Doppler approximation. We I I I I:o'n ' hall not give th e reeulting 11) expreeaionafotQ'.(E ' _ E ) here . !<., In Fig. 10.1.8, meeeured va lues of t he total scattering

K.

-,

-=-='" ,,,,,::iii}

•

r---.

r--

•

••

'",

M" !V LD

E-

ria- 10.1." '"

d.lbt_""" BoO lotaI_uerlq_

_ _ .... . . .... _ _ ... KlI·(IO.l.tll.1

crcee eootion are compared with calculations baaed on tbe model. developed here. The agreement is good. The thoory &180 eeprcd ucee quite fait h. fu lly the average cceine of the llC&ttering angl e averaged over all lIOOOndary energies as a function of t he incident neu t ron', tmergy (d. Fig . 10.1.9).

T bemodel worke exoopt.iona.lly well in the calc ulation of integral tbennalization effecta , M we shallllOO lat.l:lr 1• It seem. plausible t hat the modol developed here could also be ueed to describe neutron _tt.ering in ~t'y 1DIJtn, but t here i' the following ohj&Otion to thia

procedure. The formalism developed in &0. 10.1.2 holda for pllnl ly incoheren~ _to tering, i.e.• _ ttering in which all inter. fere noe phenomena are negleoted. I~ appli. cation to water is juatified only beeeuse the protona mainly ecatteJ' incoherently' (0'.. z= 4 0'1/ "'" 81.5 bam. O'lICIb =- 1.79 ba.m) . On tb e other hand, the aoattering in deuteriu.m ill largely cohertlnt (0',, =7.6 bam, 0".... =5.' bam), and we may expect that ' I, 02 interference effecte play an important role Ein scatUlring. A calculation of the lleat te r· "" 10.1.1. 'nII . _ _ ..._ d . l b t - . . ...... rw ..uar!IlI .... &,,0 • ing croee I&Otion taking this effect into • • __ 10; - - " " " " " M O aooouot 11M not yet been carried out ' . aq.(lD.l.tSP However. HONEOlt haa shown {taking into acoount the diffe re nce of m.... and 8C8.ttering Cl'06ll Il6Ctionj that a modified Nelkin model ootteCtly descrit - varioua integral thermaliution effects that hav e been inVNtigated in D.O. Crystalline MotkraIor,. Both beryllium and gnpbiUl predominantly lIC&ttel Deutr ona coherently. For Be, 0'..... O'lICIb =7.63 bam while for C. O'' ''"'''' O'..,b'''''

" " '"

---

ok' I:'<'

----

..

'" "

'"" '" '"

M._

I The mocItIl,-iNk 81 b&nuo JllIII' R,O moJecule at 1'0010 '-perature. , Thia. no Sonpr tnIe at "wy 'ow WDpetllturM. where paniai tpUa _Iationa are poMible. t NotAl edded in proof : Cf., howe.,., D. Bt1TLa&, Proo. Ph)'•. 800. 81, Part 2, No. ~IO, 276

( 1i63).

Neutron 8oI.ueriq: in Water, !lel'yllill.lD, aDd Graphite

10'

5.60 bar n. Th e "incoherent appro1imaUon", t.e., the neglect of interference effect.l in th e caluulatJon of th e aoatt.ering orou 1IeCtJon, will olN.riy f&il oomplet.ely in the calcul..tJon of the orou lIleCt.ion for elaetic IlCatt.eriq~ . In Sec. 1.•.3 we ehowed th ..t in oohereAtly _ttering cry.talline moderaWre the t.ot.&I tlI'C* IIllCt.ion uhibitl .. " eut-off " behaTior, whiClb rome. ..bout becau8e coherent elaetio aeattering is DO longer ~b1e for neutron w..""lenglhe that are Wpr than twi ce the largeet IIepantion het_n lattice Jlla!-. Natura1.Iy, caIculatioM beeed on the i.nooberent ..pprolimat.ion eannot reproduce uu. behaTior. Aleo the continuoua angular diatribntion predicted by the inoohenlDt calculatiODI i8 oomp1ete1y wrong since the actna! angnlar distribution of acattering from a polyeryatal eshibiu aha.rp m.. xim .. at diIcret.e .uttering angle. (in the lIenae of Debye.Scherrer ri~). Only at neutron energiN 10 high that reflect.i01Ul of many uedera on _ntially every oonceiva b1e lIet of laU ice plane. take plaoe (about 0.06 ev in Be and 0.03 ev in graphite) can the croY eection and angular distribution for el&8tio IC&tt.ering be calculated in tbe incoherent a pproxi mation with ....tia. factory accuracy . In terference effecu al80 occur in inelutio IC&ttering 6nd have been treated among othen by PL.t.OUII: end VAN Hov. and by SnfOW l and KOTIUJtl . M..t.R8H AU. and STuART have e&leulated the energy-dependent erou eectioll.l of magnNium and aluminum for pr~ in which one or more phonoDII &ttl emitted or absorbed including interference effect. and have &hown that neglect of tbe int.erfM'tlnce effeet.t C&U8N only . mall efTOn (lleveral % ). From theM calculationa and from eome quali'-tive ugumenta, one (l&Q oooelude that in tbermaliu.tion calculationa the iDelMtio aeattering croea eeetion ean be deecribed with .u1ficient &CC1U'aOy by the incoherent approximation de'l'tlJopedin Sec. 10.1.2. Aa we .hall _Iat.er, e1Mtic _ttering pia,. no role in 'pw$-independent thennali· ution problema and theftfore we lntroduce DO addition-.! errore by treating it in the incoherent approximation. However, if the \hennaliutiott pl'OOlIU ia coupled with the U&nlIport of neutrona, elutic acattering pt'OCNIM play an important role, and we m\l.lt aupply elaetic lC&\t.ering CroM _tJOQII either from meuurementa or from epec iel calcuiatioRl. Th e el.tic acattering CfOllll eecucn for beryllium has been ca lculated by B HA N DAIU end that of graphite by K UBOHA NDANI d al. To caleul.te th e acattering Cl'OIIII eectlcn In the inoohcren t a ppro:rimation, we ,tart from Eq. (10.1.30). The main difficulties in thermaliution calculations for IIOlida &l'CI the choice of the frequency diatribution function on one hand and the inveraion of tbe Fourier traD.lform, which leada from the function X(- , ' ) to the CJ'OM eect.ioD on the other. In ~'''Ila Debye apeotnun p (t») with 6 D -IOOO · K appro:J:im&t.,ea, the actual .pect.rum of the lattice vlhn.tionI rather well. The i nelutic acattering ~ aection in the " heavy erynal a ppro ximation " ia then giTlm by Eqa. (10.1.33... b, and c). Aa NKl.UIf baa &hown, the -"'ering en- eeetiona calculated in thi.a _y reproduce quite well the 'f'&lueemeuDn:!d with ooldneutfOQl., even thOllih thia approximaUon dcee nolo _m j\l.ltified becaWJe of beryllium'. , mall m... (A - e). BeeaWle of tbe atrongly anillOtropic mucl.ure of gropAiU, ita apootrum of lattice vibrationa cannot be approximated hy a Debye diltribution. In fact, we must diatinguilb betwee n the apectrum of vibration. parallel to the lattice planea of t he graphite and t he .pectrum of vibration. perpe ndicular to th em. The fil'llt exte nda to 'H
..

,

.. Tery rough firet approzimation, we can lIet

%(/11,1) - . %.1. eN.

t>+fzyc", ')

(10.1 .39&)

where %.1. and %I are caloulated with Debye.peotn. oorreeponding to 8 D of 1600 GK and 2600 oK, nIlIpectively. A much more ...tWactory model baa recently been donloped by P.un (d. a1&o T .UI:AB.ABBI). P.I.BJ[I starts from

..

f JP~J! (tK+1Hei·'-1l+n (e-l·'-1» dl dfuj ~.

X(II'. ,)=exp{: ;with

p{w, II = II P.L (w )+ (1-1')P11 (co) .

(lO.1.39b) (10.1.390)

Here l 18 the eoene of tbe angle betwHln the direction of x and the normal to the lattice planM, hd P.L (w) and Pa (w) are the frequency distribution functiOfUl for vibratiOfUl perpl:lndicular and parallel to the lattioe planea. reepectively. p .dl.U) the theoretical calcul..tionll of YOSHIM:ORl and Kn"'lfO l • _tion from the %(/11, I) lpeeified above , a phonon exp&naion wu carried out. A great many terms mUllt be included in this expansion, and the l'Il8ulting e~oM are oompleJ: a.nd can only be manipulated with large electronic computing machines. We shan become familiar with lOme numerical and

Ftl (al) anl tUen from

In order to ulculate the

Cl'OlI8

reeult. later'. In the diaeuMion of neutron IC&ttering, we IJbAll restri ct oUl'Nllvee to the above CUN . c.J.culationa of the IC&ttering CfOlllJ MCtion of &0 have been done by B~D"'RI d al., of ziroonium hydride by MILLJ:R d al., and of various organic moderaton by Bown d al. and by OOLD.....NN and FSDJDLIOIlI .

10.1•• • EKperiment&llnuIUgatloD 01 810w Neutron aeatterlng Becauae of tbe great difficulty of theoretically calculating diHerentlallC&tter. ing Cr"()R Mctiona, it _ml clear that we muat mlllll!l1U'O them . Over and above their .peeial neceuity in neutron phyaica, such etudiee hold great interest for the phyaica of the eolid and liquid lJtau. einoe they enable tl8 to draw ecnelusions about the internal. dynamica of eotide and liquida. This upeet of elcw neutron IC&ttering, however, ie outeide the lIOOpe of th.ia book'. The principle of the meuurem.entoe ill the foUowing . A .c&ttering umple which ill emtoll enough to nelude mulliple .c&ttMing ill bombuded with monoenergetio neutroIUl in the range 0-0.6 eT and the inteoaity of the IC&tteredneutroJUI measured u a Iunctlon of &Jl8Ie and _rgy. We muat clearly alwaY"oonetruot two neutron apectrometen, and In principle all the method. of making monoohromatle neutrol1ll diseuMed in Sec. 2.6.3 can be employed. Th beet luited for theee inveetigatlol1ll is the 'ilMo/./fi,1tl MdAod. which ianplained in Fig. 10.1.10. A neutron beam from .. rue-tor lim enoountera a "pulalng monOClbromator". Here neutroM In a narrow energy band arowm B' are eorted out, and the oontlnuoua beam. i. chopped into ahort pulMit. 'nl,e ,",utrou are then .c&ttenKl and can be detected by a large number

- ----I A. YOUDfOI.l UId Y. KIT.... o, J . Phy.. 800. Jropaall. 3lS2(I9&6). • PdU obWn. Jf._1.088 bam lu _

'-perMan ppbJ.t.e. in SolidI Mel 1Jqllida, Proe. of Ua. lie!

• ct.• ..... Ia.IMUo Soat.t«tns 01 Neu,"""

CbaDI: Ri.... ~ ; Vol. I aDd. II. VieDna: Intenlrotiooal At.omlo Eaers1 Apney. 1M3.

1..

anp..'..

of deteot.on pJao.d at TU'iowI --uertna: Each det.eot.or ill 00I'lDIlCted with .. iUDtl a.naIyr.er; thu detectioD. ill equift!ent t.o an eowgy meuuNment . AA the pW.ina: monochromator _ freq uen"'T u. .. dotIbl.s ~. I.•.• two ph. ...ynchroaised choppef'l located OM behind the other in ...hich p~ ahift, anguJar "locity, and mutuaJ. diatanoe defiDe the energy E' (Eo.L8'I'~, B.uoo• • ). r... frequtlntly uaod ill th e rotanng~ -Jltdromdn (Fig. 10 .1.1 0 ) (BROCX.Bom &, GunK). In thia inatrument, ...mall, rapidly rou.ting !Jingle e:rynal ia Ioc..t.ed

between .uiLable

e Qt~oo

and exit eclll-

maton. U re fleotio M of order h igher than tho fint and reflections from undeeired

.......

lattice pl~ caDbe .up~, the reOeo. ti on oondi\.ion ill fullillod only t tri ce pe r revolutionj thua pu_ of monochrom.tiCi neutrona leave the .,zi t collimator . W. , hall Dot go int.o the num.erou technie&l problem. UIOciated with such neut ron . peetrometerl ally more deeply here. The meut1ttlment. yield the tl Dergy aDd aogIe diatri bution of the tcaUered. neutr0D8 in relatinl unit.. In order to obtain the ~ ~ tI. (E' _H, 008 8.> from thMe dAta , we normalize them uaing the nJue of the ,Iu tic _ ttering emu lleCtion of l'lIlIIJdillJII (a•• _ U 3± O.02 buM). Here .,. CIon

-

neglect cohereDt effecta and caleu1ate the 1m -.ngular dbWibution of the e1utio&Uy I ..... l Q,l.lo. .. l"..,w ..... Jar _ _ ttered neutronl ,netty uaing the Debye..WalIer factor. I n the e val ua tio n of suc h acattel'ing meuurementl, it ia cUltomary to introduce an even function 8(x, Aw) defined in the foUowlng way :

----.---

a.(E' _ E , coe6.) _2x

~ e- .rir 8(x, Aw).

Here Aw _ E' _ E. I n ,t he defining E q . (10.1.40), we m ake Mide fro m the factor

V:

1lIe

(10.1.40) of the fact that

,a,(B' _ E. _ 6.)dependa only on the magnitude of t he

momentum tranafer Ax {ef. Eq. (10.1.17) and the related w.eu.ion}. Further. more. neT)' oro. IeCtion formed aocording to Eq. (10 .1.40) (that ia W'ith an &rbinry even fun ction 8 (x. Awl) ...tit.fiM the principle of det.aiJed baIaDee. Tbla th. mew ured Cr"C* .ectlon. which I. • function of the three J-l'&IDeten, E. and ooa " •• Clan be exp'! n d in MrmI of the t wo-pu-ameter function 8 (M. Aw). WhOM bebaYior ia oa&ier to IfMp. What ia more . ainoe for givea nJueI of " .M. A ~. 8 _ be determined in nrioua (ie., h1 m ore than ODe .wtable oom· binatloo of aDd 001 ".>•• cheokoo. the CODIiIteDc)' of the method of mMIGJ'fI. mea t it ~bIe. A IlfIriouI problem in enJ.aMma ~ ~• .,... oat of the fa.ot that for want of inMDalt1 .. ftI'J' .mall IWl .,." Wp DfI1RroD. e~. the B·fuDction can onl1 • atadied in a reatriated ranp of " and • •

r.

r, •.

war-

200

With the aid of limple ...umptiOIlI about th e lIC&ttering pr"OOl:l88. ho wever, the experiment.a1.ly determined S'ValUM can be flJ:uapo lated over a much wider rege of iIC and w. Thi8 h.. been done by Eo.L8T ....... and SoBonaLD a mong ot.hel'l . BaoooD baa eompiled .. II1DUIl&l)' of all kn own Belt, Aw).meMuromen u . We ou. UM u-. ,.u)t. ei\her .. numorio-J inpu~ d.." in thermaliution problem. or to check and impro ye Ule theoretical modell for calculating diHerent.t.1 CroM MetJon• .

10.2. The Caleulatl oD of Statio nary Neutron Spedn In ~. the moet impor1ant. problem in t.berm&liu.tion pbyaic. i8 th e ealcuJa.tion of the epe.oe- aDd 8Dierc.dependent. neutron Ou lJ> (r. E ) in .. finite , abeorbing, aDd IIOmetimN even inhomogeneool medium. doe to apecified IOU ~ of fMt neukou. To 1101", thia problem. we muat _tNt from t be general tranApo rt

Eq. (IU .12) and _ II; &llOlution baaed. on t he t1Mumption of either a t heoretically calculated or an empirioaJ. IIC&ttering law. Even when we apply very fAr.reach ing ..pproxiInt.ionl, .. lim ple analytic treatment is no longflr possible in t be general lpa.ce-dependent cue; we c.n only ob1&in numerietJ result. by the Ute of large eloetronio e&lonlat.ing machinM. 'The apaoe-iodependent . . . (&0 infinite medium with hom ogeneouaJ.y di .tribuLed lIOUl"C*) JW-.'- a much limpler lituation . Hero the li mplo b&!ance equ.tWn I. (E)~ (.I') - L ~ (E) +8 (g)

-

(10.2.1)

with lbe thermaliz&tion " operato r " L~ _

f I.(.r_g) ~ (E'') tlIr-I. (E)~ (g)

(IO.U a)

• L giTN the eJ:_ of DeUtron. _ttered into the energy E onr th oee

applies. acattered out. In Tiew of the principle of detailed balance, the equation

L JI _O

(IO.2.2b)

•

holds for a MaJ:Well spectrum. M (E') ""' (.tTJ" e- · IlJ'-. Furtherm ore, for any ph)'*ill&l1y li.gnifi.OIDt ~ (E)

-•

fL"

IEj dE _ O.

(10.2.2c)

S eE ) re~nta the -00."", denait1 in Eq. (10..2.1) . Binoe _ are in\ereet.ed in th e apectnml at enef'giN that Are am&1.l oompared to th e larg e energiea of the -ouree neQtrona, we ClaQ taketbe aouroe energy to be infinite . We eeed no* then inei ede the k)Ul'OfI term nplicit.ly in Eq . (10 .2.1) but munrequirem.c.ead \hat the flu ia prop-

eny oonnalized, Le.. that

-

f I. (g) ~ (i')tlB =8.

Alth ough U.e aJ*C&-iDdepeodent

• in praetice. iY treatment ia atill of greU inter.t rigoroua!y

_ dcee ~ OOCUT beeauae it pro'ridee inai.gbt into the mechanism of therm&1iz&t.ion and beca uae in ~ e.- it ia po-lible to red uoe apaoe-dependent problema to apace-independent 01*. For Wa .....(lrlo , _ li nt treat apace _independent apectra in detail ; following thia, we append. ahort diacwoaion of apace-de pende nt apectra in which we limit O1lnIIIlvea to quali tati ve oonaiderat iona.

20.

TIle CalouJac.ioa of SWioIIuy Nw troa Spectn

10.2.1. Th e Spate-Independent Spertrllm In a Heavy 0 .. Moderator

A closed an alyti c -oIution of Eq. (10.2.1) cannot be found for any of th e ecattering lawII introduced in Bee. 10.1. However , two simple limitins C&8N of Eq. 10.2.1 can be reduced to dilfe rential equation. of the -00 order whoee further treat ment hi li mpler than th at of the originaI lntogTaJ equation . The fi~t ~ ClOnOOnw a g.. of proton-. (WIOlfU and Wn.ulfS) and will not be t reoated further here. The second _ , which we Ih all invowtigato in deta.il, hi that of a t herm al g&l of heav y nuclei. To begin with , let U.ll in t roduce th e qua nt ity V' (E) =-4t (E)/M (E ) into Eq . (10.2.1). Wit h the help of the law of d etail ed balanoe th e latter beoomea

-

E. (Eh, (B) =-f ¥ (E') E,tE_B') dE"- E, (E ), (E ).

•

(lO.2.3a)

S ow let ue de sejcp ,(E') in a Taylor 8eri.M around the PJint F: _B ; th en

E. (E)¥,(E) _

.-.1: (-.~). ~!J:1 I,(a E" ,

(l0.2.3b)

Here ~ = N~ are th e momenta of t he .cattoring CI'OM eoct.ion defined in Eq. (10.1.12). I n Sec. 10.1.1 it wu thcnm that for a heary g.. the fim t wo momenta are of tho orde r of magnitude of l /A. while aU th e other momenta are of higher order . U we tab ~ a nd ~from Eq. (10.1.12), th en in Ieadini order

[

"

. .·1

I.(E)Y'(E)-fI, (2i:T, - E) ~.u:· + Ei:T, dgi- ,

(10.2.30)

Herowe ha vointroduoed tobe approaimatio n f - 2JA. Th eflul: 4tIE) ilthengiTeD by (10.2.4,) Le., for a heavy gat the thermalization operator hi a aimple eecond -crdee diHeren· tial operator. We can eaaily chock t hat Eq. (10.2.2 b) holda for thia operato r. Lot UII nu t consider Eq.(10.2.4) in t he limiting cue E :>i:T.l We can then lOt i:T. _O a nd in t he abeenee of ab&orption obtain

(10.2.5a) . hieb h.. th e aolution

......

4I (E )- tz;" B '

( l O.2.n)

.-...

This i, t he l /E.behavior of t he epithermal flul: familiar from the theory of Ilowing down . Th e OOnItant foUowl from the requirement that lim cE,E 4t(E) be equal . to th e lOuroo delWty. For all t he IU~ oonaiderationl of thia I Ubeection, we auume 1/_baorpUon , m ., I.(B ) _E. (i:T.l yD".7J'. Fi g. 10.2.1 abow. the ...a1uoe of g4t (8) obtained hy Hvawrn; tlal. for .....neu. valuoe of E. (.tT.lffE, .. funotion.t of YBI.tT;. The ...a1UM wore obtained bl na merio&l aoIution of Eq . (10.2.4) (in thia connection ell. aI.o CoHU). I n . uch • .olution, '" mua* take AI a boundary condition 41(8 =0) _0 ; the IOlution ItUi contain. a oonatanl factor .. a free

•

pamneter "hiob can be determined from the requirement I E. (E) f/j (E) 4E =

• normalization

8OW'Cfl

dtlnaity . Under certain circumstances, another may be con",niemt. We can _ from Fig. 10.2.1 that only in the preeence of weak abiJorption (L'.(lT.I<$L'.) doN a "thermal marlmum " emt. In thtl presence of t1trong abeorption, mOldof the DClUtroM are ab.orbed before they become thermal. W'" t1beorptioR.. The C&Be of weak absorption ia of gt'Mt inte!'Nt eiace in many media X.(tT.l<e E. _Here an inltruetin approximate lO1ution of Eq. (10.2.4) i.J poaaible . Let us .et

n,,·L

1/

'" 1

,

,z,(E) =M(E)+F(E) ,

~

I , 'I

~

~

tu OD'O

I

\\..

,I tr'_ •

PIc .lo.Ll. no. _,

l.IlenI:oo.l_..,.,

"~

"~ i Yk lt ~

_""""""'" ...1or

bI ..

X..(E )M(E )+E. (E )F( E )= L F.

II".. bsorption,

approximation. Then

[EE,{ (E-iT.)F (E>+EkT.~ }j .

(10.2.6c)

I

(lO.2.6d)

integration of this inhomogeneou s diHenmtial

.r.:~tl M(E)jf'~ j Y; e-" dzdy+ • •

F(E) =

(IO.2.6 b)

As long ... the absorption is small. the term on the left.hand Bide involving F(E) is .. amall perturbation, and we can neglect it in first

X.M(E) =LF =/.

In t.he cue of equation yield.

(10.2.6&)

Le., we split the flulI: into a Maxwellian component and a perturbation which clearly vanishe8 when E. =O. (We lhall return to the question of normalization later.) U we lubfltitut6 this form in Eq. (10.2.4). we find

+ {dMIE) +b [MIEl E"

(:r.II).'

The term in braoee ia the genoral aolution of tho homogeptKIu. equa tion LF=O and oontainl two free ooutanw, Cl and 6. Sinoe 1'(8_0) mutt equal seeo. 6_0. We can determine the conna.nt G from a neutron balance. Clearly,

. •

,...

Hm €I:.E~(E)-r E.(EI~(E)dE.

(20.2.7a)

I

Le., the number of neutron' absorbed per em' per aeo muat equal the aouroe denaity. NoW', on the one band.

J'!".IE.E~(El-l'~IE.Er(EI

-~ X.(kT.)8M(E) I

a . fllOb'loM Oft p. M8.

f ~ Jryze-·

vz:

. IU'.

••

dz d1l = LJ'!. E.(kT.) •

(10.2.7 b)

The c..loulaUon
203

while 00 the other hand for l /v.abeorption

,f E.[M(E )+F(E)]dE = r; X.(.l:T.> (I+a)+ ~

Il/I;" .

<10

(IO.2.7c )

"

+ X.:~J !X. (E )M{E ) ! ~J Y;e-"d:rdYdE.

,

Now (ct . Conn)

"

Il/ I ; " . "

<10

.. Jyze-'J J

!X.(E)M(E)!

,

~

"'" E.(.I;1'.)

'

,

~!JfZe-d:rdYdE

"

;~

(IO.2.7d )

V;;e--tlztlydz =E. (.l:T, )xO.7989 x 2

Vn.

' "

If follows by equating Eqa. (IO.2.7b and c) that E,.{lTJ

a =- ~

Thus we finally obtain 4t(E) =M(EI+

X·i~~·) M(E)

XO.7989x 4 .

!! ~ f ]l%e-Il /I;".

c

0'

"

d:r tly - . XO.7 089]

y_

4l'(E)=M(EI+

2~ X. liT, ) fE.

~(E) .

(10.2 .810)

Here we have introduced tho funation P'(E) = M (E)

. ! ~f [~•• , ,

which for high energies approacbee I/E. Fig . 10.2.2 abows N {H) and ~ (E) ae funotiona of E/.l:T,. Thia elegant repreaentation of the apeotnlm ia due to HoROWITZ and TaETLUtoJ'J'. It appliOll to .. bouy glM moderator for weak ab60rption (X, (iT,lIeX. muet be < 0.1). We can oompare it with the reeult of the etementary calcula tion ginn in &0 . 8.'.

There. oompletely neglecting ther ·

(10.2.71' )

,

}'%e- "'dz dy-

,••

~ x O.7989

/ ''"T....-,

~

/i"..-w 1o / , 1\ \ I

.

\

I

(10.2.8b)

-

--

--

" " ' (E)

\

M r----~ ·

",

,

I

"'-

J

" ,, '

malizatioo effeota, we divided the B.......u.TmlUo:>tf f1u>etloIl 'i.(.I) apeotnun into a l lE. put for energiea ..... 10.L!... TIlt . 1IIMluJ. _..., po .......... >.1:1', aDd a Maxwell epeotrum for thermal energiel. A ooutron balanoe wowed that the ratio between the thermal f1u.z. and the epithermal flus: per unit lethargy i. ginn by tho moderating ratio

ex./r; X.(iT.>. 'l'hia reaw. remains unaffected here; howonr. for low enersitll

the functioo 1i (E) deecribee the tranait.ion ~.een tho t.wo limiting beh" vton. The ~JX-nw.t.ionof ~ (B) by Eq. (10.2.Sa) baa ,till a nother intenwting property : In onUr too ditcuM it. let UI form lho totllJ lWWI"OII lkMiIy

f- '

~ r,I"'J f-'

f-'

,. "'" fP{E) -.-dE = " II M (E ) tlE + -,,""" (E ) es, (lO.2. 88e) , E.- o 0 0 With the help oJ Eq8. (IO.2.8 b) and (IO.2.7d) we can ea.aily convin ce ou~lve, that

f• ~ F.

(E) dE veniehee. I n ceber worda , the perturbing function docs not

,

e

~

r! •

-

_17

V

K

•

.

-

•..... IU.. ... ~

IJ&lIlO

~_ oz t-

!/

-

contribute

I ~ 1'-

"'"

.

.-

pIoI '" _

, l _

....... - . . . . .. . . . . " .

•

""""*'011"'",0' ... ......1«

J.+

w th e neutron donaity. INld t he tokl deneity

,..,luo ... tho Muwellia n portion. In our work, a _

il normalized to th e

V;I:~ ; ~-

M (E ) rlE _

o in t he work of HOROWltt and TaftUlton " i. normalized to unit y, Le., our 8tT Eq . (10.2.8. ) mUlit be multiplied by the factor ,• in orde r to agree witb the

..

V

work of t.heee authon. Whicb normaliuti.on factor we use il unimportant all long.. the eoereedenaity i. no," lpecified.. However. if &lIOuroedenaity 8 (cm-'!leC-11 i. given. the normalization il fiaed and we hue lJ>(E) = -

8

- - •.-

_~'": E. ltT.l

[

~

t- E.
1

- M (E) + - , - - F I (E ) .

(I O.2.8d)

t;

TIr.e Elltdil'e Neld"", Tnllp'MtIll't. Nen we ah&ll dillC1lU the oonoept, UMlful for pn.ctie&l pu~ . of tbe eHect.iYe IHlUtron temperatllJ'fl. For thia pu~ let .... oonaider in Fig. 10.2.3 • 1og.1og plot of tbe calculated lpectra lJ> (E ) 'flInU1 B for • .nou. amaU ..IUM of I . Il'T1)/lI•. 1$ tum . out that in the t hH'DIal tan&e the • .nou. l pectra have very limilal' ahapni but with intteU.i ng a beorption llbilt more a nd more to th e ri,;ht. Le., more lind more to b igtKor energiN . Thi l

' uggeetlJ th.t .hen E.+O the thel'lll.lJ 1pcct.nLm can be deecrtbed by a Maxwell di.ltribuUoD whOM temperature - the neceoa tempenture - ~ higher thaD the moderator temperature. 'l'bere i. no good phyaioal foundation for naoh a ''GPO

~tio n. 'Dd it e&n only be justified by it. .u~. In Fig . 10.2."', In .~I) i . plotted ~ 8 /111', for the lJ'l!'CUa shown in Fig. 10-2.3. U th e perturbed ' pectra were Mazwell dilltributiona, the plou would be .tn.ight linN from whoee 110pel the neutron temperatures oould be determined ' . We see that I~ ---.:....:..., ~' I in fact in th e energy region g < 5 iT. ~ _• • • -11tuafI/ian ~ and for .E. (.l:1'. l/e.E.< 0.2 straight " -, linel actually do occur. Fig. 10.2.5 Iblnn the ratio (1'-1',,/1'. deter.

,

~

K

r\~~~

-

~~

,,

•-

,

;;: ~

• , ria- l U&.

,

z

, ,• ,

...... -

on. D"*"oI

.,

• -.Ill,. .-tIIIII • .,. _

'

..1-+/ +-+-+---1

.,.

- . . . III ......Ior b,. al>lhod

IIUI

...... -

on._ _ r -....... btr

rla-I _

....... .,. _ ~ ~..,....

mined from . traight..line fit. '<) th e apectra as a function of .E. (1I1'.>/eE•• The b roken line . boWD in the figunI oo~ponda to the equation

1' _T.[ 1+ 1.46

E.,'r') l-T.!I+ 0.13 A E.~T.) I

(10.2.lOa)

whieb. we can oonaider ... a recipe for calculating the neutron temperature in , he..y gal moderatoi'. We can eaaily obt.ain other reoipet if we perform the rather arbitrary fitting of the . pectra by Maxwell di.tributiona in anot her . ay; for eumple. CoHn find8

T_T. { 1 +0.6A ~]

flO.2.IOb)

. hile Oova roe d aI. obtain ' (10.2.100)

... UiWlg the an, . tben:nal abeorption eroM.ection. and t.be fact that

r.- t;-r.li:T)... .Y; r a(i T,)

I'J U:.-fD.,Jf1'.... (_ below), we can write th _

"'f~

T - T, _ ~ 8

relationa in

( IO.2.10d )

r,.

where aeeording to Eq . (10.2.10. ).9 _4. 0.73 . TJ }'i = l .66 In thia form we very clearly t he cau se of the temperature riflfl, which i' gJ'tIater the lArg", the ratio of the epithermal to tho thermal flUll , i.e., the leeget t he ratio of t he "infiux " > to t he " popul lltion " . t Ned we t um to t he questio n of ,..hat .hap' the epi thermal n UI baa when t he th ermal flux ill rtlJW-nt.ed. , by .. Maxwell di .trihution 1 -- -

e&D. _

,,

It,.'''r! I/f ,

I

I

,

.mft.ed in temperatUl'fl. W e

£- •

•

.. 41

thef'tl fon!l

I Ubtnd

from the

apeetra IIhown in Fig . 10 .2.3 the Max.eU di.tribulione .... I U .. no. """"'" rao:uo. "' . u..r-I _...., _ - . . " " fitted to the m by mea na of Fig. 10.2.•. The reeuhlng functi on F. 18 1 is norm&1iud 10 that for E> l T,. ' , (E) _ l IE . Fig. 10.2.6 show. g .F. (E) .... function of HIlT (not g /lT, I). The figure aho... that. '.(E) depend- on lbe ab-orption : in .. roug h appro:rimation ,uffieieot lOf many praetical PIUplMM. we ean introdooe an average " IE)-

£I(~~E. iI (~) i ' ealled the "ioininl funct.ion" ; .. we een eee from Fig . 10.2.6. it "'a.niahe. for E < 4. kT. goea througb a maximum at E "'"8 I:T. and applWoChee th o ....Juo I for E> 16 .l:T . w e can no w writo th o total spoctrum in th o form 41(E )_

.E: , - . lt1' + A .1IE/k.?2 (t T) ' 8

(10 .2.10 0)

.f

~

Tho ....Iu. ol J. followa from a noutro n bala noo: EI. J. ... ~

~

f I . (8 )iJ(8 /.1:'1') dElE. i.o. • r;. r.,1tT)

s" I . (.l:7') + J.

1_

u:

I,

I. (E)lP (E) dE -

I

-r-i-n... 1_ ~{l7') .

(10.2.101)

u;

When I.<EI. _ caD 80t tbe lMOOnd iaotor equal to one and obtain the aimple rMd that the ratio of the Maxwell flu to the epithermal flu per 1lDit lethargy ia equal 10 the moderating ratio.

In.. joiIl.ietc , . . . .bow:D. ill fit. 10.2.1 appodzaMely ~

forl~:

J •

.il.l'/H')

E.1J')- -8-4.1'.. ,r. IU ').

tlr.

touowiIlc ",1a1OOD.

Ar. we ab..u _ lat«. the appftnumate ftpreMnktion of the .pect.rum by Eq. (10.2.10e) ia frequ ently pouible in real modon.",n. ] t toI'IlII out that the .. joining funotion" .1(EIU,) depew rather weakly on th e properti N of the modtlrato r. Th ue in many practical _ , we oaD deecri be the .pectrum enUrely by meane of th e two paramotel' A and T. Thi, praoti co ia Npocially cUltomary in tho eval uation of probe mMll1U'emeDU (d . Cbapten II and 12). Thia dllllCription of th e .pectrum. i, certainly not. very eDCt, .nd in fact th e tempe r.ture concept fail, oompl.et.ely when th e abeorption croaa eecnon ha. resonancea near t hllnn.l enllJ'iY or if E. (kT,l i. not < ~ r•. 10.2.2. The

Spaee-lDdependeD~

8pedrum In Real Moderaton

U we wiah to calculate th e neutron . pectrum in a n actual moderator.....e mutt IIOlvlI Eq. (10.2.1) usi ng a eoattoring law which ooITOCl1y dOlMlribol tho chllmical binding. In principlo, the following p<:*ibilitiM are aY&ila.ble. la) DireclNlUDerieal IntegrationofEq. (10.2.1). We can trandorm Eq. (10.2.1) into a .yetem of linear equ.tiona by dividin& tbe continuoua energy range into a nlUDberof diaerote groups in the manner of inultigroup theory. Tho number of groupe e otlld be large (26 to liO) in orde r that the energy variaUon. be well approximated. The .yltem of equatioM can tMn be eolftd on an elect ronic eomputing machine. An advantage of thi . method fa the poNibility of • •ing an a beorption croelI eeet.iOD with any a rbitrary energy variation and a ny arbitrary IIC&tte ring law, including one that may have been determined nperimentally. A diMdnnkge fa the considerable npeoditure of eHort Iaroleed in the calculati on. for the ~rum mu. t be calculated anew for each value of E•• ] n addition. we Ioee any phyaical " feel" we migM have had for th e connection between th e . pectru..m and the _ttering law . (b) A Modified Horowitz·Tre tiakoff Method. H th o abeorption i. weak a nd follow. tbe l/to.law l , we can &ot 4J (E )=M(E ) +

ju t

at

we did in Boo. 10,2.1. Here

r;

X. (lT.I t X, -".(E )

(10.2.1110)

F" (E) i. determined by

E. (E)M(B ) _

-

J;.X.(lT.) t E, LJ;, (E )

JJj~6) dE_O.

(10.2.11b)

(10.2.lI e)

•

J;, (8) muA now be det.ermined by nlUDerioal .altltion of Eq. (10..2.11b) tieing a DlMIW"lld or oa1cula&ed ICatteriDI: law ; tJU, ler.de to all e~ture of com· pu.ktional offeR lIi.mi1ar &0 that in method (a). HoweYer. we only Deed to carry out auoh a o.SeuIation onoe Iinoo theN ia a &ingIe. characleriRi.e J;, (8) fM ~h 1'I1Ie

au. DODllliioa .. _

OOIIIpIJaaIed

-- .r.+ 1/9.

- , . . but tMd.c-lp.tioa ol Jj (6) " - - ml d l _

...

Thermaliutl.on of NeutrollA

moderat.ol'I. Shown in Fig. 10.2.7 b Fr(E) loll calculated for graphite oompa!'OO. with \he heavy gu approdmatioD. There are marked diffel'enoN, _peelaDy at low _rgiH; becauM of t be Itrona: binding, the .peotnun in p phite i. much hardClr than in .. heavy guo Unfortunately, there are no calOUlatiOM of F.(E) for ot her modeeaeora. M OftlOfllf , there hall &till been no lUooellllful attempt to extract ~ ( E l from experimentally mea.eured spectra.

(e) The Method of CoRlfOOLD . As CoRNQOLD ballo shown. t he general solution of Eq. (10.2.1) can be written in the form of a power Il6ries

(10.2.12) where the ooefficientl a" depend OD the absorption erose eeeucn and on certain integrals of the function X('" ') tbat CharaculUe8 the _tt.ering law . A deeper (J.I di.8cu8aion of this physically very eleg&ll.t but / 1'-. formany rather difficult !--~ ._-_.met hod is beyond t he --acopeoftbiB text [ef. • 1/10 o P 4RK:S). Itll particularly I .__ . \ --- - ~ - - -- ----well suited to repre66nt • \1 _ _ ing the spectrum in tbe I tr&n8ition region be. I tween t he thennal and epithermal ..&rlg('lI . (d) Semi. Empirical Methoda . If the _tte.." ing law fOT a mode..ator V J S is not known or ia on!y >!'known poorly, or if the ~ 10.1.7. hi B IlIOW\~ rr.uakaII I\uWI&kIa " (~ .. _ _ pen. tuff mathematical aidll to comput ing ehe spectrum are not anUable, we can try to modify some of the method. introduced in Sec. 10.2.1 with t he aid of experimental data in order to at le&lIt arri ve at a rough dMcription of the 'pelltrum . Suob procedures were frequent ly U.\I8d in the dawn of thermalization pbyaiCll. The mOllt elementary of 8uob method. WI6ll th e concept of eHective neutron temperature. The apectrum i8 repl'ell(lnt«l by Eq. (10.2.IOe) (with an expori. mentaUy determined joining function or with that of the heavy gee model): tbe value of A can be taken from Eq. (1O.2.IOf)'. The relation between T and T. i8 determined from experiment. FrequentJy the concept of " eHeetive" maea ill also introduced. The difference between the neutron temperature and the moderator temperature in a heavy gall iI proportional to the maaa number A {ef. Eq8. 10.2.l0a-ell. If there is appreciable chemica l binding, the temperat1ll'fl dif ·

•• •

1---11

..... .

_.

,,

-

• , - """,,,,

-.-,,,.

'-

-, -,

,.

,

- -'rll.k _._hoIda of

,

......

•

'''.

OOIlnll only U Ioqu E.(.l:T.1 < 0. 1. Th o .Jowina-dowa power fEo I l l ' " be gi.en N " platea u " value, Lo., tho ..Iue i' hu abo .. I t

I

0".

... ference caD UDder oel'tain cin:um.w- be larpr IiDoe the thwma1ization ~ hi hiDdered. The "effective " maN ia tben that YalQfl of..t that we mut Rho .Utut. into Eq. (10.2.10 _) lar b 0 11' 0 ) in order to obt&in Ute aoturJ. temperMul'e difference , It t.UlnI out. t.he effective m... e.n be many li.mN Wgw than the t rue mau. Another IMlmiempirical appro:dmatioD peeeeduee ia hued: on the thermaliu·

u.at

tion operator of the heavy gat (el. Eq . (to.! .4)). U.mg an energy-dependent slewing-down power, we u n eon.tNct a more general operator (10.2.13 ) which ob-rioual y lIat.i.BfiOll the principl e of d et&iled \:l&IaQoe LJI - O. EE. moat go o ver into th e p1&t.eau value for E> 1 e.... but fOl' . mallet enllrgiel it e&ll beba" arbitrarily. We can by to apJX'ODmate the binding effects by • Iwtably ehoeen decrNM of th e alowing.down power with deereaaing energy. We C&D then determine the function J;,(E) for th e renJting operator. (Cf. C £DILB..6.C . . weD ..

Soa u ru and Au.son.)

.

A oomparilOn between ealcnlated and experimentally ebeeeeed .pecka and temperatures folloWI in Ch&pw 16.

10.2.3. Rule Faeu abeut 8paee-DepeQ4eDt Neutron 8ptldra For the foUowiq qualitative oolWderationa _ .hall u.ume the ~ty of elementary diffuaioo theory. Fw1.her we Iha.ll UN the ,..ul.... of Q1apt.er 8 to the foUowing edent, m ., we Iha.ll _ e that t M e!owm,:-dowu deoaity ia known either from m-..unmeu.t or from a&loulatioo atKw. a out.4 energr E.,> .tT. 1 and U80 it u the 'CIW'lle denait.1 for the thermal neat.rou. TIleD. for E< 8 . ln a homogeraeoua mediwn -Dr' ~ (t'. B)+E. {8) ~(t'. H) •• E, (E'_E)~ (t', 8') dE·- E.(E) ~(t', E )

=!

+q(l". 8.)/(8) .

I

(10.2.14)

..

Heee /(E) in the energy diat.ribution of t.he lOuro& neutf'OIUl that. han made their 1Mt. oollie.ion abon 8 • . /(8) i. DOl1D.Iolli.ed 10 that. !t
In other ...ordll, let. u. t.ry to n pre-ent the t hen:nal.peotru.m ... th e produot of a nonn&lir.od, lpaoe-i.ndependent l peotral function and the t.otal thermal OU)[. I TIl. llIIkJIf - u IboWd be.b.Jcb -ab thM E, (r-~... 0 for ~._~

r < ~. < • . 16

.10

I

Thi. procedure IMW to

-DIE) ....:- Il>(E)+E.IE)Il>IE) ...

-J

I.(E'_B)IP(E')dE'_E.(E)lt>(E)+

•

f~~~i I(E)

(10 .2.17 )

...

with

(10.2.16)

D _ JDIE)Il>(E) dE

(10.2.18&)

• r, -J"E.IE)Il> (E) dE. •

(10.2.18 b )

Ii it immediately clear from Eq. (10.2.14) that I1>(B) i, epece-mdepeadem if .00 onlyifthe Ow: ourvature P'lt>uJrPlJland!llr, E.)/4)u.(r) are epace.independent., Thi. it al....y. the cue if there il looaI equilibrium between the thOl'm61 neutrons ~d their 101lf'CIN. Such equilibrium typic.lly oooura in the inner regiona of a homoseD60\1.I rMOtor or at l&rse dl..ttanoeI from .. toUfOe of fMt neutl"OM in .. hydrogenou modontor. U these oonditionl are fulfilled, we can, at l6Mt in principle• •pocify I1>(B) immediately: Tho lpectrum 4I(E) ia th o ...me all that in a.n infinite medium with bomogeMOlaly diatributed lOuroet and tho eff&Cltivo abeorptiOD crou leOt.ion

z:tt(BI =,E.(E )+D(B)Bt

(with Bt =- f"'~~ll) .

Unfortunat.clly. mOllt of the method. eed reaulta given in Sees. 10.2.1 a nd 10.2.2 are valid only for l/v.abllorptioD. while beceuee of the appearance of D (E), x:'(E) certa.inly does not follow the l /v.law. All .. rule, therefore, multi. group methods are wed for the oaloulAtion of 4>(8). DB SoBRlNO and CuRE. have ecleed Eq. (10.2.16) numerically by 8fIrie. exparaion for the eMf! of a heavy gt1t8 moderator with I/t'-abeorption and aD eIl6rgy.independent diffwlion 00_ efficient. Fig. 10.2.8 aho... .clme of their I"fIfIU1te for varioue value. of the parameter DIJIIEE•. When AE.(lT,)/E.<1 and ADIJI/E.
.

.

.!:;,Tt. _0.73 '~~!W.'>

+0.70 A'?~

(10.2.10)

~

The calou1atioIlll of n. SoBRmO and Cutut were carried out only for 1JI>0 (Le., only for GUM in which there w... a net leakage of neutronl out of each volume element). Eq. (10.2.19) II allo valid, however, for negative 1JI, i.e., for ....... in which there i. a net infillJ: of neuUou into MOh orolume element. The fi.m lummaDd on the right-hand Iide i. the lame ... that which appliN in aD infinite medilUD i in addition, there oooun an lnol"flUfl or deoreue of the temJl8l"" ture aeeording to the ugn of 1/1. It i. m.tructin to bring Eq. (10.2.19) into the

form of Eq. (IO.2.10d ): Setting we obt&i.n

~-

r;.

'"

E. (U') +DlJI and

E:-IEr. -gJ."J~lA

41_ 1 AT, T-T, _ -,:8_ -. - D.8I -r,- '

(10.2.20)

Here the fint term, which ..h,.,.. ,...tly n oe«U the lIeOODd.ia id lloLioal..nth the right-hand aide of Eq . (I0..2.l0d). t.e.,the temperat.ure increue it determined ju.t .. te IUl infinite medium by the rabo of the "infiux" to the "population". The eoeeecu cn term comea about bec&uae the energy dependence of the diffuaion ooeHicient i . different from J/., aDd repreeentil a n additional temper&ture inO~ or de
"

I•

!

The fonnaliJm devel.

\

.,

/

..

oped here ia appro:z:i.mately

~

,\ "< \

1/

r-, ""!l'

[111. &MnMl

H, D

'*""

....lid even when the . pace_ J'II. lO.I.L on- _&rPI _ ...... ~f~..!! .. O'I . " . . I _.. ~~ energy diltribuUon " not rigarouaIr . parable if the Ipoot.nlm 'tWiee lIlowly with position. In \hU cue. we mUlt inLtoducre .. "100&1" effeetive a bsorption CI'OM leCtion rw.ff

~

VI/41 lr, K)U

k.- (r.B'J - ..... (8)- D(E)j4J{ "f'-;EJ tlE • Th e " lOC&1 II flU1 curvature Moat be obtained either from ealeulationt or from meaaurementA. ThiJI method it ce rt&in1y not l uitable for a elMO a nalytio caloula_ tion of lpa.oe-dependent Bpootr., but it can be UMd to obta.in pt'l:Illininary qualitative information about .. 1p&Ctrwn . Eq. (10.2.20) i. alto very useful in the analyti.

of.peetrwn and te mperature meuurementa in eetJ.mating tho change in the . pee_ trum eeceed by diffueioo effect. (d. Qlapkl r 16). Beea.o8e ol their great aignifioanoe in \he the0J7 of bete rogflneoul f'MC&Ol'I, lD&Dy methodl have been denlopod for caIcalating ~ependent. pect.ra. Some authon ky to &IN.t Ui. problom WrIT ~yti0&ll1. They may, for euruple• •tan from .. Fourier' lranldonJlatJoD. ollhoenergy~ependent diHu aion Eq . (10.2.14) aDd aolve the NlIU1tin,g equtioD in the t.herm&I nnge by e~ in elgegfunctiona of th e therma1iution operator (d . Sea.10.3). Bowont, the inVerM \ranIlonnation I. difficult and 1or.dI to oomplloat.ed npreuioDi. In oontrut to th._ few, mOlit authon employ moltipup methodl, which 06 require the

."it..,

I Th_ It ...ill be -.bo_ 1IW \h.

t.em~

Il)

chane- c.-.d

by dl1fwicm • .-.uy

va~ Jj ( ... &f?m by -7'-1'. 7'; - - --';1f1T; 1+1 7ID 1'" ; toM - ' . __ nJ..- approprilote &0 lM.pecW _

.

JD

Eq. (lo.uoj MI

01. -a.a\ D UMI ......,. I M ~. W

u..

'" of electronic computing machines. In t.heae methods, the thermal range is divided into a large number of energy groupe (up to 70), and the energy-dependent

1Pe

transport equation apliteinto a eyatemof energy-independent trallllport. equations. Various approDm.ationa (lh lh 8" , and 8••appronma.tio1lll) ere used to 80Ive th_ 'YIItem.. Some authors have al80 treated the integral form of the tr&nllport equation numerically. In every ca.se, the computational labor i. enormou. and b larger. the greater the number of groupe. A detailed dieeueeicn of the&e varioul methods, which we cannot undertake here, ca n be found in

HOMscx..

10.3. Some Properties of Thermallzed Neutron Fields Neukon. emittod by a I....t neutron IIUIU'OO 1"f111 ellorgy ill IlUClllltNliVll Dollilliullll with the ..tom. of th e moderating Ilu~t&nce and eventually arrive in the neigh. borboodof the thermal range. In thia range. they ulijmately achieve a n aaymptotic . tate in which their average energy no longor changet from oollillion to ooUillion. We cal! neutrona in thi/J asymptotio /ltate " t herma lized " and /lhaU invll8tigate their propertiea in aome detail in this eeenon. The /lpecVa we dealt with in 800. 10.2 do not repreeent thermali7.ed neueeone; rather, booaulM of the peeeenee of 8tationary aoUI'CC8 of non.thermal neutron., they are aVflrag&A over the epeetre, of neutroJl.l in all stages of slowing down and thermaliu.tion. On the other hand, there are two important ol/Jllllell of e~rimente in which pure thennalized neutron fielde do occur. viz ., th e stationary diffueion experimtlntIJ to be dillCUIJlICld. in Chapter 17 and the pulsed neutron meeauremente to be dilKlllMlld in Chapter 18. In the 8tationary meeeuremense, the neutron distrihution U Itudied at IUch largedlltan~ from the eoureeth at only thermali7.ed neutrora are JlI"l!lI'l'nt l • In an infinite medium in plano geometry, for example. the DUI: i. given by (10.3 .1)

In pulaod neutron OJ:porimenta, we Ihoot a /lhort bunt of fast neutrone into a Iyftem. wait until all the noutrollll al'tl tbormalizod, and cbeerve the decay of the roaulting neutron field. whieh folIo". the law (10.3.2)

In earlier aeetioJUl, we became familiar with certain I'tIlatioDI derived on the baais of one-group thoory between L and 0: and. the diHueioD and absorption propertiea of the medium. Non we shall ItOOy the changes in th_ relatiollll eaaeed by thermali&ation effoota and by the changing properties of the spectrum. tJ>(B) . To thi.a end., we ahall uee in See. 10.3.1 a greatly simplified model in which the 1J*lll dependence of the nUl: il given by elementary diffusion theory and ita energy depeedenoe il hand.led. by moaM of the oonoopt of effective neutron temperature. In Sec. 10.3.2 the oonoopt of effective neutron temperature will be • Beoau-eoflohe Ib.arp~of tbe noutron.protoo crou.eotioD.bon 0.1 He..., . punlly th-.l field .. reaobod in hydrogeno\18 medii. willi higb.-rgy 1OUr'OllI. 'I'b«e the primary DeUWonI from \he IlOW'OO al..a,. determine t.be IJlI"Mdin3 out of the diatribution .a ~ one m..teithor ~ollowvenerc' (e.I.• (Sb- BoI IO~) oremp)oya.-dm1lUD diff_oe IlIethod (of. 800. 11.1.1).

lie".

'*'

Som~

Propertie. of Thermalized Neutron

."

Field~

abandoned, a nd in Sec. 10.3.3 the diHuaion approximation will be given up . Thus the treatment of themalim neutron fieldi win ea&enti&lly be ezroet, ezcept for th~ question, to be diecueeed in Sec. 10.3.4, of Iep&l'at1ng the epece and energy vanahlee in finite media. 10.3.1. Elementary Treatment of Thnmaliud Neutron Fleldt: Dlflu8.lon Cooling an d Dlflullon Be.Ung At fint let ue eonelder .te.tionary and non-stationary fields together. hi the framework of elementary diffusion theory, the energy -, .pace_, and. time.dependent flux i. given by I iJ0 (A·. " .1) . CO -

l:"..(N ) f/> (K, r , I)

~

D (K) r t (1) (E, r,

IH- LI1l .

(10.3.3)

Here L ill again tho thermalization opera tor ~

Lf/>= f I, (E'-+E)f/>{E', r, I) dE' - I. (E)"'(E, r , I) •

•

No eouree term h8ll been inclu ded in Eq . (lO.3.3). In the thennalized field the flux mUlt be aeparable&llfollowlI: "' (E, r. I ) = "'(E) . f/>(r, I) . FoUowingintegration over all enllrgiel, Eq . (10.3.3) becomes 1 0 0 (" , 1)

,- - ,, - = - E.; "'(r . 1)+Dr-"'(r.l)

(10.3.4)

where E.; and D are th e usual ave ragea over the epectrum
= -

z:; E.
}

~ Jf~

+ "'I' ,'If El where

• •

E,IE' _EI"'IE')dE' _E,{EI"'IE) dE

I

(10.3.5)

~

E. =

,gE. (Ej~(8)dE

_

l·~ ~ .--

I E. (E ) 0 (&') dE

(10.3.6)

•

For IJIl-ab80rption, which we now aBaume, E. =E. Furthermore, ~

I E D (E )0(EjdE

ED = -'. ~c----­

(10.3.7)

I D(8)0(8) dE

Clearly

•

iE [i X, (E'-+ E ) !P(E') dE' - X.(E)tP(E)] dE ~

~ ~

f f {E- E')E,{E'_E)<1>{E' )dEdE' .

"

.

"'we

U now muhiplr Eq. (10.3.4) with I eed . ubt...etit fro m Eq. (IO.3.6). we obtain &her eome limP. rMn'fonIement

J--

..

(10.3 .8)

Dr; (Ell - R)- !er - E) x. (E"-E) 4' (E') dr flE .

WhenP'~ /f1J _ O.I.e .• when there b nodiffu6ioo, the left.hand aide of Eq . (10.3.8) noah.. 'The right.hand aide muat ~en aleo vanish, and f1J (E ) muet equal

(~J' ,-_It!'. (d. the footnote on p. 186), Le., th e 8pectrum of the th er. malized Deutrora mut be • true equilibrium . peot ru m . Sin ce ED never equala E M (g) "",

- otberwiM, D(E) would h.... to be proponional too lIe - tbe left-hand lid. dOM oM T.w.b if VSf1JJ" it no\ zero. Depending on the lip of V'll/)!
nlIume element either ~ or gairule_rv through diff u.Non. T hi. 8RClrgy mal t be piDed 01' loA through oolliaioN with the . tom . of th e mod~tor. t.e., IItIE ) mut dep&rI from the equilibrium apeetrum.

Aaide from. the c:liHuaioo approrimation . Eq. (10.3.8) ia euet. Now we make the &UWnpUon t.ha.t the . peotnun e&n be reprNeDted by a Maxw ell distri bution with .. IIhift.ed temperature T .pT•• m ., " (&I.... l~. e-· .....

IU ' ,

Then

E == -:-J:T

(10.3.9)


(~)'

. -.lw _ M(E)+ T; ,Tt

(;

.

- 2) M( E ).

'I'hiI equatiou. ia obt&infd. by erp&Dding lJ)(E) in .. Taylor and truncating Ule Mriea after ibe aeoood term.. Th en 1

--• • J

Ilene.

(10.3.10) around

T _ T.

J(1"-1f)E. (r-1f)~(r)4E'U

- 2';;.Jf

E'(E'-E)E. (E'_B)M(B"') dE' dB

(10.3.11)

1

- T i (T - T.>.N .M. 1&-.11._ (I~J'

fftl&' - ~·.II (rltl.{r_ 1f)~ It'~ ._ (~Jiff(rl+ K" - u r)(

If

x .II(.,• • (r_~~r4._ fJ.JI rlr-6).II(r)tll(r_6}~r~ • . ron-. & - tIM. priDoi. . 01 doMiItd bUaDet .

The Iut .tep

." (with N _the number of atome per em'), Le., the avenge energy absorbed or given up in a ooWalon 11 pl'OportJonaJ \0 the produot. of the diHel'llnoe between the neutroD temperature and the model'totor t41mperature and the me.n .quared energy ION introduoed in Sea. 10.1 [in Eq. (10.1.14)]. Combining Eq•. (10.3.11), (10.3.9), and (10.3.8) and notm& that E ... 3IeT/2, we obtain

D. ~~.

1;: (1+2~t~) "'" ik(T-T,INM.

( 10.3 .12 &)

. or, if we &et 1'"""T,on the left-hand 'ide and 80Ive for 1'-1',• 1'-1'. - T.

• lnn -ll-IJI -.iii"''-NJI. vaG)

1+27l'ill'

(I0.3.t2b)

Non.SlatioMry CaM. Here t1>(r,tl""R(r)·e-·I • where R(t') i. the lowest eigenfunction of VtR+BlR =O lubjoot to the boundMy condition that the flux vaniah on the oH&etive IUrfaoe of the mediwn. It then folio," from Eq. (I0.3.12b) that 1+2 dIAD

1'-1'. "",, - D B' T.

7Jii"1F

(10.3 .13)

N M,

Le., tli/fuNm cool,.,." OOOUR. The physical reaaoD for thia diHuaion cooling ill tbat the more energetio neutroIlol preferent.ially leak through the IJurt.oo of the medium during the deca.y of the pulsed neutron field. This preferential leakage lead. to a cooling of the llpootrwn whiob iA more pronounced the weabr the energetic ooupling of the neutron gu to the moderator. The decay eonatantot ill given by Eq. (10.3.4) all ot"","J;+Dl'IBI. (10.3.14) When the abeorption Obeyll the I/v.law. tho lint. won on the right.hand aide dON not depend on the llpectrum : 1lJ;-VL: = v. E. (v. ). 1n what. follo.-tlW abeorpt.ion term will frequently be denoted by Cle . D ·" depends on the llpectrum. and through Eq. (10.3.13) al80 on Bt ; for DIl(T) = (DIlHT.)

+(T -T,) W 1+S

. II II d Ih 10 that if we let. ( Il)(T.}=D. and T. . ?;'F '

(10.3.15&)

d In D

'illDl' NN.

0 we can combine

Eq8. (10.3.16a) and (10.3.13) and obtain Ih "",D.-OBt.

(10.3.16b)

Thus fin&lly.

ot -Cle+D.Bt-OB' .

(10.3.16)

In oontn.et to the linear relation between otand IJI peedieted by simple one.group theory. now u a oorwoquenooof the diffu.lion cooling effect. theft! i. a parabolic relation l • The downward (IUlT8ltUl'fl of the ot·VII.• IJI (Iurve h.. been obeeeved experimentally. a ia ctJIed the "diffueion eooling oonat-ant." It !a given by 0 _ I;&.. I

(1+2~r·

N.t.un1ly. hiflher tenu in lJ' &leo appear. bv.t for

(10.3.17a)

-n J1I they GIll be ueslocted.

·..

or with

D' _D,

,.,..."

c_

•Ubi ";. )' .NJI.

(10.3.17 b)

The quanUty c_O/D,ie alllO used frequently; it i . given by

Il (I+2,fln 'mIl)' C- nry; -r- . In graphit-e and beryllium D (E) it nearly

(10.3.170)

conlltant 80

lli.mply ~~ -

that

~ _ 0. Th en 0 Ie

nr:JJI and c il ~ . Some ca lculated vaJU(III of 0 are given "...... I I in Table 10.3.1; obvioualy they depend IMlnaitively on the tb ennalization model eeed .

Table 10.11.

~

F_ _ for C ...~."f ...... JldAotl of liJltd.... N","",- T ....~ (., IO "C)

Ji,O . . . . .

:a. (1.8I 1Jor)

•

I

o.I U

0

0.1

eeeo

'

&alioNuy CGH. Hen!

xlO"

6. 1)( 10'

0.81

IO.l xlO"

• x l OO

~ (r. ' ) ~R (r)

w giftD by

..'"

UOO

Nelkin Model Eq . (IO.U6): .4 _ 1 Eq . (10.1.16): A _ 18

Eq. (10.1..330) Eq. (10.1.16): ..4 _ ' ..... HodoI Eq. (10.1.15): .4_12

with V'R -RIV =O. 'The kmperaturo lIIiDl)

li 1+ 2 "lTiIl" - ,.; - --U NJI. 1'-1',

,In Il

(10.3.18)

1+2 "i li-"

- r;; .- -Ii JI, .. " In other warda, d"JuWm Mati"l1' OOOllnj It dON 10 beceuee the average energy 01 the neutron• • trMming into eaeh volume element is IAtg(If th an that of the neutron. lMIing abeorbfJd there. The diffuaion length i. given by

Il. V_or:Il ... ; r.l;J-·

(10.3.19 a ,

In analogy with Eq . (l O.3.U b) (10.3.19 bl

and therefore

v _ ~ + .t
(l0.3.19c)

C/.X.(. )V....C/D.-c

V -.c: +c.

(IO.3.l lId ) (l0.3.1ge)

'" 10.3.t. Treatment.f tbe DlffaI.IOD

Coolln& EJfeet'l the MethfMt

of Lagutlml PoIfnomlalJ The ne utron Soempen.ture model i. oen.inl,. . u.itable for achiering q ualitatift underat&llding of diHuaion effeet.l in therm&liud fiekla . but.W1fommat.eJy in Tolve. th e auumption th at .p (E ) can 1.1_,. be deeeribed by .. Ma:rwell di. tribution . We .hall now free GUBelve. of thia UlIUDlptlon . In illutrating how we do thi. , we ahall only coneider the ONe of diffueion oooling in .. pulsed Deutron field : t ho treatment of diffu eion heating i.II enti rely analogous. Let us aga in start with Eq. (10.3 .3) .nd again .tOt tP(E. 1". 'I _ tP (E) .R (,.) . ~-·, ; in view of th e fad. th at ViRIR _ _ /JI. (10.3 .20)

In t.be int.erMt of brevity we ha"lIl &Uumed E.._ 0; .moo .. l /_ _rption daM not. inlIue noe the .peeVtllD in .. pule«l field bu t ~y inere-N the deea,. conAant by t.bo amount «t -~. W. &NUID.ptiOD oeca.eion.t no 108lI of generality. Eq. (10 .3.20) de£UlN &It eigen value problem ; th ere ed .tI a n onlite . poctrum 0' eigen valuM lI" .nd eigenfunctioM tP,(E ). Here we are only inte rested in ]the Iowellt eigenvalue.nd ita MllOci..ted ene rgy epectrum ; in Sec. lOA, we thall return to tb e question of tho higher eigenllOlutions, which determine th e approach to the tb erm alit.ed ltate. In order to eolft Eq. (10 .3.20). we attempt to trand'onn t be intqra.l equation into a lyatem of lin.,.,r equ at ionJI. Thie can be done. e .g.• by the multigroup method. i.e.• by dividing the oontinUOUI energy int.erral into a n um ber a of di&c:rete energy group'. More elegant and. lOueh more tr6et&bJe, bowner, i . the denlopment of l/1(B) in a lene. of orthogonal functiOnJl . Let Ullet (10 .3.21)

where th e A. are ooratantll ami t he L~l) a re t he ltIIOCiated Lagueml polynom ial. of the finlt kimJl . Th o latter are given by

" " I,:;t:_ ) ~.+ ~ (i+I)1 (- . 1zIP

~

and in particular

.-.

I

4 "1<) - I 411 (~) _ 2_ ~

(IO.3.22a)

(10.3.22 b)

r.r1 ( z) -3 - 3 z + ~f2 .

Furtherm~re. th ey aat.il.fy th e following ortbogonality relation : (10.3.220)

L1

1 ) are partieulr.rly well suited to BecaUlMl of W . orthogonalit y re lati on. the t.reating thM10alized neutron fielda ; a nd . .. a matter of f&Ot. the equatioR.l take

_.

.

~

I

- - "-Kia"- T r -ndenC&l. F1!not.iou ", Vol. I, p.l7t , Mear...·Rill Book Co.,

Ct.• .....

New Yorlr..leM .

.IS a .err &imp. form. i.J:l the cue of .. heny gu moden.tor. U". lIUb-titu~ Eq. (10.3..21) into Eq. (10.3.20), multiply br l111 (EJ1T. I••nd integrate OYer tht' enerv. ". obtain (10 .3.23)

Tb_ equat.iona nlJW-Dt .. homogenooua, liDoa.rlyatom of equatioDl for the A• . In order that there be .. aOD·triTial 8OlutioD, th e determinant of the coefficientll mu" ,.&niah : l«lh-lJ'D•• + ~ .l -O . (10.3.24)

Eq. (l0.3.U) ~ an algebraio equ..t.ioo that det.enninM the eigenn]UN. Before ... 801", it, ""' mun ev&l~ the mUri% elomenta D4• • an d ~ •. To besin with,

v.•.

• J , I4l1 (-;'.') ~M(B)L4ll {-~;)n - ~ ,f•L1 y;,-.L~ll(,z)h

'1. -

(10.3.26.)

U(: )

with

"1'- ~ .

1\ then follo_ \hat

r..- ~

Yf; r'I-v..- ~v..;

,

fal- 'Ye..

(10.3.25b)

H.i.nu and Dull..... have 1h0lrD tout in general for i :iii I:

1 ~ Tli-l+l)1'(l--,+tW(I+'1

V.

•

(IO.3.26c)

J•L1"(-.{.;) D(E)M(E)L\" (-,},) 4E .

(lO.3.26d)

It "'" -,. .,. l~

Seoondly,

D.._

(i

1)111:

l)lll

•

In the important. apecial ONe that. D (.&')it oonat&at, which we ahaU be oonaidering &om DOW' 00 . we fi.od uiDg Eq. (10.3 .22(1) that D,, -(i +l)Dc\._

(10 .3.250)

•

1,,-

J1, (.},){LM(")L, (~;)}4E .

•

In the ~ cue of .. *YJ gu moderator, th e ' ono . 10 lhi8 _ , L iI gifllQ by

~. take ..

(10.3.26')

panicu1ar:lylimple

, IfE-cT.ll1l'(E' H8iT. '·1

L~ - lZlE.

tl~

{d. Eq. (10.1.4,)). No...

:~ !(lr- l) +lr l~l.t'.-· Lin (.t') _ :- t .ijU(*)z. - -.

(10.3.26g )

"'

Le., the fuDet.ioM Lj,ll(ZliT.).lIlH) are &ge nfuDetioDi of the hea-.y gu ~enuliu­ tiOD operator oo~g too the eigeo't'.lue - 1: E. _ Ulling Eq . (10.3.220), we therefore obtain

(lO.3.U b)

I",=-l(J:+l IEE."'l

for the hea"Y gu moderator. h cao eMily be , bOWD th~t even for ' D arbitrary acatt.ering law all the Lu and Lu "aniah ; furthermore

.-- -,Nil,

1.,

(lO.3.26i)

"hen N ~ th e Dumber of _ Uering aklma per em' and 11, i. th e mean aquared e!l'"ll:11ou (01. &eo. 10.11. General .~. 101' th e ~t hue been ginn by PuROHIT and by TAK,RUBI ; the higher moment. of the _ ttering kernel.ppear in th_ e:zpreeaiOD'. In order to IOlve Eq. (10.3.24), we ,han break off tbe f1J:p6n&ion of I1>(E )/M (E) after the l-IA term . Eq . (lO.3.24) ia then of loth order in L For I _I and 2. we

find. reepect;nly, t hat

IIY,,-IJID..- O Ol e:

..

B' -D.. v.-

.

%

D - = tlrll' =D,lJI

I

1_1

(10.3.261)

V

(I0.3.26 b) 1

0'

with

(10.3.260)

Thi . re-ult for C ill idontioat with that of Eq . (10.3.17_) (for connant DI. w)uch "' .. derind from the neutron temperature model. We can Mtily undentand thiJI re.Wt with the help of Eq . (10.3.10); ... IODg •• we oonaider only term. up to order lJI, the 1=2 appronmatiOD i. identi oal with the repnJMntatJon of the perturbed .pectrwn by a Maxwe1l diltribution with alhifted temperature. W. C&Q DOW in"YMtigat.e what erron _ make with thMe appronmatioal by taking more tel'ml in the e xpaJ1lioD of t/)(K)1M(E). For a hea,.,. p i,

,

Al)I -- '~

..,(1_1)

,,I

-

Vi

ler.

H... wm. at order hiper ~ /JO M", beea IIlllIIecce1.

(lO.3.27a'

Thermaliution of Nout.rona

since M. = 8aJA. . Aoeording to lli.nLJ: and DRESNJ:R, for arbitrary I AD":/: ' ,

~l• "'I"! --

2' '~ 1 ! (h-l)lI j' - :/: E. -- - · £.. .(.+11' (b) 11.

y"

, .-,

(to.3.27 b)'

AD' ----,."2'

Table 10.3.2 tho'iVS {jJ)/--2~ as a function of I. We see that (jJ1 converges slo'iVly ; its asymptotio value at"") it very clue to

e

AlJI - =V:r

C<"") = -f2~~ ,

(I0.3 .27c)

and is ecme 33 % higher than th e value given by the 1=2 approximation. Eq. (10.3.270) ean 0.1110 be obtainNl without tho UIlO 01 tho lAguerre polynomiAl approximation by dire<:t integTation 01 Eq. (10 .3.20) (cl . HURWITZ and NELKtN as well as BIWKUBTS).

dO

Tablo 10.3.2. -(-~ ; y:~ '~;) aamdi"1 III Eq. (10.3.27b)

,-.

,r, s

1

l

0.1566

0.12ro

~

l

0.1603

J~1t

0.1667

1

ll

1

a

7

l

8

I 0.16.'19

1

l'

0.1660

I

U

I0.1861

u 0.1665

0.1623 1 0.1635 1 0.16'3 IS

1(1

I II

0.1862

T.&X.&B.am has lItudied the higher-order approI.imatione to 0 for eryetalline moderaton. He ueed th e heavy crystal approximation and aesumed D to be conetant.. He ehcwed that one must go to ' -20 to get good convergence of the value of C. T4J:4B+1lHJ lound that CO:'~lOl _2.f8 oo:'~ tl lor grapAi4e Imd 00:'- 101_ 3.36 (J'Il-tl lor btryUivm. With thellO adjuetmente, we can use the values of (p-Il in Table 10.3.1. These results a110 ehow that in oryatalline moderaton with strong binding the neutron temperature model il a very poor approximation. Finally, we wish to note briefly another prooedure for calculating the diffulion oooling oonatant. The 8OIution of Eq. (10.3.20) can be written generally in the fo= IP(E) =M(EI +B'IP.{E) +.Bt4>, {E) + ... \ (10.3 .28) « ::aD.B'-O.Bt +F g+ ... . U we inlltrt these expreuione in Eq . (10.3.20) and note that the coefficients of the varioul powen 01 B muat separately vanilh, we find that

( ~. -D(E))M(E)+Lfl).(El =O , -

~. M (E)+ .!l.IP,(E) -D (EllP! (E)+Lfl), (E)=O

I zll _z(z _2)(z_4) .. . (Il or (I).

(10.3.29&)

(IO.3.29h)

'" ete . U we integTate bot h of thOM equationa O1'or aU energy. it folio... that

J• .p(81 Jl(&)~K D _ !....-- -- • • • ( ! M IE}U ,• •

(10.3.290)

• J ( ~' -D(E)) ~.(E)" C-"'-~----

elnce

•

,JLf/) dE .

(IO.3.29 d)

O. In thil cue, we do not eeed to M)lve an eigen,..lue pro~m ;

m.tead WII muat dotermine ~I ( g) from Eq. (10.3...29&) and then the diffuaion cooling conR Dt from Eq . (10.3.29d). For .. heuy guo we can eMily integrate

Eq . (10.3.29.); when we do 10 we again obtain Eq . (10.3.210) for C. For other scattering lawl, we ca n IlOIv8 Eq . (10.3.29 . ) by the multigroup method or by expulsion in Laguerre polynomi ..t.. The80ll methods are 1.180 euitable for deter. mining th e higher terms (F etc.) in th e lI: (B').ltllation . 10.3.3. Traneport Tb OOl'J 01 th e nerma!bed Neulron FIeld

Now we mm innetigate th e erron th at ariae from keaUngthermaliud neu Uon fieldli with e1emMltlq diffuaion theory. WII l tan fro m tbe t.nDaport equation in a ecurce-free, ieotl'opieaUy _ tterin8: medium in plane geometry :

I

1 ~T( E. ,.. lI'. ')

-;

il l

1

(10.3.30.)

.. +1

- - E,(B ).I - ,u

:~ + ~!

E. (E'_Bl F(B' ,jl' , 7:, 1) dB' tip' .

('!'be assumption of isotropio eeatt.enng will be dropped later.) In the . tationary ct.D write for the therm&1ized neuUon field

cue, in an infinite medium, we F (B.p. ~.') - F(E.I') c - ·· . Then

.H

·-.

JJ

(1:,(8 )- Itp )'(8 ,p ) - -}

E. (1l' _ 8 )J' (8'.p' )dE' dp' .

(IO.3.30 b)

In tbe norf..tauonary cue, I'(E,I"~, ' ) = 1'(8,1" %)c--1 a nd

(r,(EJ-

: } F (B ,p , .:I') _ _

·H E.(E'_E )F(E ',p', %)dE' dp' . (10.3.300)

-.J

P ·~~·+ -~-J

,

Theee two equaUona are eigenvalue pro blema for the rela:u.tion oonatant It """I lL and the decay oonstaDt Cl., rwpect.ively . In Eq. (10 .3.30 0) boundary oonditiona moat be apeoi6ed 011 the aurlaoe of the medium. Before we ~ with the IOluUon of t.h_ equationa, let UI oonslder &Do elementuy argument. oonoerning

'"

al_,. ;;:0.

the exiat.ence of an upper limit for the eigenvalUN. 7 {H, p) and F(E, p. z) atl Furthermore. the in'lOIottering inUilgral jI.(E'_E)FdE' dp' j ~ al....,. ;:0. It therefore folloWll from Eq. (10.3.30b) that Z',(E)-"fI ;;:O 0 1 since p S.I that (IO.3.30d )

Thu the rela:u.tion eonatant can never be luger than the minimum. value of the tot&! emu lIllOtion over the enUre energy range. It followB similarly from Eq. (10.3.300) that (IO.3.30 e)

provided that we tint &88ume 1p&C& independence (8F/a%=O) . Naum hu .hOWD that Eq . (l0.3.30e) &lao bold. when 81"18%+0. The intuitive grounds for th_ limitation. are a8 folloWi . The Pf"'l'noe of an uymptotic .~ in either apace or time me-De that all groUpil of neutron., no matter what their enel;:hM,deoayequally rapidly in either 1p600 or tuu.e, reo 23 epectively. The intensity 18 of neutrons with energy 0e.e.' E cannot decay futer than t-r.~" in tbe .tationaryca.ee and ,-'L'o(6)' in the non..tationary C6&&, fOf IIE,(E) i, the mean free path between oollillior1ll and 1/,,~(B) i, the mer.n collieion time. Thill the neutrons for which .z; or ".1; hu a minimum limit the tAte of decay. (" The grade ecbccl c1a.M cannm proceed faator than the Ilowest pupil ." - ColUlQOLD.l In HID end D.O, the minimum. value of I,(E) occun.t E' =oo••nd in crystalline moder.ton immediately below the Bragg cut.off energy. The toinitnum of ,,1:,(8) IlIways ocoun at &=0. Table 10.3.3 .ho1l'll .,...luCl of

1

Tr.>-....

I

and ("X.)m1.. for eome

moderators. If the abeorption ie inereued1 (in the stationary ease) or the geometnc dimensiona of the medium. deorea.aed(in the pulsed c.ee),. or «will increue,reapecti'9'llly. They appro.ch the limita lpClcified by Eqs. (IO.a.30d and e). There arilMl8 the quMtion of.bet.her theM limiting valuea are achieved uy:mptot.ically, Le., in the limit of infinite E. or lJI.orfor finite E.«E.l.) or lJI «lJIl.) , If tbe latter is the cue, then for E.> (E.l. or lJI> (lJIl. there will be no solution to Eq., (lO.3.30b and 0). In other wordl, under thMe ut.rtIme clroumltan~ there e:rlet.e no uymptotio .peotrwn of thermllill&ed neutronl. There are indicationa from experimenta and oaIeulationa t.bat indeed the latter i. the e - but 11'& are still far from. .. complete underetanding of the beh.. ..iour of the neutron field under .uoh extreme circumltanc.. A. .. rule, for _II: abeorptJon or in moderato,.. that are not too .man we are well UDderthe mtioallimiw, and we eh&ll now try to treat Eql. (IO.3.30b and. 0) further in lheee pnotioaJ ea-. I

e.... by

~

the ~...n.h boroo (af, &0.17.1).

..

'"

If we introduce ~ (R) - f F (E.,u) dp into Eq. (I0.3.30 b), ..& oMain .. ,"lilt.

-,

&imilU' to that of See. 5.2.2, viz .,

, ("" <)+') !•E. (E'_ E)(l> (E')dE' . ( I0.3.31a) • A reduction of Eq. (10 .3.300) for th e non ...tatiorwy to .. l imple intotgral ~ (E) = bln zt (K)-.

caM

eqll&t.ion iI not poeaible beceuee of the apaoll dependence . We een, however , ca rry out a Fo urier tranaform . let us introduoe F (E, p ,B ), th e Fourier trand'orm of F (E, p. r ), defined by

.-

--

F (B, p . B) It. ..ti.diee the equ.l.ion

(Z;'E)- -;-)F,E.p,B)

f F (E, p . z )e- Ul·d'x.

(10.3.31b)

I

..,-,

+f JJI . (E'- E) J' (E', p ', B) dB' d,u· • - +1

- i B,uF( E. ,u. Bl

(10.3 .3 10)

If we now introduce ~ ( E, B) = f F IE. ,u,B) dp. ~d proceed exactly ... we did in See. 6.2.2, we find that - 1 flt CE, B )=-

~

B

&I'$D \

.1 J-E,(E'_EI~(E'.

1:1(6)- .

B) dE' .

(I0.3.31d)

•

We can eolve t he integral Eqa. (10.3.31&"00 d ) numerically and thereby determine the e1genvalu8lII If a nd (l and the .pectra OOrTeepo nding to ebem u fun eti ona of t he abeorption and Bt. n.pect.ively. Such D1lmBrical C6lculatione have beee done by HolQCZ.. Aooording to Nn&Ilf, however. &II. analytic aolution in \he form of I. powel'lleriNia aho poMibl.o. an d we IlhaJ1 now find Ine h a lOlution for Eq. (IO.3.3l d ) for the non..t&tion&ry cue. We mall negIe« the a beorpt.ion (l/_ t.orption onl y inOl"O&SN tho decay ool1ltant by an amount ~ =- E.(tI.)tI. and dcee not affect the . peotrum) and write Eq. (10.3.3 Id) in the form

\

B1.man(

.

B ,) - E.tE)I ~'E. B)- L~ &"C .,--.

(' O,3.3")

where L ia i.8&in the UU&I thorm.tJiAtJoo. operat.or. Now let u.n

U"

~ (E. B) _Jl(E) +B'(l).(E) +~41,(E)+ ...

( IO.3.32b)

_ _ D. ll'_C ~ + J'.8I + ... .

( I0.3.32e)

wbn.itute tbe.e oquationa in Eq.(IO.'.32a), ezpand

and equate the coefficient. of li.ke

(J ~(~

po1VOrI

of B, we get

B1arct.an( &,,(8)B-

, ). -

..

- ~)J(E) _ L lfI. (E), (IO.3.32d ) ( . ~,<) - ';') ~.(E) +(~ + •.;,<) {!t'- - ..~,<) })M(E) - L~.(E). (,....S,,)

Thermaliaation of Neutmluo

Eq. (10.3.32d) il identical with th e retlult (10.3.29a) of elementary diffusion theory: in P-orticulN'

J D, _ 3E: 11') I

JI(1') 48

_

(I0.3.33a)

f + JI( 1')4&

•

COrTMponWng to Eq. (I0 .3.29c). I ntegrat ion of Eq. (10.3.32e) over aU energiell I.ado to

f-

•

(IO.3 .33 h)

.~- Jl(i')4E

.

c.

c.

Here the fint term 18 identical with tb e rwult of Eq. (l0.3 .29d ). The eeoond term obvioualy bu nothing to do with diffuaion oooling lince it. doN not contain 4)• • Instead, it. ftIpneent. a purely ttanaport t heore tio ecreeeuce to the ftIIUlte of elementary diffusion theory. The ftINl:JfU fot t be a p~ of IUch rorTeCliotul were already diacuued. in See. 9.2 ; theahove rwult, bow ever , ia different from - and more ~ rate th an - Eq . (l:t.2.1li d). For energy. inde pe ndent. crooooo

_,ion.,

0. ia aimply - DJ I6 X: . In a hea ...y g.., on the ot.her hand , 0D= wilb th e rwult th a t. 101' ..4: .>1, CD> c. .

Analoioua corWd_~

are aOO pouible in the

CUll

~i~

•

of a niaotropio lCat.t.ering

if we I18t.E. (8"_E, Q '_Sl) _,I. E••(E'_Bl + .'. E.1(E'-E) (lOll 8, + .... We &haU not. write down hen! the very complicated lormu1a.l that ft\8ult but ehall inetead gi ..e 80me numerical ft\8ulte of HONEOJ[, who, in addition to the direct numerical eigen...alue calculationa mentioned ahove, alao carried out calcula tionll of the fint few ooefficiant. in tbe /l (B')..npanaion for IIOveral lubectaDOOIl (et. Tabie 10.3.4). In tb_ calculaliona, t.he aniaotropy of acat.te ring Wall taken into aeoount byinclu.ion of Nterm. in the expaoaion of E, (E'_E. (lOll 8,) in Legendre polynomiall P'(cc.8,). on.- rMU1t.a will be compared with nperimental reeulte later. NlM the amall OOIItribuUou of 0. to the diffuaion cooling oortIta nt. C. Reeult. analogow to ~ found here al80 hold in the . t&tionary cue.

Do t.... __ ·1

h_

Hs°{N -3} DIO {N_J)

.

c_c. +c. 1-' - ")

3.70&8 x 10" J.878 x10" ! .Oftf x 10" ' .lllll x 10"

On.pbite (1.8 I /oml ) (N _ I) J.l18 x 10" 2.467x HI'

""'. - 0.067 - 0.10&

' 1_

__"1

., x , O'

-

Nelkin Kimel. 3.73 xIO" Modified Nel.

... ......

-0.008 - 8.3 x l O' ParbKemoI

10.3.4. BeDlal'U

OD

the aume Pro.lem for Thermal Neutron

In the treatment of pulsed . therma1ized. neutron fielda. we have hitherto tacitly made the aaaumption t hat a aepan.tion of the.paoe and energy dependenOM waa pcselble, Le., t hat ,p(E , r ) could be aet equal to lP(E) .R( r ). where R (r) i, th e lowest eigenfunction of VI R IJI B = O. [ThiIaaaumption is equinlent to the Fourier trulaformation of Eq, (10. 3 .31 b).] Now we shall inveRigate th e Y&1id.it1 of thi.. _llmptinn. Let. WI finJt. OOIlllider th e general Milne problem for t.hermal neutrona. Let. the half..paoe z > 0 be filled with a non...beclI'bing material 01 t.emperattl.re Ta• Let. th ere be a aouroe at Z = 00 th at emita neutrona with an equ.ilibrium energy dis tribution. Let th e hatf-epaoe z< 0 be empty (vacuum). The aaymp toti o flux distribution at large wMtanooli from the eurfeee t hen haa t ho lorm

+

,pIE , z)_ (z +d)J:l(E )

(10.3 .31-)

--

with aD energy.indopeDdent e:rtn.poIated end. point d. For monoenergetic neutr0D8 d -O.7Ilc. (Sec . 6 .2.'). In media with onl1 weakl1 energy· dependent ecatterinj: CI'OlIlI ~. the behavior of t he therma lli:ed. neutron fiold will not dillor much from that 01 a monoonergetie liold, and d lltj O.' I I... On tbe oth or hand , in the ca&O of a 1'1.1. 10-1.1. Tho II 1 em from t he eurface ; d ie 9 % greater than t he e:rtn.polated endpoint obta.ined by averaging I /Ir. (E ) over a Maxwell dmribution. For dlnanooe < I em from the aurface , th e . pectnun deriatee from the Maxwell diatribation. The deviation hecom... etronger .. we approach the aurface ; at th e .wfaoe. the . pectrum ill IIOme 20% "hotter " than in t he interior (d . Fig. 10.3.1). UnfOl1unately. there are no ei.milar ealoulat.ioo. for otber moderaton (exoept gaeeoua moderators). In beryl lium and graphite 1 th_ edge effects 1ri1I pI'Obably be leu marked than in water ednce the eoattering craM eectJona depend only elightly on energy. We can conolude from th_ resultl th at in the interior of a pulaed block of moderator whoee linear dimecaioDl are larg e oompared to the a. orase tn.naport mea.n free path then ill a epaoe.ind ependent . pectrum of tbermaJized. netl.t.rona.

_t__.. .

-a-

I K G_ ",". M low te mperat.uI'M. "o;Md " _ lroDa, le•• _trllIw.nb belGw ta. Dna IIaktff - e...aicb ba... 'fWJ Joac _ " . pHU. Jeed. to VfIrJ larp «lp~­

.......

""""'IN.

1I"..uuol

1'10"

16

... nu. ocmc:iI.GIioa » nrified by ~ of

Q:D.UAD aod D,nu, wbo numeri. e-Dy caIoalaied 4Jo(.I'. :c) a.nd • for u.b. of wat.er by • multigroup method in 1ht p'...pprorimMioo. without 1oD.1 additionaJ uwmptiool .bout eeparability. OIlLlWl.D aDd Unu Cl&loal6t.ed the e:rtrapolakd endpoint d u . function of th e Itlab ~cm- a by _ gnlng to oa.ah Ilab • JlI IUch t.h.t the dinlctJ.y calculated

~ of« equU ..+D.B'-OB'-+ ..· . J'ig. 10.3.2 M01B • CUI'nI of 4 (B') denTed from. t.heee calcu1amm.. For IP= O. 4... 0.76~. in good AglMmenl. wit.b the ~ of KiD'Jlnn. The cum of 1II1B') for apherM W 'terJ aimil&r to t.be carnl Ihown iz:r. Fig. 10.3.2. Fig . 10.3.3

BmoeJll- (1I:2lr. d"" .}(:-a).

Ihcnn -

o/.~~~~;>;;

'"

I •'"I'"

~.

......

functioa of :c for a Ilab 6.1 om. W eir. from these

_.0'. ., • Q',"->. - -

- Jt.....

1m

K

~

JW!tDIf~

... ...

U

.,., IU.I.. 'nil _

U

r-

~

U

..~

/

-

I"

'",

-.0 -

....

~

u ~u

......................

••. . - .. -....--

O&.Icu1ationa. U tP(B, z) were rigorooaly Mp&rIble and P-R+ BlR -O, thi, QQ&Qti1.11hou1d be • oonatant. In \be in\orior region of the alab. t.hiI ia ~uany the cue. Irr. ooalnA, DeU' the rurlaoe the IocllJ. l1u baekling dependa I trongly OD pomioa.. In t.IWI region, the .~ aDd energy depondenoee are DO longer IOp&nloble.

A detailed Itudyof thMe problem. can be fonndin

WO I I AMA

1M. The Approach 10 Equlllbrlum In Pu1Jed Neulrou F1elda In tha. MlOtioD. . . Iba1l

dod,

how aD inoomplet.ely tberma1ized DeUtroD. &Jd ..~ the equiUbriam diatriblrlioD. We ahalJ. limit ouneh. to ~ the approach to eqWli.briwn te tJ,me. Tb_ conaIderatiora are im. portaDt f~ iDterpntLng lKNDe of the eaperiJnenta too be diacuued in Chapter 18. In .wlitioa. they pn good iDIigb' Into the meobaniam of DetltroD t.herm.liz.a. Doa. ~ of &he epWa.I approach to equilibrium. ma1 be found in KOliWtft,. Sm.uOVT, Ktl1IOlI&o aM ot.ben. ~ of tbe appro&Oh to equilibrium. willlMd UI to 1M higber eipo...... and eigenfunoUOBI of Eq. (10.3.300) ; the Mympkltio ataM t.rMMd in Sec. 10.3 correepoDd8 to t.he IOWMt eigennlue. However, before _ embark on thiI formal ueatDlen'; t.he ....nee of t.he approaeh to Itqllillbri um. will be Itodied b1 m _ of the limple temperUal'e 00D0IIlp'.

Matbem. tbU, . the

CICIQIid,w

'" 10.4.1. ElementarJ Treatment 01 the Approaeh to E,aWbriam We limit oonelvN in the following to ~ in infinite, non-abeorbing media.. .A. in See. 10.3.1, . . _ t up .. neutron brJance : (10.4.1&)

Eq. (10..... . . ) deecribee the th ermalizatioD of .. puI.e of neut.rone of energy Eo Ihot into a.n infinite medium. at time 1=0. It follows by integration over aU ~

eoergiea that

,,~f ~ ,,(g" l U= O

•

for 1>0, i .e., that the tot&J deMit,." 01

DeutI'ou ~ DODAuIt. in time. With th. _ DOl1Il&1iutiOQ ehoeen abo• • , it; i ' equal *0 ODe. H we molt.iply Eq. (10 .4,.1 . ) by g &Dd integrate all E, it \hen folla". that for 0,

0"'

'>

~~

-"g. - f f fg'-E)<1'(E'. ,)X,IE'_E)dE' dE .

(to..... tb)

••

Here we have tn.nd'ormed the integnJ f BL~U euotJy .. in See. 10.3.1. Eq. (l O"&.l b ) ..,. that the eDelV ginn ap per em' and Me 'ria oolliaiOD.l equ&l8 the decrMM pot _m \be mean llnefJJ' per om' , Eq. (t o.4.t b) i' exact. When 4)(B, ,)-M(B), the right-hand we ...anW»e. ; in

uu. ONe . . are dealiDg 1rith the

.ymptoUo ltate', for .hich B - 311'.12. Nnt, we ag6in make the ...umptiOD \hat in the neighborhood of thiI equilib. rium. ltate the lpectru.m can be reprMeDted by .. Maxwell dUtribution with .. Ihifted temporature. whiob in thia cue muH depend on the time. Beoauo oft.be DOnu1juUon to oorwt&nt IMl1t.ton deraity. 1nl -* ~ Ig

W'

) -.e -tITti)te Z . - ./U''lI) .

(10.4..210)

.1

U we aublltitute tWa e:q>l'flllioD into Eq. (10.·U b) aDd rewrite the right.hand aide with the help of Eq•. (10.3.10) and (10.3.11). we obtain the foDowing ,..ult when 7' (')-7'.<7'. : ilT(fJ - -zl -

_NJI,

J

(T il)- ~) •• •

(10."'.2b)

(10.4.2c)

n folJowl \bat

7'(') _ 7'. _ e- llfr•

(IO.U d)

The neutron temperature thue approach. equilibrium e:rpououtially with the time OOIlItaut 1/". . We Ihall oall the quantity". defined by Eq. (10.4.20) the

".

" te mpera ture relaution time " j the quantity ,,/- I {'r i. frequently ueed in tbe literature. Tabl e 10.4.1 oonta m. lOme valuel of 'r , By oompa ring Eq•. (10 .• .2c) and (10.3.l 1 b), we obtain t be important relation

--

•

H,o . . . . •

U Z%

NeIk.ia )(DoW Eq. (10.1.16): A _ I Eq. (lOoU~I : A _ IS Eq. (10.1.330) Eq. (1001.l5): A _ 9

'40 Il2

Eq. {IO.I. IIi} : A _ 12

3 .!

M il

Be

(1..651J~

•

Orapb.it.t (1.8 ';-.)

C -{ (~~~).",

(10.• .3a)

between the diffulion cooling con.tant and th e te mperature relasation ti me. For an energy. inde pendent D, we haTfl llim.ply

...... """"

0 _ _~lr_ .

(10.•. 3 b)

Next, we nail calculate th e te mperature rclant ion In a finite medium. In thi. _ , we find by a .tralghtfonvard generalization ofl E qa. (IO.3.12a) and (10 .•.2b) with PJtPll1J = -lJS that

_ dT(11 =- .!... (T(t)- T.)+ nll lJS.!.~(1+ 2~1~~). if

Thia equation IeadI to

.In~ -

3

'1'

,..»

1+t7Jii'Y )

P(')_P. ( I -DB' ---y~ _t

_P_ .~~ II+ .... !I]) I nat'" . Ir

I

(10.. .. .)

(IO.' .fb)

The tempen.tlU'e nlaObed lit the end of the th ermalization prOOeII illower than bec&ue of the diffuUon cooling effect. [d. Eq. (10.3.13)]. Owing to the pnof. erential Ie&bge of t he m Ote eoerrtio neutron' . tbermalizat ion proceedI futer tban in an infinite medium. •

r.

10.4.2. Tr-eatm.ot 01 Time.Dependent Thermalls. UOD u an Eigeonlue Problem

Afte r iDjection of a neutroD pulte iDto aD infini te, Don.at.orbing medium , tb e followin&; equation holda [Eq. (IO.• . la) for ' > OJ:

.!..•

8"'(B, I) = L ill

... .

At. geDel'a1 8O!utioa of thiI equ&lioa IhouJd be poeai.bJe iD the form.

l1J(E. f) -l: B.tP.(8).--'

•

......... where

Cl.,

(IO.• .6a)

Nd <1>. (8) are reepeotiftly the eigennlUM Nd eigenfunotiou of the ( IO.• .l5 b)

In order that. the aolution can be ..mt teD in the form of Eq. (IO.• .6al, th e tP.(E ) mUit fonn & oomplete , orthogonal . y. tem of functioDl. We can euily prove the

orthogonalit y of th e eigenlunct.ionB of Eq. ( IO.• .5 h) l. Bo.enr, Ule queetiOll. of th e oom pletenMe i' ...ery compleJ:. '!be di8Cl'ete eigenval ue. obey the oooditioo ..:it min {III, (vl) (d . Sec. 10.3.3). and we c.n Ih ow that. iD the nnge min (.I , ell)) < <<<m u (oI , CII'I) there emta a continu um of eigeo'f'aI UN a nd eia;enfunetiool (d . CoIlNOOLD, lIJCJUKL and WOUllU.lOf .. W'011 .. K OPl'.L). hmead of Eq . (10. 4 .5 &), therefoR , we mUllt wri~

rz, (E, tl "" 1: B. 41.. (E)~ -'" ..

1IlU (•

.I:'.l

+ J B (a:) F., (E)e - ·' dfl, _ (.E.)

(10.4.3Cl)

and only if we inc1udtl both the di screUland oonti nuoua eigenfunction. ia the ' yltem complete. H owev er, we are not interested here in .. complete 1IOlutoion /,fl(E , ' I valid for all ti me. after the pctee but rather with only to know ho. the apectrum a pproaehelJ the equili brium .ute long .tt.er the pulee. F or reuon, we on ly need to kno w the lo_t two eigcn llOlut iona of Eq. (IO.4.5 b). The equilibriu m , tate oorTellponde to th e eige n n hle « =0 and t he eigenfunoLion M (E) ; for long t.imN we then have

uu.

'1>(E , l)=B.JI(E I+B,.tP1(E )t - · · I + _n , 1 t.enne decayi ng eve n m ore r.pidIy wi t b time. + auu.itiona

I

(10.4.6d)

Now we shall try to calculate llJ. ..nd 4'1 (E ). 1/llJ. il cl6Mly a characteristic meaaure of th e dUl'l.tion of th e thermalization proeeee, we ithaUeall trA ... 1/llJ. the" t herm..Ib...• tion tUne" . By use of the form (IO.4.5 d) we I.nl tacitly m&king the 68IIumption that thoro is in fact ..t IOMt one othor diecrete eigenvalue beUdOlll _O below the limit min (.,Z',) ; thi ' ......, rtion hal not yet been proven in general. It will tum out, indeed, that the TalUN of .." caleu1ated below will be of the aame order o f magnitude &I th e critical. .....Iue in gnphite and oven beyond it in beryllium-. We aga.in U IlO tho Laguerre-polynomial method for the ca1cula.t ioM. Let III lub.titu te

4' (8) _ tA..LlIl (~_) M (E ) a- a

into

Eq. (IOA.lIb), multiply

by

(10.3.21)

•

LII)(~J ..nd

integrate over ..U energy. Tho

i, a linea r, homogeneoul eyatem of equ..ti on l for the At ; the condition that th ere be .. non-tririal l!lOlutio n i, th at th o dotermlnant of th e coofficientl vaniah:

~u1t

Ill Jlj..+ ~ a1 - 0 .

(10.4.6 ..)

In the 1"",2 a pproJimation, Eq. (10 .4.6 a) beoomN

IllJ!i..Y.. llJ!i.I +LUIllY..

J~: tK) -; ~.(K)IIi.l'-O. 5

I

When .. ... "',

0

(l O.4.6 b)

H_ tZl;1~.ia aD eigenflmriioa of the aqualioa

• it i' lJiv~D by <11.. (&)/.11 (E) . adj oin t to Eq. ( lO.Ulb);

. It should be noted here that lor th.. bNvy IU mode1 min("E,)_ oo, t .e., the ~ time.ia sera . 1'01' tb" reMOII, the .,....m of diM:nlt4Ia~ is OOIIlplek lor thillIIOdtl. 'I'bt heavy au mode! is obt&iDed bl ~ A._oo whilt Iteeping the quant.it,lE.-JEJA fillite . nltu DO OCIIIclukJu about Lime-deJ""'dellt t.boermaJiutioa ~ IIboaId be dnwnlrom thf; bMYJ sa- model. I

tOO

when we factor out tho tririal root « =,0 . Thue 1It ... ~! ,­

".y"I-"1,

Nil.

2:

- "i "r - , - •

r1

UO.".6e

,

tu- -;;-

71rm "''''lT.ii;~N-N

ym;.

I

In the 1_2 approsimation, ~ widentioAl with the t.emperature rel..otioo tim!!' introdaoed in Eq . (10.4.20)1 Uling Eq.. (10.3.10) ud (10.4.2 .), we caD euiJ.y ooa...mc. oaneI..,.. that the t.emperalw1l I.PP'OJimatioD ia apin identical wit.h the appro:limat.ioo. of ooly uinc t wo L.pe1Te polynomia1l, jun ... it wu in t he

............. 01.........,_.

By bepilla: more I...guerre poIJDOmialt. T .... 48Rl in-mg.w how good an appro:Eimation the 1=-2 cue rM1lyla. It turna out that good OODTMlJIlOoe of 011 ia notl'NChed UJltill =20. T~BI find, that

~_tDl _

-riat)

fora hea"1lu.

«f-IO)- ~

«f.. IOI_

for gnphite

.t.2O-o.

~ for beryllium" 2O -c•

•••

The 1_2 appromnation thUi oonaiderably overeetim.tee Cli l Npeoia1ly in media with Rrong biDding. Thia IDMM that the effective neutron temperature tlOIlCOpt ia unauitable lot the qUDUtali.. UMbnent. of timo-dependent tben:rWiut ion proulII . In order to olMin the right therDWi&&Uon tim_, mUlti mult iply the telppentUJ'8 re1n'Uoli t.ime. p't'eD in T.bk 10.•. 1 by t he h.cton . pocififJd abaTe. 70r pphite .. \bQII obYoin ~ P.MO from the Parlu model, ...... Iue which la jut .bon t.he oriUoa! limit, while for beryllium Cu._tOO JlMO foliowl from the Debye model. Howenr, aooonling to Eq . (lO.3.30c) and Table 10.3.3, t... for berylliwn .hould be no IMi tha.n 260 .... 1eO . Thill lhe e..leul..tion. of Tll:AIUaB1 _m not. ~ be .. pplicable ~ thY euo. and indeed • calcul.tion of Coo-

w.

r......

Ula& in beryllium DO cfulcrete mode .bove the fundameneziIY. The fa.etor bu not. been calculat.ed for B,O. but the nlue 1.28 I.ppropriate ~ the heal'}' p i model ma,. be • Petv.l .pprolilll.lotion in thia -.e. AI "15 caa 1M b,. (l()ID.puieoD of 'EqI. (10.4.9 0) .nd (10.3.260). wben the dif. huion eonat&nt ia independent of lhe following rel..tlon exilltl betwoen 0 and~. in the 1_ 2 I.pprozimation :

GOLD and 8llUDO .ho.... ..... ODe

-'1J'. c-

oe~ .

(10.4.7)

'I'hia equation OOI'11lIIpoOO. ~ Eq . (l0.4.3bl of the k1mporature 1IlOde1. In p.aeing from 1_2 w I _to, both aod t... mor-, aDd by1"lJlII8hl,. the ..me f&etor. inwpecti.. of the model. Tbu Eq. (10.• .7) remain8 IpproJ::imakly . .lid. T A..... 'R! hal Ibown withoat refereooe to an,. lpecial thermaliution model th.t Eq . (10.4.7) ia .PlJIOZimately true.

c

a1_,..

'31

Chapkt10:~

We can a1ao oa1oul&te ~ for "av. media ; PnoHn' baa done tim in the 1... 2 appro:limation. An increue with lncnuing JlI ill ..ideni .. it. to be expected from the qualitati,.. conaiderat.iooI of the toemperatu.re model (d . Sec. 10.• .1).

....,.,

Chapter 10: Ref ereDeeI

AJw.tJr. E. : 1Do. (Ii,-. . .peoialIy I8&-UI-

eouoow. N. (eel): ~jnp of th8 BrookhanoCoo'- OD. Neatroa. Tbenn ,Jj,Woo. BNL-'JlII (I tIe2). PooLa, K. J •• K. 8. NUoKDl. ~ R. 8. STotrS: 'l"h& 'l'booory and ~t of Reactor Bpectn., Progr. Nuot. Euerv aer. J. Vol t. p. 91 (l eM ). !tonal, L. 8.. &ftd V. P. DC"OQAI.: ~ of Thwmal Neatrom from 80Iida aDd t.beir n......a1j,... 1on N., Equilibrium., Ad_In NlIcker 8oimoe &ftd Teclmorloltr. Vol!, p.186 (l ll6fo).

....,.,.

eo:.:-. E. 1l : o-n laM, Pj8U. Vol 6, P. 631.

} DUD a., G• • . .c) JI'I Ph,.. Re't'. No 1m (19M). NeaiIoQ ~ by. a.WN "'" E.P.. ....s J . E.. W~ : AECI).!2'l'5 (Itu). of Atomio N-w. W ~ J. 1 : CP·U82 ( ltu). . Baa. .. H .D.. &ftd D. 8. St. J'omr l DP-U (1966). Jl"Dx l, E. : m-- Sol. I. 13 (1838). J.-m, J. A.• ....s A. Xo"~ I INU.tv.te of NaoJ.r Cnoo... Repon No. SD6 (1953). XaaIoa, T.J.• N¥l M.Nm..:m: Ph,.. RMo. lot. 2QO (llIll7). ~tablofh­ N~. :W. OOA·Ill8ll(I960). Neat;ma &at.. Puc&D, G.: Pb.Ja. N , m (lNS). Pnour, B. N. : BNL'JlII. 203 (19ll2). WiIlcbJ~1 BaaDd Atom.. PnouT. B. N•• UMi A.. L au.AOOP.iLI BNL nil. us (1ll8! ).

l'tl,..

....tio

Be".

&ma. R. G.. UId E.1'Iu.D: Pb,..IWto . .., 18(1"1). s,,0Luba, A..: Ani" ~ U . 318 ( l ll6II). Vount. H. c.: Pb.,.. Re't'. Ill. 868 (UNllt): l 17. 102t (1900). WlCS, G. c., PIa,.. RMo. M, 1m (19M). ZPua. A. Co. .ad It. J . Gu.no.: Pb., .. Re't'. III. 118 (ItlIe). Do",- V. c.. V. c. K OI.DUl" &ftd D. E. PdUl BNL 711. eo (1982). GoulIl.U"Jl. D. T •• aad. 7 . D. l'aDazo..: BNL 711, 100 ( 1M3). ~.

G. : N'IIk1BoDlk f . 1I0 (1110 ).

Bo--. H.: TraM. Am . NuoJ. 800. 1. I, 41 (1962). KIu.o.. J'., R . L. B..... UMi W . J . & .._ , NAA.BR-lI40 (1982). NKLKI1I. IL: PIa,.. RMo.I1'. 741 (1980). Bnnl'oa, T. : NUIeocUk " 110 (11181 ). BUlIDUl" R. c., J . NuoL EDqy "'''l
x-v

s..

NIlLJJlI'. X . : NDol BeL

z-v

:r.a,. I, m

(1967). p........ D. Eo: GA·!43lI (19611; NId. BeL ED( . II, JOe (IMJ). PI.t.~

T· .. •••

G.• UMi L. "'U' Bon: Ph,.. Re't'. II, 1207 IleM ).

H.: BNL.7111.12ll9 (19M).

Boo."

I

Baoouot, B. N.: Bull. Am. Ph,.. m (1868). A .. BaoQO'" R.lI., Uld. J. E. ETU' 1l NIIol. llMtr. Ket.bod.III, 16 (INI). B~re, EomMUJ'I', P. A., J. Cocu1l'o, &;nd J. &"UlflID: CRRP· I078(I~). .Iutlo:ru:n.t;ma GLl.p" W. : Neatroa. Time of PliP' :w.t.bodJ, p.IOI , Bna.e.: Eunotoli:l Le-..lR.

1"1. I a. foot.oote 011 P. 63.

-

.....

... B.-ooo... R .!l.: :BNL71V. 3 (1962). EoDBl'UJ'. P.: Nual. &i. Eng. It,. 260 (1962). Eom.eul'l'. p.. and P. 8c:BOruLD ; NucJ. Soi. Eng. It. 260 (1962). lUTWOOD.S.C., and 1.)1. 'hOR80lI : BNL'l'l9. 26 (1962). StNCLAD, R. N.: Ioe1utio &.ttenng of Neu\.tom; in 801m and Liquica, Vol. 2, p.l". Vienna , Interutionat Atolll.ioEnergy Agenoy,IOO3. CoRall'_ E. R. : Nuol. Sci. E'.nf!;. 1:, 227 (Ji57). eo,..,.Otl. R. F., R. R . BAn, and R . K. OsBOIlJ'I: ORNL-I968 (19M). Spece -Independent HollOWITZ, J. t aDd O. Tanruon, EANDC.(E)-I4 (1960). Neutron Sp&Otnl HtrltWlTZo a, M. S. NII1.n1f••mI G. Y. Hdnua: Nile!. Sci. Eng . I, in GU80UlI 280 (19M). Modonton. WIQ"'" E . P., and J. Eo WILIU1fS: AECD·2216 (1lH4). WILUJI" J . E. : CP·2Mll (1lH4). C&DIldI.AO, H ., fit -.I.: BN(..7UI. G9 (1962). CoPOOLD. N.: Ann . Phr-." 3tl8 (1969); 11,338 (1960). ~J*'&-I~(I=:t P ...U :I, D. 10:. : Nuo!. Sci. Eng. t. C) (1981). ~ In . kln Smu.u.... O. W.. and K. Au.lIo,r: BNL.719. 614 (1962). with Chemll:al Binding. HOK-. H .: B.'lL-719. RD.l (1962); BNL-821 (T·3Ie) HucIn1&, O. r., etaJ.: BNL-719. 706 (1962). S~Dependent MJ:1JILf.a.. P. : Nucl. Sol. Eng. 8, 4:6 (1960). Neutroa Speotn.. n. sc..INO, I., &ad!l. Cuu; : Nuol . Sot Eng. • ',388 (INI ). B1IC:Il.11KT!I. K. H .: Nucl . Soi. Eng.!. 616 (IN1). BIlCIUTlI. K. a., z. N-turfoncb. lb. 966 (11161). VOl' DUDlu.. O. F . : Ph~ Rev . .... 12'72 (19M ). HbKLK., W.• -..d L. DaIl31l'o : Nucl . Sci. Eng. 7.30((1960). Theory of tho Hu..wrrz, H.• -.nd M. 8. NKLUlI: Nud. Sci. Eng." I (1968). DiHUBion NEL&DI. Y . 8. : J . Nlld. Energy A 8. 48 (19M). Cooling Effect. PnoIllT, S. N. : Nud. Sci. 161 (1961). SllI"OWl, K . S.: Aniv Fyliik II. 386 (1960). T.I.1I.OUID, H. : BNL-719. 1m (1M2) . ) HOll'IlCI. H. : BNL-719. 1186 (1962). NKLIllll", M.: Nucl. Sci. Eng. 7.lUO (11l6O). orn.n.pon'I'beo!'y of tile NEL&DI, K.. : PhY'i<* n. 261 (1963); GA-3I22. 3m (196%). TbermaJ Neutron Field. &Olnu1cD, N. G.: Azoki1" FyIiik II, In (1969). Conn, W. R .: Nucl. Sci. Eng. 7, 296 (19M). GnaUD. E.}I.. &nd J . A. D.1"1I : Nad. Sci. Eng. II, 237 (1962). SpaOII-Energy KlDuuD, Eo : Nad. Sci. Eng. 18. ofOf, (1964). Se~bllity in the KLAIlxm, R. : BNL-119. 1211 (1M2). Tbwm.m..ed Field: KLulIftlr.. R., aDd I. K08OD : Nuo!. Sol. Ens. II, 1'9 (1962). the Kilne Problem, WIU.J..UQ, M. X . R. : J. Nud. Energy A&B 17. 60S (1963). OoPOOUl. N.• p, llIaum., aDd W. WOLLIIUInII : Naol , Sci. Ene. U, IS (196lI); BNL-11', 1103 (1M2) . Time-[)"pendent OoUOOLP. N., aDd C. S. SIUI'IBO: Nud. Sci. Eng. (in JlreN). KOI'I".Kt., J . U. : Nud. Soi. Eng. II, 101 (1963); GA·2988 (1962); Tbenna1izI.tioo. BNL-7UI. 1232 (1962). PuwIllT. S. N. : Nad. Sci. EDfl. t. 161(1961), BaJOfJ:fT, R . A., and R . E. HmlIZJulf: Nue!. Sci. Ell«. 8. m (1963). Neutron DiffWlion KO'lTWlTZ, D. A. : Nud. Sci. Ell«. 7. m (11l6O). iD tho Neighbor. KOlleD, I .: J. Nlloi. Energy A.tB 17, '9 (1963). hood . . . D. WOO\llClLUD'&, J. R. 1.., and I.. K. Gaou."H : Nud. Sci. Eng. 12, Temperotlll'll 238 (1962). DilIooJIt.inuity; the S1l1&1I0UT. D, 8. : Nucl . Sci. Eng. t, 9f (1961). Spa.Ua.I Approach T.u.AB.tBID. H. : BNL-118. 1299 (1982). we... a, &ad W. Str•• a.IU : Nu1UBonib I, 243. 0643. 691, 7Of. (1961). w Equilibrium.

1

119631'1

Eoc.'.

I

I

The Determination of Flnx and Spectrum in a Neutron Field 11. Measurement of th e Thermal Neutron Flux with Probes The neutron detectors di ecueeed in t he third cbapter a re not Al wap auitable for det.6nnini ng the int.enait y of ...t.ationary neut ron field , but we can al Y' uee to advantage the .a-ealIed "ndioroctin indieaton", i.e., '"' can ..I ,. aetint,e .. probe in t.be neutron field and then oount ite radio&cti'rity. I n t hi, ...ay . _ can achieve very peeciee relative .nd abeol.ute meuurementl of neut ron inteMitie.. Since there &nl many probe . v.bet&ooee _ boee activation croM ~ depend in nri0U8 WHerent wayJI on th e eeetroe energy. '"' can &lao derive information a bout t be energy dinributioo ill .. neut ron field from probe __ aurement.. In thiI chapter, we IhaU examine probe measurement. in purely thermal

neutron field. . In the prooeu, queet.iont will be enewered which are importa nt for . 11 probe m~menta. The following chapten will then be devoted to the meuurement of th e epithermal neutron DUI: with reecnenee probN a nd the IIepuation of th erm al and epitbertnlol a.etivat.ion (Chapter 12) .. well ... the U8fI of th reehold detecton for th e detection of t ut neuteone (Chapter )3). Cbapt«ll 14 and 16 ..leo touch in placea on th e UI6 of probes.

11.1. General Facti about Probes II.l.L AeUVatlOD and Aelh1ty We ahall begin by . peoifying the geometrio form . t hat will interNt u. hereafter. I n pr &ctioe, the mOlJt Widely used probN -.re f2E!, i.e., probell made in th e form of a thin, dilC-abaped foil with a . urfaoe area in th e range O.l -IOcm'. Prot:in the form of long lop" or vriru are OOOMion.Uy ueed to meuu", n Ul: profilM. For foil. and "pel, aeti n tio n and aetivit y a re usually el:J'X'6U6d per em' of aria, while for wiree they -.re e~ per em of length. The GdilKllm a i, t he number of radiOACtive atom, form ed by neutron capt ure per IIeOOnd and cm' of probe area. a dependa on t he inteneity of the neutron field F (r , A . E), th e thick.- 4 of the probe, and itA CI'OM Metionl I ..(El, I . (B), and I . IE). We .ull ieam th e general oonnoetio n between th_ qUlJltitie- and in See. 11.2.1. For (I.{J')+ I ,(E») 4< 1, Le., for thin prot:-, the oeutroD field In 'l'e neI the probe praetica!ly nnattenuated, aod

a

-

a _ ~I....4.

(11.1.1)

Here ~ a. the thermal Ou l aDd E... the "a"rage" aot.i....tioD 0I'0IlI aeotion i.D tb e of &to. 6.3.2. The aotintioa of thin prot- u. thu proportional to the f1l1:l or to dle deuity; if Z'...(-) obe,. the l /__Iaw, &hell O-a"iE.. (ii)4 and we can obt&in the DeuVoll deuity from the aoti ....tion with0t3ImOW'ledge of the epeotniiii. The "., ~ B(II u. th. DUlQber of ~" at.c>Ju preeent per ~I of probe an&. U 1 u. the decay OODItaDt, tilen

II8IIMl

B (,)- i (l-, -lf)

(11.1 .2)

for a ClOn.d.ul.t act.i.ntion Lllat bes&.ll at t.ime ' _0. B (I) rMOb. the Mturalion nlue 0/1 after an .. infinitely Ioag" irradiatioD. After &II irTadia.tion time of 01lCl haIf·life 2'. -0.693/1, lSO'rt of tho ..turatioD ....Iue I. reaehed. after ten haJf·U...... 99.n• • After tbe flId of tbe itndJation. B (, ) lalla off aooordinlJ to iIie

liw

.. .-

(11.1.3a)

'I

whore '" ill tbe itndJation time and I. tbe elaJ*ld time .mce the end of the irftdiation. FinaUy, the at.htiV, A ill the DWD.ber of atom. decaying per see and eml of probe nrfaoe. A _IB aDd A(IJ _ C(I_,-lf,),- .U, or

2' _ I

....-11,·

(11.1.3 b) (11.1.3e)

In onler to make different probe meuuremenw oomparable, we cu.tomarily orJouIate t1MI acti....tion by m11lt;plyiDg the meuand actinty by the appropriate time factor. Note that A a. the " true " probe aMinty whereu aimple counting gi... IUl "ap$t" actirity \bat it IlD&1ler than tbe t.nIe activity owing to tb e .u..beorptiooiIle emitted r.d.iation and to geometrical (lOlid ' a.nsIe) effeeta . 11.1.2. Malerlalt for Thermal Probel Table 11.1.1 containl lOme uMIfuJ. data on IUbitaDON tb.. t (lin be IUOd for tbennal prot-. The &rat three oolllDlDl contain the DQlllber of atom.I per pm cl the IUt..taooe, It. abeorptlon orc.a eection, aDd iw IIOIttering crou MOtion . In 1Idditi0ll. the table~.u.. acti....bIe leotopee, their akndaDoM in the element. their aoti....UOll ~ Mll'tJODI, the ldopee produoed. aDd the laUer'l half.li.... n. ~paI 1Cltl~.'1 ,"en ill ~; kDowIedce cl the oUl.et' aolltlrritiell u. imPJ"o&Dt lD order to enhIate the probe mewurellleDW ""' mUtt !mow how' Ionj: to wait for the ehmJind aotivi\iee to die oat aDd how lIIuch back. pvund to w~ to &oCIOQDt for the loog..J.intd &CItivit.ieL

.moe

MAfII""'I bu the lpeoial adTaD\at:e of being a pure l'....beorb. in t.be tJier· mal energy nap. ID pnatJoe, it ia mainJy U.Md in the form. of lII&Zlg&OMll.nicbl foile eontaininc about 90 wi- ~ of mangane&e (~ _'1 .6 glom'). Neutron oaptUf'lll by Ni'" l-u to a perturbing activity which coineadentally hie tbe MlDe haU· I

Kano Pf"'C*l11 ~ I . . . . . of the &hena&I flu 01'tI' tbt pt'Obe ,urr-.

... &I Mn" hal only oontribute. 0.01 % .. much .. the principal adirit,... MaD. gaD.eee-niokel foils are n ry .table ADd can be rolled .. thiD .. 011. 8 mgfcml ~ 7.9 xIO"'mm.. The abeorptioD and activation coeHicienta l &re lAw -O,I36 IIII.d P.. _ O.l 30 om l /g. Such foils are .wyble for IDeUW'8JDenta in DUM greater than 100nfom.' /_. Fig.Il.!.l iho_ tbe

life

decay ii'ibiIme Of maoganeM. 8-.. well

Jt....,~

Ult'W

\\-~,,------'~

u,

..

~.nd.i&l.ion OOOUJ'!., aDd both can be

u;;a

for counting. Mangan_ i. abo wed in the manganeMI bath method of integrating DeUtronfielda(cf.8ec.lfo.3.3).

1:1 ~;~\~=:-

,

J)~

.-

J 'lI,,1t..LI.

-

~ ,,00

no..,-... o."

T.b'e 1I.1.1. 8""*"- for 2"--' p,«-

....,

Nit" 10-'"

1.007 1.022

•• 1....'

_. (\lull)

lU±G.1 3T.l±l.o

r.o~

(u.. ..

~ll " ,

1.. - tIOO .....'

U±o.J 1lD." (100)

,±,

a,w

(100)

A,I

.... Ibual 1la4IoMIl.. l ootDpll .._ DlXl IaI_ 1 ( H••,W.,

lU±G.l

IU±I.a

"oM

""'-

(U" l (10... m ill)

(10.4 0lin_6.J8,

(99 %1)

.,c.

....

...

aD1 / .A.

0.... 0....

II 0....

0.37. 0.....

"'1-

1. IIH± I , (0.0263 ....)_ 1.01M

"'±"'

,....

7.f±O.e Cu- (18.11 Cu· (30.'1

I±I

Ac- (6U

lI l

A.,- CU,MI

t .I±O.a hl" (01.13) lDw lOO±to

(9ll .'77)

cIea1iDc wiLlI t-b-

t.O±O.e

....

U±I

b i ll

11&.1 _I

~.

( 1.3 mill) (lMJ"')

...,.

U;tO.41 IOG±l6 lI8± UI

l llO± 1

DT"" (1:8.18) ....±... 8OO±lOO

1liU±0.4 U±l.o Auln (100) ,(0.026311'1')-

_11ft

(U8rJ (11.111)

~,-

AC'"~

(a.I4 11R1D) . (1.3 ....)

(W dl

1"_)

b,m -. (dd)

m_1

l .w" ( 64.11""'JI)

ur-

llU±o... A.'·

(t.m ~)

h " ~ to \he ~ iD w- of tM _ . bMllIl(,.,,1(&,1-J> radlw t.bMl1D CD; t.b-. \he __ ~ -"~iIa, EJ, I --',) .pp.n III pi.- of &lie - p i a ~..."...r,..

I~IJl

fac.e __ IA.-

=.1

S.81±O.03

2O.2±U CoOUl±o.JO e.U±G.4 Co"

MNoIIuremen\ of \he Thomnal Neutron Flux with Probee Beceuee of ite long half·life and it. small activation croas eectdon , cobaU is ueed for the measurement and frequentl y for the long-tlme integration I of ex. tremely high neutron flUIes (I()ICI-IO" n/cml/sec). It ill used principally in the form of wires . The irradiated probes are counted most conveniently by means ~f their r·radiation.llinco Co'O emit. only very weak p-radiation. OO'fl'lWr like manganell6 bee th e advantage of being a pure I/""a bsor ber. Be. ceuee of their longer half·life and smalle r activation erose section, coppe r probes are lesa sensitive than manganese-nickel foils (flux range above 100nlcml(sec). Pure metallic copper (e =8.90g/cml) can e&8ily be made into thin foil~ or tapes with good mechanical properties and surface loedlngs &B low a8 5 mgfcrnl . p" = 0.0361 em1fg and Pv.t = 0.0289 cm1fg for

'1Z

•• ,.

~ I

(0.2 1)

Q.l¥1Ib (Q.1f u~ /

1.1,/

..

UI U!I

r.rz

~ s,,"'_

J'1C. 11.1.3 .

Tbo ~1 odlemoo of

la"'" _ la"'"

la"'.

••'"'.M"

U£ !It:.

"

_ ___..!:..f1!....

""

I I I I II I I II I II I

". ~

~

dS'

1-

L

,. ""'"

ria. 11.1.1. Tbo '**1 oMo....

of

111'"

the 12.87·hoUl' activity. Fig . 11.1.2 ehowe the decay scheme of Cu". The probes are beat counted by meanB of the p-. and po-radiation or by me&D11 of the positron annihilation radiation. r-r"ooiDeIi:feil oi"methods may be used here with advantage.

Becauae of ita abort half·life and high activation CI'Ol!8 section, Alver - and a lso ,hodivm - is often used in demonstration experiments involving neutron activationj _ have included it in Table 11.1.1 only for the ea.ke of completenl.lU . The M-minuk'l activity of iMivm ill frequently used fo r the determination of low fluxee. The crose section does not follow the 1/v-la w j in fact, there are reeo, nances in the evrange which can lead to a strong epithermal activation of the probe (d. &C. 12.1.2). Be&dea the activities given in the table, a perturbing ' .6-hour activi ty is produced by an isome r of InlU that cau be excited by the Inelaede scattering of f&llt neutrons (el. Sec. 13.1.2). Durable foils can be manufacturOO out of indium metal (e= 1. 28 gfem.l ) with thickn68868 down to 10 mg/cm l ; " t hinner " foilll ean be made by evaporating indi um onto auitable backing" or I A 1ong·time integnltion ia_ r y tofind the toul dosereceived by. umple irradiated in • reactor.

alloying it 'llritb tin . For indium met&!, ,u. -l.OIG cm'ja: ADd ...... (M min )_ O.flO4Sc:m I/a:. Fig. 11.1.3 aho... the ~y -cherne of InlM; ob'rioualytbe activity can be oounted by mMna of either the p. or the ,..radiatkm. ~"". i' ,1 10 auitable for the meuurement of low Dentron Oux... The .ctivati on CT08I lleCtion deviatM alightly from the 110"' ; oompared to indium , the contribut ion of epith erm al activation ill amaller. Dyaproeium iI lOtDetim.. used in the form of dyeproeium metal . in tho form of dyepfOlium-aluminum a)loy, or in the form of dyepreelum ondo Dy.O• • which i, deposited on an aluminum ba
1'•.

",.

r-

'

I

e~ '"f--+-~7i<'----+--1 ., J'II.' l Ll...

laJ

no

DJ

f(l)

TAx:

'

&II

, . . - '"

.--.I _ . . - Id.. .... tU l .....-u>1

6 rngJem' can IUily be rollod ; thin foila can abo be made by e.....poration, p. -,u.", =0..3008 oml/g. Fig. 11.1.6 . ho. . the decay Kheme of gold; p. all well .. )'oOOunting it poeaible 1 • ~~ Fig . 11.1.6 ehow. the neutron.temperature-depondent g.f&eton of eeveral probe l ubetanOlll. Data on the Ol'OM leCtionl in the epithermal range follow in Sea. 12.1.2.

11.1:3, The Detennln&Uon of the Probe AdiritJ We must diatinguiIh between the problema of determining th e .beolnte I.Pd th e relati ve probe activity . The former ill neoouary whon we want to
&II ~ ~ ~

owina: to

of 30.000 - - . With"", hiah _troa Lbe Jo. 01 ~ ll.ueW by _ k'Oa e-ptara.

D~

per em' and per 1OOODd. ill thus equal to h&lf of the true activity (aaide from th e fact that eaoh decay may DI» produoe exaotly ODe ,... ray ). The meuurement thua yioldl a oounting rate proportional to the true activity. Many probe aublltanoet< do not emit &I1y UNable ,..-ray.. In thi. cue, we must determine their activity

U .. (.MoM)

1J:.i GUS lJIiq (bW

" ')

U. 1.81

0.... 1.333 1.173

• IS

ue

1.61 I ... 1.10

0."

0.411 0.14-

"Au

.-... .""

S.eeGd

0.4112

(Au l l l)

•

"

.....

...

...,. 0.... 1.00 0."

0.81 0.141

,.Dy

1410 lJIiq

S.d&6d (Au''')

0 .0Q'l'0

0."'" 0.192

.",

.

...

"A.

0."""

0.30 0." 0."

e.eee ( 19%)~)

(Dy'M)

0..... 0..... 0.....

0.'" 0...

12.87 b (CuM) (In.. • ..)

0.0612

0._

u 1•

. ' ..... 11""..)

.. r!i~ I 'N 11.t±1.0

0.673 (38%) 1.00 0.87 0.60 1.28 1.111 0.29

(IU %) (28'fo) (21%) (86%) (13 % ) (1.41%)

0.967 (911.0%) 0.296 (I %)

8ol~ '

... ....... "

2.86 (60%) 1.l16 (241 "') 6.76 (I e ", )

6USmil:l

....· 111 Codr\olloat

0.0f,11I (Mn) O.OWlS (N i) O.OAliO (.Mo) 0.0&36 (Ni) 0.06lllI (.Mo) 0.0&&6 (Ni)

0 .1611

(.MoM)

,.c.

II ToW lll tbi

)

~

...

"'=

. ...... ..... .... ....... --,

.

"co

"

_

-.... ....

:U.9 ±1.6 26.0 + 2.0 12.6 [Eq. (11.1.4))

2O.0 ±l .0

-

'";' ]101 ........ i [

U>o

7.'

.. "

... 12.6

..

by melln8 of their P·radiaROD . Even with .ubataUoel whioh emity-ra}'8. p.oounting ill 1Omet.lm1!lll prefetted becaUM it requirl!lll a muoh lhD&1ler expendit ure of ap' puatu. Here. however, the appwent acti'rity ill unatler than half the tru e aoti'rit1 heoaue of the Itrong p..-ell-ablorption. In p&rticWar. the appuent

aoti'rity may depend OD the fon onentition lD the neutron field... we lhall show

... in Sec. 11.2.5. ~..beorpt.iou cau be ~ approzimaWy by &11 e xponentW 1&",of &"-eD1I&tion : H .. ft.emi.tkr.1M»e ~ deDait1 i, negligi bly IJm&1l i. covered by .. layer of thiom- .., the prot-biM,. ICI lz ) that .. dee.y elect ron penetratea the layer I. given by

w(:a:)_.- ··.

(I Ll.4.)

limpl. decay lMlbeme (gold); for I U1wta.nee. with .. OOIDpleI ~-.pectrum. l ac h indium, it DtlCllMU)' to take the _"-eDuation of MCb iDdiridaal component. into aoooUDt through n. own IIepaRte mau abeorption coefficient «C. For mod_ a.ocuraciel. ho""YeI',

Tbia law hold. qui t. aocurateI.y for n b.ta.DceII that h

~

-

•

,

.....

<,

w

r•

'"

r- ..• " :::: r--..

r":

w

..

,.

d._

'"

III

Ill .." .

"., II.U. • __ .. -"W rr.. .... ...-... n. .. . t--.or.u.-- ,...,. ' _ ~

~

GIl

~"ua-t-IAl~

" ',

"

~

III

JI)

d_

III III

,._liIOIIlW ....... n.rw__ - ' _ __...,...w .. J'lI. 11.1....

101 ).

_ _ fill rrtueu.1lI1bo IhloII:-. III 11M _ - . l .. Ibo ~

J. ~

CbaII . . . -

~


ftl.-

UM of &11 avenge « it uaually lufficieDt . Table 11.1.3 oontaina data on the p. _ rgie. .... well .. tb e m&8I abeorptioa ooefficienta of IIODlfl probe nbst&ooee. The foorlb column CODWn- mua . bIorption coefficient. lit determined from the initi&.l portion of an abeorptJoo cune me.aured for aluminum (el. Fig. 11.1.7). lit t. given ratber aoeun. tely bylhe following empirical IIJIl'N&ion fOLKdO}(): «e-=17.0 E~ [oml/g] (B__ in MeT).

(ll.l.tS)

When th e lpect.rum ia complex, ~ for NCb component ma R be calculated from Eq _ll l.l.6) aDd then &ll the ..m- annpl. The fifth oolumn p .. &D average abeorption ooeffi.cieIl.t thU i8 clK.eD .. that the ~ at.orptioa ctIn'e (with th e probe nbetaDoe ... ab-orber) ia fi\ ... well .. poeaible by an expone nUal func\ion ...... in the raoge 0 -100 mglcrnl (d. Fig . 11.1.8). We _ that II and or. do pot differ ItronglY i therefore. Eq. (11.1.6) Ga P be used to Mtimate II for ItIbnaDoN where po meuurementl ha .. been reported.

11.2. The Theory or the A.dh1ty or Nelltron.DeteeUng Foill In order to take full adY&ll.t.age of the high

aoouJToey aUainable with probe meuuremenw. we mPH bow the theory of foila tha\ will be developed in thi. and the nut 1OOtiOD.. The theory will be deeelcped in two ltepl : Fint we .hall di«nw the relatioMhip between the aotintion or actinty &ad the peutron iDl«1I:ity inciden\ OIl. the foil. In thia di8cuIaioD. it will be euumed the, &he

/. .

... eeuerce field ineident OD th e foil U t he ume as t bat which wu pl'ClWlnt before the foil wal introduood. In roality, each foil perturb. tho neutron ~ and it ill DeoelIll&l')' eo a pply foll ~iou. We _hall go into thie latter problem in detail in Sec. 11.3.

ll.l!.l. AeUnUon and th e Nrutrun Fin Let U I firet oonaider a mon()(lJlergetio neutron field. Let tb e vector n ux 00 repreaented ... uaual by a series of Legendre polynomial. in COI (J, where {} i¥ the angle to t he field axi. : F (r, 0 )-

1 i"i 5' (21+1) Jj (r) P' (COII (Jj.

(11.2.1)

t.::

I

Let " be the a na:1e bet ween tho ' iekl . zi. and the norPl&1 to the foil (el. Fig. 11.2.1). Then in ••yete m of polu' ooordin.f.tea uouoo th e norma l to the foil, we have F (r, 0·) - "I,.

~. (2l + l) F,(r) !P'(COI (J- ) P' (ooa If) +

.-1 ::+

+2 L 1:~

(11.2.2)

pt" (COli 6*) pt" (01)> ¥lCOII-IF-I·

We _ume t hat Ule F,(r) do not change appreciably along th e foil . urf ace, with the l'OlIwt that we can henceforth omit th e argument r . Furthermore, we UlIume th.t (al no toattering OOCUri in the foil (p. =,u.),.nd (h) no neutronl enter th e foil through ita edgN (01', in othor wonh, tho ndi ua of t ho foil ia vory largo com. pu'ed to it. t bicu--). Th o Dumber of noutroDa th.. t are ..becrbed per em' per see ..t deptba bet_ D % ..ed z + dz i. then

P"' (z)dz -

(Tf'.

....I' -#)

F tO·)t - - .....- eln {).

cti~(o.)e ~;'r

d~· d'fl.) J'.dz +

• •

+

-

lin {}'

d~' d'fl')lI.dZ.

I

(11.2.3)

Tbe LiJn t.enQ on tbto right.band aide cle-oribea the oontributJon to the .. b.orption of the DeUttona incident " fro m below" , the aooond term the oont ribution of thoM lncident " from .. be ee ". If '"' au_t.itut.e t he vallMl of 1'(0') from Eq. (11.2 .2), the t.ermaoonWning the waoeiated Legendre polynomi ..l, vaniah upon integration Oftr 'fI' .. nd we ohtai n ~ (z) = i

r• (21+1) Ii gj(p.., z. 6) P,(00fl ¥)

(11.2.4-. )

1-0

(1l.2....b)

. Tho Theory of tobll

In pu1oicu1u', if

01»"

Aotl.-~y

'[.--,. . .-.,+.--,-~' l ..

= "

, ,(JI.,:c,d) r::p..!

_ ",'[ E.(p.(~-.)) +E,Ip,..)],

'It--.- -.--'- jdl

1

flCA..:c.

d>-.p..!

.,

of Neutron.Det.eo\lD«Foi1e

I'\o(' -a)

....

I I

(lI..2•• e )

(ll .2.' d )

_ ",[E,(p. (~ - >})- E,Ip,..)].

' 11"..,:C,d)_Atji (S,I_l> [.-"'('I-a) +'-"';']d'

- !'t [3·E,(p..{6-:en + 3B'. (p"z )The funotiona E. {z ) _

B',{pw{d-

~})-E.{f4.:cl].

I

(11.2.• e)

i'::' tl. _j,.-.,-T

(I U .• 1)

III

have been tabulat«i by Puozu: among othera (el. Append..ix ill).

.....

u

Fig. 11.2.2 .ho_ ". i l' and g, " All " wit.h even I remain innri&nt under the tn.nlfonDAtion :1'_6- *: on the other hand, " lrith odd I ch&oge their lIign.I. The ~V&tion C i8 obt&ined from NnnoI*,fM!rIII

"""

~

...

-

if.

"

~-/U

-

-:;'- 1.8

I ''1\

r

t"

If.'<

-

~ ,J

..

~

, ~

ll.l.L

--

,,1/1*"

•

4

no. IIlWIooI

~

<-m.

,. . " ...... ..

U

fU

%/d_

JP"I') h ~ ,.. i ~ (21+ 1JJi 9'l CAo dJP,(-

I

~)f_~

•

~

-

/ ~

U

fU

U

"'.""_41.1"'1.0

,

=-

.I

rlI-1L1.1. 'JIM r..-.- Il
P"'(:c) bymultiplic&tion withp.-Jp. IIDd integration onl' the foil

0 _ "::

,

tmem-: (11.2.6a)

,,). .f

9'I(p.6) -

f•i,(p.,:I', a)dz

•

-p.!'[Ii,,- ""C _..' -.) P'(coef"' )ain"-d"· +

(1l .2.6b)

. ......f p,(coe6· I . in (}. dIJ. ] +.!.,-• •

We

~.

..u t bo 9'I(p.6) with odd I varu.h.

immed.ialoely that

ClIoD _

dz . F or "Ten I on

the other haDd.

..-

.

( 11.2.60)

IU - -- -

/. ~ I~

..f-

- -f-- - - - - - -- f---

,

,

r J Pal=-

,

,,-

., .

J

"..110 1l..LI. 'boo . - . . . - . . . - -. ,..... .. t..." fII. .. ... ... - . ... n.u..I

<,

IJ

<,

as

"

U

"'-"".. . IU -.._..an. u."wt... . ..... - . .no . . . _.-... M III . . -.... 1leId


_ _ I .... "' .

'h(p.61-1 - 2E.(p.6 ). 9'. (p,,6) -

i + E. (p..6)-

(IJ .... d)

3 E.(p. 6).

fll .2.Ge)

In the following, we . hall limit aurae!v•• to neutron field. Wh086 vector flux i, deeeribed by only the lint tw o te rme in the eJ:panaion in Legendre po lynomiaL!, Then, &imp)',

a_ "'j.-. - -t-

,... . (r)

,.. Cp. 6) ,

(11.2.6)

Le., the a.eti...tiOQ ill independont 0 1 th e foil orient&tion an d proportional to th e flus:. The cw nmt term. affect.l t ile local dilkibutton of th e activation in th e foil

- ct. Fi g. 11.2.2 - bu t not t be t.oW.l actintion integnted OY.,.-the foil th iekn-. Sinoe t/)(rJf2 i. the number of neut f'OlUl falling on both , KIM of t he foil per em' , ~lP.a) can ObTiouaIy be identified .. th e ab8orpt~n probability of th e foil in an iaotropio neutron neld. Fot' an Utnlmely thick foil (p",6> 1) we have flt(p.6) - I ; thlUl 0 _ ";;

..!t'. On the other baod, for an e xt.l'emely thin foil, flt(p.6) -

21"006 and (l L2.7)

The Theory ot th e

...

Activity of Ne lltlol!..Detectill& FoUe

[Ct. Eq. (11.1.1).] Fig . 11.2.3 aho_ 9'e(p.6) aDd th e .uuple a pproximation

21'l.6/(I + 21'l. 6) to it . which _ ah.u ooca6ionaUy ute later. Frequently, Eq. (11.2.6) ~ written in the form ... ~ dl

C -Jl..6 -2~l - ~ (r) .

~;~::! ~(r) i.

(11.2.8)

tb o averar Dux in t be foil ; Fig. 11.2.4 QOw, the 8lllf-ahielding

faetorl 'l'; ~::l .. a functi on of p.,, 6. 11.2.2. Th e Erred of Suttenn« In tb e Foil Non we eball inveatigato th e enon introduced into tbe calculation of tb e activation by tho neglect of _ tteriq in the foil. Sinooin the energy range under conaiden.tion tho acattllring et'ON lOCtiolU of moo . ub8tanee. used in practice are much _ than their abeorption c:rou ~ , it ~ enough to cooaide r only the 6I'It and eeeced collieiona in the foil. Since in tJU. _ at.o the C'W'Tent term doN not contribute to th o act.i.vation. lot u.t oonaidOI' tho activation in an ilM> tropi e neutron field . The num ber of neutroDl which . uffer their Lint colli.ion at depth. betwoen %and %+4%~ [d . Eqe. (11.Ha- en (1J.2.9)

Here Pc=Pe+ p, i. t be total com-ion coeUieient. Of t beee eolliuOfl.l, t be fBction P..cJPc leade immed.iatllly to activation. Tho fracti on pJPc reptMOlnte _ ttered eeueone, whoee hietory we mlllt follow further . I n eo doing, let ua UlWDe th a t tbe acattering i. idropie. The probability tbat a neutron ieotropieally acattend at r l!IlICaJ- from tbe foil without making anotber collision i. l (E,CPs{6- r}) + E,l,u.:r»); therefore the number of neutrona which make a eeeced oolliaion in the foil it

• ::J ,

p' (r )

The fun ction

(1- -~- [E. (p.{61

r}) + E,flV») 4r _

lit )p. 6I (p.6).

f'

I Cp.,6)= ~2",i , (E.(p,{6- z}) + E,(p,z)] X

I

(11.2.I0a)

(1I .2.IOb)

X12- {B, (p,{6- z}) + B, (p,zl}!",dz it .hown in Fig. 11.2.6. Sinoo tho fractiou ~

IIwJPc

of th o eeoond com.iotl8 Uo

to activation, the activation of the foil reeulting fro m the fint and IeOOIMi

collisiolUl it

c.: ~ ,...

I.j

P'{zldz +

~i" "'6 Z(p,6Ij \

(11.2.1 1)

~ ll'l

- - ,..- ' - ,- [.. (p,61+ ",6Z(p,61]. I Thill _It-ehioldina ~ ,h'. &he ....tM> of the ....orap flus ~ in the foil to &he .!UI' perturbed inciden' nUll: ~ and M \0 &be nus ~. OQ \he foil 'urlMe. ~. Ia uoa1Jw \haD II I ~

~. - ' 2 - [1+ .I'.tu.' lJ

[ct. Eqa. (II .Ha and bll.

".

... W ._

111 Fig. 11.2.8 it GoWD the qaantit,1 01",-, 6 til cUcult.ted aoool'ding ~ Eqa. (11.2.11) aDd (11.2.8) for ftriou; foil thiclmelle. aDd for p.,Jp.,-O.OI &Dd 0 .1. thal the effClC\ of ~ itl lma1J. : in moo CUM ~tt.ering O&n be neglected and lb• .et.intioD. calou16l.ed. aooording to Eq. (11.2.8) . neD when JJ. ill not negligible oompued to p. . The phyaioal teNOD for the , mall role played by acat tering ~ bI the following : The average path length in the foil .ubtlthoe of Dor. mally iDdd.ent neatnJIuI ia ~ by _ttering while that of obliquely incident DeUtroaa 18 deer eed Ira tim approd11'" , - -,--:-0.:- '-, ::-, matioa. the t woeffeew e&DCJeI Mcbeth.•

•

.,1/ ,

f,/

./

••

U

. . . .. _-----........ D

. . 1l.L&. •

4IJ1

,u,'-

~

o.tJ

tJ.1S

., ....otbIooI4lac . - . ~

In all the eouiderationa to follow, we . hall neglect _tlering in th e foil and Itart from Eq. (11.2.8) for the .." t i.-B Uon.

1U.S. A.etln&.ID. Thenul NNtron neW Our ~ ~. ~ 1 Eq . (11.2 .8), mut DO'Ir be _"rap! oYer the Mae-ll diAriktion of neutron energi e.. I'or th in foll_, naturaU1...e amply hayo

C- ltl'u. .u-. 6- 41...gIT)

v ~e" YI~E.. (O.0263 e't'ld .

(11.2.12)

The tempen.ture-depoadeo.t ,.t.cton of IMIft1'&! foil •• t.tanoe. "fen Ihown in P5g. 11.1.8. For thioJr. foib, _ aha1J. I"llItriol ouneITM \0 au. where the rati o PwJJJ. de- not depeDd OQ the neutron eDe!11. Then

~-

.

-W• ' --.. .,(p.IZj6) Wa · J •

(11.2.13.)

(1l .2.13 b)

I"ig. lU.1lhon ~ .... function of 0"-,1:7')6 in the important.peoiaJ._ of 1/.... bMxptioD. (p.(B)-lll'j). Tbia OUI'1'e wu obtained by numerical integratioD of Eq. (1l.2.IJb) ; the fuootlon ~ hu been tabolat.ed bl Ildmfu

... among othen. AJ.o .hown in t.he figure it. the a ppro :rim&tion

~

..If..(p.('T)6j

(11.2.130)

which repecdueee th e 00l'I'eCt value to better t han 0.6% for p.(iT)6
IeCt.ion *,IM .. a function of tho thickDeel 101' T _M.6 -X:j th_ obtained by numerioaJ. integration.

..

u'.--,_-,_ --,-_

crOM

O11n' N WN"O

/

'r

/

/ , / -> I, / - ->,. """"

..

'1] d'~

_w.. . ............ (U

""-

..

(U

J'lI. ll.t.f. ;;u;;Jl" for l i _ b u loIIo III • Mu· wellIuo

Tho do""" 1 I a I _lIa .....pIo

=-:-:d-

.~! .. 11<01) b Gold ud lD4I 1oUo ili a JltoaQIlIq.,..,.. W1t1l 2' 1:

J'\c. IU.s.

n.!.•• ActlTation .,. NMitroDJ lDd4eot Oil. the F.dIM of tile FoU

Up to UUa point, our ClCKWderatWna bold rigoroulJ only for infinite foilil linoe the &eti....tion due to neut..rou which enter the edgN of the foil b.. beeu neglected. The ClOnt.ribot.ioll. ol .nch Deutr'oM to t.be act.intion of di.c-ahaped foil. baa been e&n:lfully in.-t.igat.ed by HUlfA.. In & tIl~1l l.at.ropio lI "utron field, IlA IHiI. writ-

0 _ 0 (1+ . )

(1l.2.1t)

where C il th e activation of th e infinite foil and 2..,.6

2

r =- ~f..u. 61 "'ii" {1(p",6)-it(.p.6)J.

fll..2.Ui)

_ _ I BId (R _ t.be foil ndiu, cI _the foil t.h!.clmeu) aDd

• •

.d(. 11. 6) "'" f E.(.p,.a lin f') d cot! 'P. 1(p.6 )-

'0[ 1+'-"""-"..

~oli J

"., 1

~" (l_e-·i . )

Bin1 e d'6.

(11.2.16&) (11.2 .16b)

o H£JfI'A linda .4( 1)"" 0.2258; <:1 (2)= 0.0881 and <:1 (" ) - 0,1)236. For foil, th&l.,. Dot too thiek, 1 (p.l5) ia giYQ approlimately by 1(p.6) _ 1_ -~ p.6.

(1l.2.180)

...

....urement of the Therm&l Neutron Flu: with Prob.,.

.

In practioal a - , we can uuall, neglect ..d(".u.l5) eompered to 1(p. 6) in Eq. (11.2. 16) and obtai.D

'....

(

.)

. .. ..Ul.6j liB ' l - ep.6 .

(l 1.2.l 7)

For a gold foll with 11 = 0.26 mm and R _ 6 mm • • = 0.016; tbua t ho correction for neutr0b8 eat.ering the foil throup it.. edgM Y indeed . m&1l, but in preeiaion mee..t11nIment. it cannot be neglected. According to Soc. 11.2.3. in .. thermal PlIutton field • mu. t be replaced by

.- ~r:.k ". (1~ -(I'" -6 P.·) 1....111')1

)

I

(11.2.18)

... t't lhCli'}.ij IiR 1- "3- p. (!:T) 6 .

1l.U. CaJeul,UoD of the A",,",nt Adl,.Uy Taking Aeeount 01 ~.8en.AbtorpUOD Stnee for t.hick foil. the IWltivation i . not uniform ovor the thi ekn ellll Z - d . Fig .n.!.! - and etaee in addition th e probability that .. decay electron lee ves th e foil d llpeoda on tbedist6Dcetot.hofoil .urfaoo,th eapparent p.llCtivity A • of t hick foil. ia no longer proportional to th eir acti vat ion. However, ...t· can Cluily be caloulated. on the bull of th. o%pOnential law of atten uation for p-radiation introdu06d in Soc. 11.1.3. n we tAke the vect.orOu to be th e ...me IUt bat used in &0 . 11.2.1 and neglect _tterina: in the foil .nd neutronll incident on th e edgM of tb e foil jUlitIU..e diel there, then in a monoenergetic field I A •.. :S :t

-.

f-;'" (2l+ II F,(r)f'I(••I1) P'(oe» , )

...

IWt. . :tTI

(11.2.19)

giTN t be . ppaft'nt activity of the upper (in the IOnlO of Fig. 11.2.1) I Urfa.cel . Here T il tbe time factor, . =p.6, . nd 11-",6. The functions f'I (., 11) are given by

f'I (" fJ) -

f•e" .. gl(p. . %, 6) dle

(11.2.20)

• witb ~(p" . e, 6) given by Eq. (J1.2 .4). Th e integration giVOll "1 (',I1) = ~ · (1- , - of) (I +

E,(,»

- ~ !E(,+I1>+ lh .!.j.P. +.-"{E (" - I1)- ln I':; "'}J + I

1

(11.2.21&)1

+ .~- ( I + I~,,) g.(,,) . 9':1 (' .11)-

i ["'1",/1) - {- (I+ ' - ' )(1- 28' ('1)).

(11.2.21b)

I Eq . (11.1 1' , follow. jm metliaWl1 ' " - mvJt iplieatioa of tIM .. an. of ~ (z) gi... in Eq. lll..t.h.I by . -.fh"J,..) uwl i>I~ _ the foil ~. la .ddiWa, the lime Iador . lMf. be iDeiuded u wall .. a facWr of t t.h&$ -.n~ for tlM r.ct lhaI IWf of \he '*-1 eieccroGIlI'I -.itled uJ"fll'dl uwl balf dQ.....1l'da. •

• FOIl' >., .I("-IJ) lIIun be

-.J~ 1II. ill&blllated

~ by - 6i·~-.) wbenl.ri' (z )_

In J&R:JI1[.. Eln~.·LOec1I. Table. cf HieI- l\motiotY, Stuttpl1: B. 0 , Teubner It60,

"'<" fJ) Nld 9\('" p)for gold Nld iDdium foila in a tbennaJ. neutron field are m oWll

in Figa. 11.2.9 eed 11.2.10. Tabular valUM for man,. , and , can be found in AppendU IV. The apparent. act.iTit.,. of the lower (in the ~ of Fig. 11.2.1) aurlaoe ia given by 1

1-

..4.!. - T ~"iT 1'.

1L_. (2 l+ 1) F,(r)(- lt",(" P) lHoot ¥ ).

(11.2.22.)

We can draw the following important ccnclaeione from these farmul.. : a) In an Ieoteopie neutron lAw field, the apparent activity it I.QZJ alwa,.. proportional to the 1/ flux 4J. However , th e propor. tionalit y COfIet&nt dON not AllIS inCI"NMI indeLinitely with in· creating foil th ickn_ , hut (J.Q1() in.te&d gaM through • mad. - - - - ._ .- - mum, which in th e ¢a.8O of In, dium liea in th e neighborhood IJIXJt -- - . - - "rp.II)- -- -of 6-120 mg/cm l , It i. im. practical to use foill! for flux er OJ meuurnment th icker than that wh ich ~pond. to tro. I'll- lU.I. " (" '" ..... " ('. '" lor 0014 IoIlt maximum IIentiti 'rity. Th e decre&lle in fuil llflD.i tivity at la rger thick. ln_ eomee abo ut beceuee in j1-counting (J.47 1- . fi only. t hin .uperficial region (of t hick. fUJI neu ...,I/Cl) i. eUect ive and th e proM- IUJ!J Ibility that neutron. " from behind " IlDo reac h t.h.i. region decreaaM with in. UJ creaaing foil t.h.ickne. . . 0b) Even in weakly anisotropic neutron 0-n field . that can be deacribOO. by only the sr lint tw o terml in the expaBiion of the vector flux in Legendre polynom ial., the apparent ac ti ritiN . nd ..4.!. thow lOme dependeaee on t he orientation angle ¥. ThiI dependence is more etrongly marked th e thick er th e foil ; for very t hin fow. it. can be neglected. In principle, tWa dependeooe.boald ma ke it poeaible to determine tbe neutron. O1lJTent, hu t. th e effect il too Im..u for e:uet meuurement.. c) AI a rull:1, _ are only inteftllt.ed in kno wing the neutron flux and mUl t th l:lrefore eliminate thl:l CllI'Tmlt term from the ap pa rent. activit y mUiured with a thick foil . Thil can euily be done by counting both the upper and lo_ r . idel of the foil and adding the f'l:Ieultl; wben we neglect F. and highe r even terms. t hil procedure giVCll

- --

--

,

.

"

•

-

-

,-

f-IL -

. ,

.. .....'"

- -- -

,-

~

. .. . .. •

.of:

I'..a I ~ ("l A'+A' '"1 + - _-o ".,. -'P' 0- I - °. I 1_' 1'

(ll .....bl

... .moe all the odd teI'UUI eeaeel .

~ fJ- O. "' ('" fJ) a pproach•• f.(p.d) and . e obtain ..4t+.4!. -CIT which i, the _ _ .. Eq. (1l.1 .3).

n ahouId be DOtcdiD ooncJuaion tb a t. Eq.. (11.2.19-22) Wenl derived wit h th e

belp of th e exponential law of atten uation for {l-nMiiati on . While thia law holdl very .ewrat.elr for th e at tel:luau oa of fJ.radiation in thinly depoe.ited abeorbera, i.. integratioo to determiDe the apparent aetivit1 of thiclr.er IoU. y problematic aiDoo in auah lon. .iectroD bl.cbcatter prooeuM occur t.I1M are diffi~ to deeoribe.

n- .ffeok increue the activit1 Ofti' ..bat ",ouk! be e&k:utated uaing the theory IIahtI*Wh1IJ

of thillIOOtion. The f'eoIulw of thi,leetion thua buo

oaIy qu&!j\&til'ellignifioanee; for abeolute meuure-

t

ma w the fJ_U.•t-orption muat. be determined ezperimeutally. W. ahal.I ntllm to thi, polDt in Seo.1U .3. 11.t.8. AdhaUon of. CyUndrteaJ. Prohe

Oooaaionally, we use prcbea in the form 01 long taPN or wirM for th o me&lurement of Ou profilell. Whereu we u. n e.leulat.e th. aotivi ty and &etintion of tape prot - wit.h the form olu already developed - n eep t for the inoonaidorable ecereeucn for neu· tron- &hat enter the Cld.p of the foil - new OOIlIid . eratJoo. are DOCeNU'J for wiroI . Let u oaleuJa.t.e the acti. vaUon per em. of length of aD infinite oylinder of radiua R in an Pol.ropio M aVon field aDd let u Deglect --UeriDa: by the probe eul.tanoe. 1M u. begin by coPIidering an incident Deut.roD ..nth. the flight. dif'eotion (8. ,,) (d . Fig. 11.2.11 ). If t.hi8 neutron

it not. .. !»orbed, it will tnnne .. OigM path of length 1_ 1 ~17:" ., before it bYfll th e oyliDder1• The ..t.orption proba bility i, th en 1- e - E.I(• • pl l.Dd h

C _ 2x R .r.. E.

..!.. 4Il'II

-II

JJ (l_ e-

r . I(• • pl)ooafJ ain fJdfJd" .

(1l.2.23)

••

Here the factor 2111R take. into aooout th e fact that to 1 om of oyi.inder length aorf'MpoDd om' of..m-i we ha .... ....umed th at the incident flux doe. not. nzy Ofti' tb4I cireamferenoe. Ncnr ... write

' "R

C _1I R . . >0/1

JJ

r.... r.. T X. IE• R)

(11.2.24)

I r,.Jt_ .

_ .-)00.'

Z.IE.,R>- ~ (1- . - t - . .. ain (} ti (} d". (1 1.2~) - .,."".,.....,- ,.'Tbil ~ I~ '_ be ob\&lAed .. 1oDo• • : Ia ~ ooordiD&teI, la-- Rr+ ,a- B-

.. nrl'_01

:1' _'_'. ,_,IiD.•• ."._IIiD.'_" II .-,-.-1-0. '_I. I JB'_'+I'ab:l ,-,,, 1-ainl' _I., '

. t.M ~ fw~

_

~

eqoaLIou

oyIiIMIw

at.

I~ ~

u_.

~ ey1iadlt&Dd.

iIl

_troa tnjedooty. 0.. poillt of iIl~ of the Jine and th. ud the ot.her at. n"" 1"_z'+V'+,o_2R a-+,o_ aDdl_

... Clearly X,(E. R) 11 the abeorption probability of neutl'On.l with an ieotrOpio diatributio n of nlocitiee inoident on the infinite cylinder (linoe "R ~f2 II the n umber of neutrol1l incident per MO OD a l -cm length of the cylinder). Th en

~

X, (L'.R }= l -

a..a _ .

, . _/I

f J,- ~..."". cx:.61hJ. 6 d6 d" .

••

By introduction of the .ari&bI8ll, _ obt&in

.

. __ 0DlI~

-

l-ela,,_I.

aDd s _

_ 008'

•

•

l.otegntion yielch (d . Cd., D. H onJUn'. and Pt.t.or.u.)

X,(I .R)- ";" (X,R)' {2[X.R{K1 (E,R)II (E. R )+

+

K

IE R) 1• IE• Rl) - 1'+ •• J

_

_.

"1_lin"_' " ,

a ,lI, •f Vl..'"- ~ f ,-Ir. r' t'r'- ~ .

X, (I. R)-1 - ;'4

(11.2.26&)

K , IX.R j 11(E. R) _

we

(ll .2.26b)

I

EI R

(11.2.27)

- K, (E,R)11(E,R) + XI (X, R) I ,(r , R)} where I and K are modified B -1 funotionl of the fif'lt and aeoond kindt, , . .pectively. X, (EaR ) is ah01l'D. in Fig. 11.2.12. Th e funcUon appro&ehel unity for large valllM of E.,Ri for .mall I, R. I.• .• for thin

•

cytindera. z. (L'.S).... 2E. R and thu 0 _ 1IR'E..", tl'. Th e qu&llUty l - X. (L'.R>! 2E.R ha. been tabulated by Cu_, p a HOFnUlftl' . and Pt.t.o~u:.

.

I

/ /

I

,, /

~ ., ­

I

I

I

, r

I

I

•

s

,

Since we w-ly know the abeorptJon probability for Ilab. and cylindera. for the aate of complete_ _ ahaJl DOW gin it &l.o for .JIb-. We can derin the following formula for the act.intJol:l of • ".phece probe" of Miua R in &I:l • tropic neutl'on field in . m&IIDeI' eimilar to that ued abo•• for oylinde re:

C_ b R' ?

..r

~ J J (l _ .-.z:.I(·»ooe" .m." li''' d" .

••

(11.1.28)

toO

Here C i8 th e toW acuvation and not the activation per em' . Now 1=2R 001 {} (d. Fig . 11.2.13) and tJwefOl'8

X.

C _ :tR' ' 1:.--cfI p. (E.R)

( 11.2.29)

with the abeorption prob.bility

".IE..R)"'" 2:

DI- c- u . a.• .,

• ,_u;..

= 1+- r ;ll- -

00II {J Iln (J1l{}

( 11.2.30. )

l_ i - IEo _

- :l ' (E~ ))I - '

(1l.2.30b)

Fig. 11.2.14 Iho.. ,.(E"R). For E,.R
"II~ - ~

"

.

I '" ''If-H -/--1-/ lJ ftt.

'

. '-

...

'1 •

l.J.S

-

i

~

lI.Ll.. no. .~ ~tJ III. ~

"

.. _I.""......

. rt1.~-

"'. ll.' .U. ... _ ...-", doI . ~ ...-

Inf

_ I II

...-.

, _. ... ........ I' - lol l....

0)'---'

aDd .phe~ (gi...en by Eql . (11.2.6d ), (11.2.27), .nd (ll .2.30 b). rw pectiv ely ). The

.beciM. i8 the quantity IY""...olume, S = .urlaoe) ; ~; ia 2d for aI.t... 2R for cyliDders and t R for _phe,.. It tuna ou t that with tllia ehoieo of ab.ciaa& the tIu'ee CUM' . d o DOt differ hom one another very MUch .

!; r.

11.3. Tho Theory or FoUPorturb.UODB In See. 11.2, th e probe .cti.....tion or activity wu related to the incident neutron Ous:. It ... &IIIIumed th at th e introduction of th o probe did not chan ge the Olll[, Thil. would be 110 if there were no "b6ckaoatterinl" , t.e., if each neutron OIl. ita way from birtb to ab-orption would only crou the I Urfi oe of the probe once. I n ptaootioe, ho.e.-er, a oorWderable baoluol.tterinl prob&bility emta, and neutl"ODl that a ... abaorbed in the probe are aMent in the ~ttered nUl[. The actual &ctintion C iI therefore IDl&1kor than that lCti.-ation C. thlt wou ld OOC\U' in the Ilnperturbed field. Let UI le t

ce-o - x.

~c ' -

(11.3.1)

.~ X. ., the "act.i.-.tion oorrection" . Xc iI. function of the propert.iM of both the probe aDd the IUJ1'Ound.iD1 medium. In Sec8.Il.3.l -1l .3.3 we Ih&ll itudy

oaIouIatinfl: j1,l. I III s... 11.1.1 &ad 11.3.2. .-.n.- eaIouIationi ill ---aetioo f~ will be ClOD' lidlndl iD &0 . l U .t, thl .-U'kHI. r-we. .iII be CI;lmpared with one Mother. AoUn /Jon ....riODI D1ean1 of

~

in t.b-a.I neutron fieldt will be OOftIMienIll in Bee. 11.3.S.

The Theory of Fo U PertW'''t~

'"

The following airaple consideration, ebcw t he qualitative connection betwee n the b&cbcattering probability. the at.orption in the probe. and the activation

correction. N, _F ~ neutron.

croelI

a ,urface F per eeccnd in a neutron field .

Of th ese, ~ J) Cf'OB8 for th e rU8t time an d N:,!)= N.- Nd ll for the l1OOO00. third, etc . t imtl. If p i. th e bAck_tklrillg probability. Le.• th e pro1ll'bility th at a IH'U l ron that h.. er'Clllllld th e .urface once wiI1 crwa it again after one or mora collillionl,

"'.n

(11.3.2.)

-.

N._= "" NjU

(11.3.2 b)

Let lIS now replace th e lurfaoe being ec neide red hy a rea l foU. The numbe r of neutroJui that are incident on the foil per eeecnd iI N _ NJIl Nl2). The number of neutroftl incident for the tint time iI naturally th e ..me .. without the foil. In oontraet , NUl:;;; NJII eince only a fra ctio n of the neutrons incident on the foil can penetrate! it and become available for back.acattering. If WCl &8IIume that t he distribution of directions of t he incident neutrons is a1waYI isotropic. th e prob, ability of penetrating the foil il l - f.fp. 6). Thus

+

NUI _NJU(1- ",fp.6)]p+l'o: U[1- ft(p.6») · "+ . .. N.fIl

(I - ...

c,.."l1,.

"J

(11.3.2 0)

_. • 1- 11 -· ". "'. <11'" or

(11.3.2 d)

It then foUo.. tha t

, '+-, '

(11.3.2e)

)1'. _ I: , f.fp.6).

(11.3.2f)

-.

9'.(.ao.")

or with C/C. .. NIN, that

The activation oolTllCtion iI proportional to the ab lJOrption in the foil and ill.rger tbe larg er the br.ebcattering probability. Beaid. the perturbation of the aoti,...tion . which we mutt know when we mak e ..b.olute m~nt.l or wben we W"iIh to eompano Ou. meuuremenw made in diffennt T1Iedia, _ oceaaionally wMih tD know the denaity or 00. per· turbationa. U 1',1). feprNenw th e unperturbed Ou., the flu. 1',1) (1') in the oeil!:hbor. hood of the foil illRD.aUer than 1',1), owing to the additional abeorpt.ion. We call )1'.(1') _

tIl'. - 41(r)

$ (r.J

(11.3.3)

the " flw; eoreeeuoe". ~ _it the average Ou on the foil . urface. We moo know • • (1') in order to lIIt1mat.e the mutual influence of HverrJ loU. u poted llim.ultanec:nwy. In 8ec. l1.3 .4, we ,ball familiarize ouneln, with -ame relultt 1 WedetermlneC, _ O(!+ We) from t.h.uperlmeoMI nlue of 0 and. n lueof-.oalculat.ed by the met.hoda of th. tenioD.

for _.(r) , the flus 0CIft'eCti00 x.(O) on the foil aurfaoe, on the other bud, can again be NtUnated hom timpJe ooo.tiderat.i0n8 Iimilar to those we used abovo for tho activation conectJon. The flU.J: at the upper .mace of a foil is composed of DeUtron. from above that are directly incident on the foil and DeUUoU from beJo. that han peoetrated th e foil. AOOOl'ding to the conUdentiou given

abon, the CIOlItributioo 01 the

ftlf1!l,el'

to the flllX iI -:-

I :~' that of the latter

~· 1~"; (l-f.(p.6). The Ow: at the foil.mace iI thua pYeD lII(r.) _ I:·... (1- {

by·

f .(p.6») .

(11.3.4 a)

". 1+ .... " lr.1 col + -.
(1l .3.th)

(11.3.ic) B-UMI of the .. Madowina: effOClt" of the foil. the Dux oorrect.ion at the foil nrf_ iI al..aY' gre&ter tha.n the activation ~ D ; in faet. there ia eYeD a flu pert.orb&t.ioD ..beo the baeboattering probability vanilbee.

n.3.I. Calelll&t.loD of the Aet.l....tloD CorreetlOD with Elementary Dlffullon Th~ry

Ltt QII COIIAider a dieo-ahaped foil of raWu R Wg. eompanld to th e tnn8port

me P"h in the wnoundiDg medi1UD.. In the interNt. of aimplicity . lot. ua tUe uu. nmoUDding medi!OllD to be infinite. In the al.eooo of the foil there iI a homogeneou. DeUtron OU.J: IJ). everywhere . U 11'0 denote tho pert.urbed eeueroe au bylJ)(r), then (r) = (r). Aooordiugto elementary diffuaion theory, tho nux pert.urbatioo.d If.l(r) ia given by

meut

I

.d1J)(r) = "'iiD

f

Q(r)

"'-"1 .----. 1"_'1

(11.3.6)

d B'

..here Q(r) dB' ia tho Dumber of DClUUoU abeorbed per IeOOnd by lobe .urface element dS' of the foil. The iDtegrat.ion OlIteodi over the entire foil (d . Fig.II .a.I) . lnRoMd of Q(r) ... oa.o al80 UMI the "1ooaJ " acti1"&tioo. e (r) _~ )Q (r) ; in tennI 01 e {r), e ia pYeo by e _/C(r) 4B'JdP. Wo mut DOlt" • . . . - C(r ) in W1U 01 the perturbed DeUtron field at. the foil aurf.aoe. w. tUe .. the Teotor flu of the perturbed DOUtron 6ekl the eIJll'l*ion I

,

F (r. O) .... "ii F.(r)+ i"i J;, (r) 001II. The a.ngIe '" betweea the field am and the DOI'Ulal to the foil now d.pend. on poIIit.ion on the foil wrf&oe (el. Pis, 11.3.2). Since .... erythitta: iI.ymmetric in the foil aurfaae, ... CUI limit our ClCNWideratioDl to one aide . Neglecting neutroDl that enter the foil t.hrough iu edgM and Iporing IOatt.ering 10 the foil, we han C(r) -

---

11I_~.

1'1'-:

I'-t

l

..

".lp.6) + I

:

J; (r) 1ft (p.,6) 001,,1

_ (r.l_ I+ ,;.' · ·! U+ E1 t-. 611(d.

l~

011 Po 101·

(11.3.6)

'" f t (p,,6)

E112

=

f•1'(1-

•,

, - ", ,,,) 4' (11.3.7.)

T -2E,(p.6)

- } (1- ' - .... ·) +

~.! (1-

".C}l.6)) .

When /1.6<0.3.

tF: (p.6) ....T1 9'e(p.6).

(lU.7b)

In contrut to Sec. 11.2.I , here the tmn cont&ining 1;. dOCll "'" Yanilh ainoe the C1UTent ia directed at the foil fro m both .idee. U _ no.... note that J;, (r ) co." •

,.;"" dtWdiM J'l&,ll.LL

~ ",u.ftu""'lM__

J'lI,Il.U

. W C_ tooDl

o-U'J'''' _ ...... _DOW.

tho normal aompouent of the C1UTeot directed at the foil aDd that the totaJ DOI'maI CW'nlOt 21;lr )oo." diz-ected at the foil from both ~ eqaala the Dumber Q(r ) of DeUtron. ab.orbed pet' unit Itlrfaoe. we C*Q write O(r )- 7 ; . U we now

exp~

'~"I . 9't(p.6)+

f C{rl9'~ (p,,6).

(11.1.8)

Fell') in tonal of 4>, and LlIJ) (r). th en it fol1o_ with the help

of Eq. (11.3 .5) th-* II.- . , 1 a lrl - ~ Tf'w(p.d)- i'iD

JCIYl1". -

- .... /£

1

48'1lt(p.6)+

,

1

-

(11.3. &)

+ ... O(,.)9' ~ (p"d).

Now Cp."Jp.J ~. ,.(p"d) ia the aoti.,..tioD C. that woWdooourin the unpect wbed ne utron fiefd. Thtll'Bfore the aoti....tioD depreMioD ia ginn by 1

0,- C(r) - 'sicli

J1

O("')c-I. - " I£

By int.egntion onr the foil

""""""'"

fJr(

"_ r'[

.urf_

And di...woll by C ... obta.l.D the aotintUI

C('I"').-tr-f"'ll! . ~8.8'

1 I) ~:. .., .. -~ - h D Cil'}ilS C -0

3

d S' ",(p. 6)- "4C(!")f~ (p.6) . (11.3.10&)

s

~(p,,61- -(,.:'(p.d). (ll.s.l Ob)

2M

HeuUftlID6D t

of the Tberm&l N"",tron FIliI: with Pro bell

We C&n obtain an appronmate aolution to Eq. (1l .3.10b) by neglecting tho position dependence of the activation on the foillurlace; then

x._-.,

-D

' f 211'rdr f r'dr' f d'f .-' ''._./L y '-.+ '''_'-_ B

t.

.R

911(;.1.6)- . .

,II" 3

,",+r'"-2,"_ 9'

•••

- -,f , t (,...6).

I

(11.3.10.)

In general, the integral CIonnot be oanied out (lJ:plicitl y, a nd fot' thill. relUiIOU we oonaider two limiting cue8. The fint it th e O&M B :>L . Here becaUlle of th e rapid decay of the uponential. the integral

• fh .- y"+?' -i" '_iiL f• r' d ,'• d'f --y,o+ 2,,..._,,. r' "-

11 independent of position, on the foU surf&oe and can be replaced by ite value b:L(I_e- R1L) ..... 2nL at r =O. Th en X. -

I 3 Tl L D- , .(,u.u )- -j

' / ..

I

9':i .,..u).

(11 .3.11 1.)

The eeccad J.imjting ouo hi the cue R
.R

t.

-'... f2n'd'f ~uf ·_y;r· '+"·_2rr'_" ~_d" __'_- = 2n R W • • •

with

W =3~ =O.S6."'. b&oomflll

, R 3 x. = -" 'Jj Wfo l.u.6)- -, -"t (p,, 6j.

(11.3.11b)

An interpol ation formula that h.. the correct valUelI in the limite R> Land R < L and iI probably a good repreaentation in between iI

Jf.-.:-{{; (1- , -JrItIL),.{JJ,,6)_ ,r (JJ,,6)} .

(1I.3 .11 0)

WebavereplacedDby A,,/3. Forpraet1oal Talue. ofp,,6we ean applyEq. (11.3.7b) to tr6n8form thiJ reeuit into

""='.3{ - 1,-; (l - e L

- JrR/L

I}

)- ·If 9'o{JJ.6).

(1I.3.l1d)

At R approache. zero , Eq •. (11.3.11b, 0 and d) predict a negative foil correction, "hieh iI natura1ly DOD8eIllll!. TWa abaurdity oocura because our reeulta haTe been derived 'lrith the help of elementary diHuaion theory, which iI inapplicable in tbis limiting C&lMl. ]t will be .hown later that Eq . (I I.3.11 d ) it quite .ocurate if the in the braoea it dropped. Farther oalculatlOM of foil perturbatiolUl baaed on elementary difhuion theory can be found in B OTln:, 111 TrTn.:I, and in VIOON and WIBTt.

t

11.3.2, TranIpOri-TheonUe Treatment 01 the Aetlnt10n Correction

SKY1Uf. baa oalowated the actintion correc tion for disc.shaped foill by me&llll of an appronmate IOlution of the transport equation (fint-order perturbation

The '1'b«lr7 of Foil P"rturbootw..

theory ), He findll that in an ieotropietJly ecatwring medium 1 (11.3.12 ) Here l _I/E, ie the mean free path in th e medi um , r ie th e ratio lIl ,. R ie th e foil radiua, and L ie th e diffusion lengt h. The functiona S(z ) and K (%. r) are . bown in Fig, . 11.3.3 and. 11.3.4. Io'oumall 1'& 1_ of %, ,

1

8 (%) = ; ,. %- 8" ~ + while for large . :

,

1

"" ~- I";z'o + ...

(11.3.13 a )

• (I + z'I + ,..).

8 (%) = 1-,.. "i"

I 1j

The term. f K l%, y) is a1war- < 0.1 and. .. a rule l'epre.enUi a negligibly am&l.l ool'nlCtion.

,/

/

(11.3.13 b)

u

0/

u

I

/

u

, ,

•

..

,

•

I

1-

I

P\l. 1I.u. n. ..." . . . - .fld{wIaon . _!:)

,-'"

/'

..

"'-

~ •-

/

i

_. %:'

~

~ /'

• -; - /

II

,

II

y

..

/

r- y- ' /

/

l---- ..-

"-

._ -

, ,

.

• "'''

<-

hi. 11.1."

n. "-~ I

.. .r- -;;;' J .. ,., (...... - T

", " •

II

R-

"

UJ

,, ~ U

J'la.1 l.s.L ... _ ~ .. _ )IOftarb&~ "~ "r~._ '~ tMcIo' .

-.,111-1.111.>,

tMcIo'

e-- .: -., (U....U . L

w

"r Cht c.n. ,,:

.... ~ - - . . . 0...0 , : "'UUUI

~

In Fig. 11.3.6, we OODIpue Valli. of -.t",(p. 6) appropriate to . a te r (L_ 2.76 em, lv - O." em ) that ba,.. been caloulated with Eqa. (lI .3. lI d) and (11.3.12). In oompuiaon, A w.. replaoed by .a.. in Eq. (11.3.12). We _ th at diff ueion theory alway. "vel uuller valUel tban SltYIUI. '1 tbeory, but that the nriatJon of HJ'h(p.6) with increuing R/~. ill very aimilar in botb

uu.

I

s.'RIU',

~

nnlt.I had

~

form ill

hI RrroirB and El»UtlGJI.

u

original workl bere we _

tbe notltaoa

theoriM. Thill ia euily undontood if we I10UI that fOJ' the ranee of foil aizeI important in praotioe 2R/L::;1 . '!'hu according to Eq. (ll.3. ll d )

~.:;'6l

t.

- ~ [1--;:. ~ + ...]- T

(11.3. 14&)

"bile aooordiDB to Eq. (11.3.12)

_".(.1.1.") ..- - .!.11 "!'[ l -'!!!' _R. + ~ 16 L The tint tenna (i.e., the expreesiolll

",1- -!.. 2

K

(1l .3.14b)

'

~

tr [1- .. .]) differ only Ilightly for III1Ir.1l ValUN of R/L. They l'epnllent the .. ..ymptotio .. contribution of widely ..panted foil element. to the activation U pertarba.tioD, ",hioh ia reprodlUled quite .•; wen by dilfuGon theory. The _ntial difference. in the DOn...ymptotlo temY

""

': •

"

,

,

J

...d -

•

I

an.-

f

1/V

f;o

JPV t: l- f!

S

..... 11.......... _ ~ .... ....w..JI3cIJtdp ...... loll ~ fIX Ieftllllt IoUo \II .. ~ _~-.4hl.

.1/ Fr

I, ,"

r&L

..... 11

•• If

' . ~ OIl

eoJ4 _tao_leo,

~.'D~.Nl4OUQ ...:

_ _ aq.(U.l.lt)

and IE. We obtain quite good agreemont bet",een the re-ult of diffuaion theory and Eq. (11.3.12) if ""' drop the additiTO in Eq. ( l l.3.1 l d ). The ooneeponding CUl'TII ia alIo Ihown in Fig . 11.3.3. The nlidity of the BIl:YNO theory for large foll radii hu been studied by RrraJrm and EDBIDO., who obtr.ined • very e:u.et variational aolution to the tranIpOrt equation for an infinite foil in an ilKltropiotJly IO&ttering medium. In Fig . 11.3.6". IhcrIf tcJ<". the ratio of the Ritchie.Eldridge activation oorreotlon to that obtained from Eq. (11.3 .12) by letting R _oo. From tim curve "e _ that It:," 11 IOmtnrhat peatei' than "., partioularly iD atrongly .b.orbing media and for thick foill. In praotical cues (Lll> 6, flwc);SO.3). bcwevee, the diHe1"MlClll .. lIDa1ler than i5%. When the foil radiu .. finite RrrcBD and ELD· :amoz reoommend iD pl&oe of Eq. (11.3.12) the rolation

term,

a1".,..

(11.3.16)

"bleb p.... the right ,..aU in the limiting cue R-+oo. ThiI-eaIled. "modified 8kpme formula" .. nry widely uod.; tbedifferenoe bet_n it and. Eq. (11.3 .12) it ...ery amaIJ..

The Theory of Foil Perturbation.

'"

D.6.LTOl'l and OllBOIUI have oa.rried out an independent traneport-tbeoretic invelltigation of probe perturba.tione and probe acti....tion. Theae authOnl ltart from the integral. form of the traneport equation in an infinite medium (d. Sec. 6.1.3) in which the Dux before the introduction of the detector iI 4', . They them calculate numerically the Dux ~ averaged over the entire probe volume. For a foil, c=~~lJ . In this treatment, the Ilelf.lhieJding effect that wu diIouMed in Sec. 11.2.1 and the activation perturbation are taken into aceount IUn wt&neouely. If we intend to compare t he resu.ltI of DUTON and OsBORN with thoee already obtain ed in thie eection, we mWltexpreaa ~/4', III foUowa: ~

",0-.") 1+ ,

'1>, "" - 211, 6

(11.3.16)

1+,.. .

In Fig, 11.8,7 w• • how 1 +.... for lIold foUl of Tario ua lili.ln water (from H.....) i the pointe have been obt&ined with the belp of Eq , (11.3.16) from Ow: ratiol , calculated by D.6.LTON and OSB OIUI. AI The eclld curve W&II calculated aooo,rding os to t he modified Skyrme theory [Eq, (11.3.16)]. The lame values of the ~ scattering and abeorpti on erose loct ioDl of gold and water were used in both ':;f calculationa. In t he Deuon-Oebcm calculatiol18, the aniaotropy of _tter•• OJ ing W&ll taken into account approxi_ mately ; in analogy , A,, =.l.!(I -.u) Will I'JI, II ..... TIM f1uooUoD alJo.' ) used Instead of 1. in the modified. Skyrme calcwationa. With the azception of the Imallest foil radii, where a ma:rlmum deviation of 20% occurs, the agreement between the modified Skyrme and the Dalton-Olbom caloulatiODl ill 8atiefaetory . We conclude from theee coneider· ationa tbat in the range R,;;:.J.,.,., Le., in the range of practical foil me. in hydrogenoua media, the modified Skyrme th eory with 1.=A" iI a U8eful ap_ proximation ; thil &8IIertion holds for the ume being only in a monoenergetio field . Since Eql. (11.3.12) and (11.3.16) differ only IlightJy, we can 110180 WIll the simple Skynne theory or the relIwt (ll .3.11d ) of elementary diffUlion theory, although in t he latter ClIoIIe we mWltdrop the additive term i , I n the range R<~ .. whioh iI the rangll of practical foil Bisel in graphite, beryllium , and 0.0, there ha.e been .. yet no comparieon of the modified Skyrme t.heory with numerically caloulated Dalton.Oaborn valuee. However, in thia range a complete &Glution of the tranaport equation iI not neoeua.ry for t he ca1owation of the activation correction . Rather oonaideration of individual colliaiona luifiOlll:l. all BoTaK hili already Ihown . According to M:USTKB, when R<J.. in an i80tropically scattering medium

I

:e

,

•• ,

Po'-

R

Ie. "'" 1i • ~ ,\ (p.lJ) 9't(,u.lJ) .

(1l.3.l7)

The function A(p.lJ) ill plottod. in Fig. 11.3.8 ; it3 value at the origin la '\ (0)_0.828. WhIlD R< A.. , ""ft(p,,~) dependl much moee Iwngly on th e foil thlclmeu than in the Ritchie- Eldridge oalculation for infinite foibl . In Fig.I I .3,Q we companl X. BecIlortolWIrt&, N.~u.>a l'!oy"

17

SOl

oUoa1Ic.ed aooof'diDc to the mod ified Skyrme theory aDd IIoOOOf'dina: to the MeiM,r theory for fWI 01 ,.1rioolI . . . in graphite. The lleilrter theory alwar- pVM the IID&11er Tal_ 01 JI,. but for thin aDd.hen RfJ.~-O both oalculatiou agree . AA we Ihall IM later, Meiner '. theory reprod uOM the aI~ foil pertu:rbatioDII better than the modified Skynne caJ.CulatiODI " beo R< J..r. ;.:.. Comp.moo with the difhWoa.theoretio nlRl1t lot Mao in the ranr of amall R/~ ill DOt. -' meuililgfuI linoe "beD R<.a.. eiementary

,on.

... ..."" I ...

L

~

-......,........,·P.....

/

AM

f,

~

11.1.1. Aethatloa Cor!'eeUOD In th e Thermal Thill ~ it I / / Up to thia point, ..n our rNwtl held only in .. monoenerget1o fillld, and we IllWJt now / /. in't'ediiate how 10 ."enge them over the ~ ndocity diltributioo of die DeUWna. 111 ordm' to aniTe at u. uact &nn'el 10 thiI quemon, ... IDDA 101... the tnn.pon eqUlot.ioD of the foil perturt.t.ioD problem takin8 ., AI into aooount eoergy ocbange between the ,..dY1I. 11..... . _ . . . - 11II """ ~ thermal neutron. aDd the ICIottering eeb~ u.-,. aad Cbtor7, . folI There by been no euoh treatment ~ I' " nuu)J; -_._- _ _ lkoIU""Tl) to daUl. aDd we mtiA be content with 80me mOl'lI qulitat.ln &rJWDNlt&. ID Eqa. (Il.S.lId. 12, 16, aDd 17)... II the prodD~ 01.. flu porturbt.t.ion and u.. at.xplioa probabilit1 ,.~6). nu. au pertw'badoD. II DClnUliIed to (.....uTe) anit «Im'Oe~: ita ~ ..... the.~ proemer in the probe. Let 'I2).(B) , the Dntroa IpeotnuD in the 1lII.pertarbed fWd before in . tzodvotioa of the foil, be .. JbnoeU diRribatioa M (Z ) with the moden.t.or temperature. 'The enetK1 dietribution of the 80~ of the Ow: pert.urbl.tion iI the ume .. the energy diatribution of the abeorption prooeIIN in the foil: &II .. rule, it ill DOt. .. :MuweU diltrilxlt.ion. For uample, in thin folla with l /...bIorptioD it Ja - JlIZ>l}'B. WIUeh eDerI7 .peotnam the flu perturb&tJoa h.u depemJ. on the tbermalhatJoo prvpertJ. of the IIlOder&Una medium .. well .. on the Dumber of "on..... . Dfitron espedeoc. beforel\ ill -uered b.cJr. in\o t.be foll. U thiI Dumber 11 1arp and H \he medium \hermalizel Deutl'om efficiently. \he Ou penurhaUoa will hne . Muwell eoerv diatributioD i aud m.te.el of ,..{jI. 6) we man ue \he . . . . nJue ~ gino. in Eq. (11.2.l3b) for the ablorptioa probability. On the o\ber baud. H the baoboatt.ering t&kel plaoe In .. lingle oolliIioD (R
.m

II

hfP' /.0

"" ~ ,

v:

--

.-

_"RII. .tan...

1 / - - . tb& u..pdoo .....bIlity ".!.~; b> &II pnotlO&l _

• &ad ..• ~ . . ........ pn>b&hlIit, ohooId 11& ..........(p.6)

. th.

". Since the diHerenoe betweeQ ~ two limiting ClUN ia not very Iarae aod .moe the nux perturbation in maR pncticaI CUM " ~11 ret.benn&liud, we IUggNt that the value " . (p-d) given by Eq. (1l .2.13b) alway. be u-I for the at.orpt.ioo

probability. For l /,.abeorberl that are not too thick ~ in thiI cue ..leo

ltu:3J9'. tJl:6j-

~ 9'.(.u.(kT)6) j

Yj. A(p' (iT)6) 9'-(.u.(.l:T)6) . which il worth

DOting when MulTu 's Eq. (13.3.17) ia being~. AD additional qUNtioD it how to &'t'eRge the ....Jue of -J'Ft(p..6l. OJ' what iI th e MlDe queetioo••hat ....,Jue of the tnlwport meaD free path to tile in the formulu

for thi. quantity. Sinoe J,.. depend. weakly on "DerBY in graphite &Dd beryllium, we kno. without any det&iIed mTeeti,:ation that we O&D tile the D.lual tberm&J &'t'erage of..t., in th_ Illbl tao- . In hydrosen()QIII moderato,. the energy depeedenoe of At. fa much Itrongerj l.n ".ter in particulu .t..(E)-yE (01 . 800. 17.U). NOlt' for practical foil mca.urementl in hydrogenoUli media. R:>..t..., Le., mo.t of the bacbcatulnd. DeutroDl hav e expcrienood ee...enJ oolliaiOWl &nd. are eorrectJ.y d eeoribed by diffoBion tbeolY. In addi tion . th e energy coupling between DetItroIm and hydrogenou media it TfJrJ atrong, and M UtroDli from the 1OUl'Ce of the perturbation &l'e .err rapidly therma1ized. 'Therefore, one eM tile fOl' uu. _ aI80 th e u ual. thermal n enge of the traDlport mean free path in t he cal. of ><J.. fp.6 ).

"""'lion

11.3.4. c.IealaUon of the nu PerturlJatloD Let lUI oonaider .. diac..beped foll in aD infinite medium in whiob the flus. hu the bomogeDeOlll ,..1ue 41, before the inUoduotioD of the foil. Let polar ooordiDate. ' ,1 and try to calculate th. oorrectioD

_."_,1 _ (

0

)

au:

",-"(r- O,.} " (r, ' - O)

on the foll &XiI. Uaing Eq. (11.3.lS), which il4I. we h....

-

gi""

U

InkedUtle

4:1 " (r- O,.)

" (r,'_ O)

the diffuaion.theoretio nloe of

(11.3.18)

and therefore

(11.3 .19)

EnotJy .. we did in the calculation of the lWlti...t1ora OOI'Z'eCtion, W. DOW eeplsee the poIIitioD-dependeoi flu: ora t.b. foil nrfaoe by ig ..~ ...1110; b1 iDtegratiora we theD. obtain L [ JJI+iI"] "~~:"';" •• (r - O, I)- .• 1;' ,--tL_. _.!.. J,

(I U .JO)

...

w-r-t of u.. Th8l'lD&1 N"vl.ron FlIn: witb

Prot-

WhonB>L, (11.3.2 1)

eed when RR,

I lit . ... ...

J(I , - . /1.

J.,.. - . -

.. 1I<.1l1 I-ht1l<. 6' .

The Ou perturbation al tho foi! aurlaoo ill gh 'en

(11.3.22)

by

I L _ ~l.\ . . ~6) ••<,._ 0, 1_ 0)_ 'i" ...~ (1-, I ~t f1 Uo. 61 .

(I U .23)

U Y O 00IIIp&re tJ1iI, format. with Eq . ( ll.3.11d)....hioh __ aI.o deriYed with elomenLuy diffaaioD theor;r. we find that for am.up.6 Md with W _ I,

If.lr - 0. *- 0)

-,. , +I". 1.u.6).

ThiI r.ult iI analogoua to Eq . (11.3.• 0), which . u derived at the beginning of Sec. 11.3 in an eDtirtlly diHerent way.

11.3.6. AdlfaUOD Perturhatlon of Tape ProbN 10 far in thia I&Ction hold for di8e., haped loU., whieh are the kind ffi08t frequently used in prattiC(!. Occuionally aquare All the considera tions adva noed

toile are aJ.o uB«l ; for them the foil perturbation. prob&b1y about the .me ... that of .. di&c:-ahaped foil of the ..me area, Le., if cJ it; the length of the foor. aide, ...e 11M the formw.. for .. diac foil with R - a/ fi. FOI' upe prohN that are much

longer in one diNction than in the otber , lUch a ample IkIaptation iI not pc-iblo, and we ah&U try to _tim_toe th e perturbation uains: elementary diffuaion theory. Let G be the width of the probe ; let it. aength be lur compared to L. Let the neutron field be OOI»t&nt. e"erywhere befOftl the introduction of the probe; afte r. ward. let ua uaume tha~ i~ dON DOt yuy in ~he long direction of ~he foil. The calculation of ~he activation correction then ~ exactly aa in Sec. 11.3.1 exccpt that in place of Eq. (Il.3.I0bl we obtain

•• 4~DJf CI"lK. (~-!1)d~Ib:' ". _ C'CC _

0 0

".(p. dl - {"f Cp.6l . (11.3.24)

•

I C I~l d~

o Hen

i~ K.(~~). the

now ap1n replaae the probe. _ obt.aiD

diffuaion kernel of a line

poaitioo~pmdent

8OW'OO

(of. Sec. 8.%.%). If _

actin tioD b1 it. aTonge 01'V the

•• ..- . 1C~Df!K. ( I" L" )cI%d~ 9'w(p.6)-

{pt(p.6 ).

(11.3.25)

Now we .-ume further that the width of the tape • emaIl com pared too the

diffuaion lengt.b ; then K.(l~) IIIIf 10 oon.etant, and the int.egratioD. Jo-dI too Ne D<

.:l;

(In ~~ -

"1 1"'lL rl '

.~-) f.l,u.6) -

where y -0.677 • E oua'.

.}

"t (,u.6).

(11 .3.26)

un.

'"

cue. In analogy with the lIituaUon for di.8o-abaped rolla, tho formula derived from dif. fuaion th eory ought to give the right value for the perturbation if the term !f:f}I.~ l in Eq. (11.3.26) is dropped. Eq. (1l.3.26) Ihould .180 permit roagb eeUmation of It. lor Q. infinite cy linder (wire probe) if II is replaoed by nR eed ~ (,u. 6) by x. IE.R). No t.ra.Mport-theoretio cUeu1ation of th e perturbation exiata in

11.4. Experimental Studle. or FoD Perturbation. 11.4.1. Method' of Beuurln, the AeUvatlon CorreetioD The only pr'O
Nevertbek., moet e:llperim&nte for the determination of ... are dono indirectly and without the UIfJ of a cavity. In th ese experimente, &lfJriN of foilt. of ..rying tbiclmese are ,uooeWvely irradiated. at the ..me place in a medium and the 1p&Ci.fiOactJ.vity. wbich is proportional to the lpecifio activation C/6. ill deteemined . Tb_ meuuremente determine the ratio 0"10"(6-0l-~/t/JI from whieb x. can be calculated. Uling Eq . (11.3.18), 0"(6 _0) ia the epeoifl.o aot.irlty of a foil 101' which there ia no perturbation - neither ..If.thie!ding nor an activation per. turbation. 0"(6 _0) is obtained either by meuunment with utremely thin loill (p.6 <10··) or by e:lltrapolation 01 th e nhll!lll of 0" (6) meuured lor the thicker loill to IoNO thiclmeM. Rax».u.L and W aLZD. hue ginn .. epeclal procedure 101' the determinatio n of C' (0). In oontnlt to the cavity method. in thiI method we moat m _ the tzue acti"rity. Le.• we blliat count tho loil'e ,-ctmty or eliminate the p-U..beorpt.ioo by calculation or flVeD betw by meuuremeDt (el. Sec. 14.1.31. In m.JUn1 eleen meaaunmeote of th e aetiyatio n colT'eCtion, . . moat .c a.m.nr tho irradiation appuatWl that tho unperturbed Ow: ~. dON not. 1'aty DOtic6ab!y oyer t he face of the foil. The neutron field Ihould a1llo be iIotropiej in .,.. . . han to. we CloD a1wa)'l utiafy thiI requirement hy auitable rotation of the probe. The DeutroD epeot.rwn of the UDperturbed field ahouM be .. M.uwelI d»trihoUoo witb .. temperature u cIoee u poaihle to the moderat.or temperature. Bomettm-.

e-_

it may be

D~

to oheoIt thiI point by • neutron temperature meuurement

(of. Sea. 15.3). If the epectrum OODtaim an epithermal oomponent. it mUllt be eliminated WIiDg the cadmium differenoe method (of. Seo. 12.2).

11.0. J.etintioD PerAlrWIon in Graphite (S
-,

t

Lf'

-,, II

.,.

,,\ ~r

11

.

It

,,

'"

11.4.3. AdlYatio. PerlnrWIon III Wat.er aDd Paraffin (S';;:;>...,.) Fig. 11.4,.2 ahowI Jbr.8'rKa·8 meaaured ValUN of Ht for indium foill in para.ffi.o (e&vity method). The CltlI'VN reprMtIDt e:rt.rapolatioIll done aooording to the formula 1t. _00Mt.~. The valuee of H~ determined for ,..rioua foil radii an compared in I'ia:. 11.'.3 with O&1eulatioIll dODe with the Bkynne formula [lCq. (IU.12)]. which In thIa J'&IlI8 of parameter. wpractJ.ealIy identbl with the modified Bkynne formula'. Agreement 1& very good. and we
R.ce.......

aM." "._.• "*'L (1_e-O.&U/.r;)~

(IU.l)

which ariteI from dropping the noo...ympt.otio term -t in Eq. (ll.S.lld). Vf!IlY oarefulltadiM of the activation CCll'l"tJCItio in water by meuurement of ~tP. han t-n carried out by Snuu. by H.ulf.m dol.• by ZOBUo, and by W.A.L&D., RuDALL, and S'rmllOll. Thelllfl meuuremente pretty weJ.I oorrobonte MmaT..•• renlta, i.e.• they &how that H. ill accurately given by the Bltyaxa or modified Skynne theoriMulona: u the thermal.verap ~ of the at-orptioD l The ~ . . . perfonMd with tJuo n1l* .Iv_ I.e em fOl' pphit.e and O'.{IIOChll/_I_IOOblmll b iDdium. A _sroa MII:lperMwe _ _t a& UuI foil pomdoD P" 1M - " 'l'_UO -X MId ,,-(tT) ........Iuated at thil t.emJBUun. • .Iv-0.H4 _ _ L _1.15_ for panffiD.

probability is UlIOIl and the thenna1ly averaged tnnIport meu free path .l,.. ill U&8d for A. AI an example, we mow in Fig. lU.• valu8I of ~/t[). for indium foila (R _I om) in B.O taken from the work of W.lI4EB. Ruu.&.LL, and 8TnI'SON . Th e solid curve waa calculated from Eq. (11.3.18) UBing valu8I of Jt. obtained from the modified Skyrme theory [Eq. (1l .3.ltS)]. The ~ment is exceptional.

'"

1/ /

"

."! 4

r/

f, 1/

• o

--_...-(1.1

................

,u,.d-

7\1. IU':' IDdJam l<>lI

I

~

1/ It

'.s

'""

7'

o.z

().J

~.

,x laJ.n... t.." / ;;w., I

.r /'

/

_.-

-

- " ' " br

"' ..- - * . ", Ch')

".-.-- ,-.-- ,--,

1\

/X

-j / t

) ,x t/

,

V

If ,

o

_Z

J

..

'1",--

nc. 11.U. ~u..'l for IDd1Ilm lolIoIa,...m. cu...

r-

1llunII..

~'"

II

_~

8 ~ \bIIGI71Ul4~ .

to """

(11.'.1)

"",---.---,-,--,

o

IJII1I

Q.fJ(2

I/Illt , . . h , .

t-

nt. 11."'. i /o. fr.ir 1ll4I-. foLlt(M-

1_) .......

••••• :1.1;1*1-'" of W.u.ua, ~aa4l'nnol: - - 1:" (lU.1t) wlUl.. troIl> Kq, (11.1.11)

, , nc. 11."'.

TlII flu;

z-• ~

,

~

Ia till 'I1cUIl1, 01

IIlIURIR folll ",-, ... o.lIl II pt.ploIM. f~ ~'"

OD 1011...1\11 .11_1.I NIp. OJ _ ; - - - Kq, (11 .1.10)

11.4,,4. HMIllU'ell1ent or the nn Pert1lrbatlon Fig . 11.4,6 D OWI the flux perturbation in the neighborhood of indium foila in gnphite aoooIding to MKUftB. The perturbing field,..... mNlured with Tflry 1ID.u dy.proaUIQ foill WhOM own oontribuUon to the perturbation wu negligible. Alto plotted II the fhu perturbation Jto(&l oaloulated with elementlory difftWon theory [Eq. (11.3.20 )]. We 108 immediately that II8U' the foil Eq. (11.3.20) predict. valQ81 for the Dux perturbt.t1on that are much too 1Dla1.I; thiI oomtI

... aboo~

beoaUMI for I <.t. elomentuy diffuaion theory ill not applicable. For larger *from the foil. Eq. (11.3.20) iI qui t.o aoeurat.e. Ttu.l'Nult iI of practie&1 lignific&noe, for in practice Eq. (11.3.20) 11 ueed only to inaUUI that the diatence ~

between foU. ill large enough to make their mutual influence negligibly sma11. According to Fig. 11.0&.6, diHuaion theory ia lIUfficient1y accurate for thia purpcee . Uaing Eq. (11.3.20l. we fiDel that 101' indium foila.nth l ud ace to.dingaof l oo mg/eml aod radii of 1 CIIl at a di.ta.noe from NCb other of ....16 em in graphite and ... 8 em in litO. "'. (1)_

< 10·',

Chapter 11: nerereaeee 'I'rrn.a, C. w., NUNeorUOIB, No. e. e (I~I ); ibid . t oNo.1, 1IO (1961) (General F&Ot.a About '1'berm&1 ProbN). HVllll.... U. J .. .....J 1\. II. "","W"'''''' I bNL.:sift (1I1t.MI. } At>lJw..u..1l er.- Hool.l.,... a mi WIIIOTW'I'1'. C. H . : AY.CJ..1I01 (IMO ). , .l"-.oton. &no1lJ1l0Q. D., J . M. Hm....... DUlo and O. T . 8LUOIW' ReY, )lod. 1'!lyw. ... .sllli (1968) In-y and r-EDerJi-. and HaU·U_). OLadO • •O. I.. J . D. T.uLO" and D. L.T..... e N~ ' , No-a., IJ (I It6I).} , .8eU. )lmwr-. H. ; Z. H. twfoneb. 122 (11lll8). • Abeorption D.t.n uow. C.K.. ud RoD. Ev An ' Ro... Xod. Pb,.. rt, 7V111l6J) (,..Abeorptioa CoeffieioMIw). Born . W. : Z. n ,.. Itt. set (1M3). CAlIa, K. .... F. III HOrnl.&lIl1. and G. Pucz.u.: IDtroduoUon to the Dleory Tb of Neutl'Oll Diff uaiorl, ~ &ient.ifio LaboIlltory (1t63). of HUll" G. Nucl. Sol. EDjj:. Ii, S25 (1983). A . . !Ua'rul'a, J . 8. : UC'R1r8626 (I N II. otivat lOD. VIGOII.ll. A. : Z. N.twfc:lncb. 8 ., 721 (1t63). P1.I.(%ft. G. : NRC IM7 (11161) (6. ~). Boraa. W.: Z. Pb,.ill o&3'l' (1M3 ). OoaDrALDISl, Eo: N _ 131 (1lM8). D.I.L'I'OJI'. G. R.. ud R. K. OeMlUl : NId. BaL t, 198 (11M1). D.IJ,'I'OII. G. R.: Nftl Sci. Eof. ... 190 (19&2). llmrr H.: Z. N.e.arfoncL ( IG611). llun H.: Ibid., II., aN (IW). o.Mlall . R. K .: NGoi. 8eI. EDc. U, us (1M3). RmllDIt, R. H •• ud H. B. ELDmDOl : Nile!. Sol. EDg. S. 300 (1960). 8lI:n.w.1t, T . H . R . l UKAEA-Repor\ MB 91 (1H4). ",printed INJ . VIOOII. K .. IIDd K . WDn"I: Z. N.twfonoh•• ., 28G(111M). Ouuo..... T . 1..1 N"eI.. Sol. Ene. 110 (1841)~JI'UII, S. A.. T . K vlt'u.... and T. V. BLoMu: OR." 'fi,3Itl3 (I N I). KLaIu,. E. D.,'" R. B . am:m. : ",,.. Rn'. 87. I n (1162). JUIn'I::a, H.: Z. NMufoncL II., . . ( I ~h Ibid. II., S.5II (IW). 8l:JIA. A.: N~ 18 (3), 78 (lMO). STKu.... K. : N1IklBanik 1. 10 (ItII8). TaO~II. Jl. W. II. N'OdoII.r F-v t. 28e (1866). D. TaoY-. A.• 1IDd T...u:uum: BIIII . A-t.. R. B6lg., Ct .al.. 8lIr. • • 880 (1063). Ta u..y. D. K.. T. V. Bwua, and G. M:. EaT....aoo. : ORNL-2M2 (1968). W...l .I:o, J. V•• J . D. R.&JlII.I.LL, and R. C. 8TDlsolI jr.: NlIcl. 80:1. EDg. 15. 301 (1M3). ZOIIll.o W.: OR.~L-UO'J (1M3).

SabemN."

I

.1..

c.:

:r

r..o.

no. am...

Ene.

n ., sn

a.

o.c.

a,

...

Aet JY&1ioD by Epitberm&I NeuWoDa

12. Activation by Epithermal Neut rons The oonUden.Uona begun in Chapter II will now be extended to the practi.caJly important 0&fIe in whicb th e neutron field oontalra both thermal and epithermal neutronl. The ...ery I&me acti-rity that can be esci ted by a blorption of thermal nentroll8 in the probe lubetanee can alao be exeteed by abeorptfon of epithermal neutroll8 , and it iI neoeN&rY to Iflparate th_ two partI by additklnaJ. meuW'tl· menta. In doing 10 we simultaneoualy obt&in Information about the epithermal neutron fius . We Ihall fint diacuu in Sec. 12.1 activation by epithermal nentrotY without oontideration of thermal activation ; Mlxt in See. 12.2 we .hall diaeuu the Iflp&f&tion of t he two parta of the act ivatio n. The related problem of mea-uring re.. 011" 11.... IlIte,.;, ,,I,, "'III I....Ii".. u_d hi HN. 12.3.

12.1. AetivatioD by Epithermal Neutrons

..

Let U I conUder a foil in a homopneoua Dtlutron fie1d whoM energy diatribution wgi....en by 4>(E )dE -4>", (K)

7'

I.od let 1t>... (E ) ....ary 110wly with enefJY . If we neglect _ ttering in th e foil , th e activation (diareg ll'ding the foil perturbation ) it gi.... en by

J•• <-

I C~ , -

<.

~ (R') (E ) ,.. (.<)

'O\tI.... (Ej ~) -. - . tIE

(I n

l)

Th e enet val ue of th e lower limit will be.pecified later; the energy E.- .u. 'the highfllt energy appe&ring in the IJ*ltruIn and can frequently be taken .. infini~ witbout appreoiable error. In genenJ, the energy integral .u. dUticult to carry out uaetly, but in many ~, .. we .hall ahow, it can be e.....luat6d with the help of limple appro.dmate prooedurea. U:.l .l. Approximate CaleulaUon or th e AeU,auon In calculating C, we make th e foUowing aaumptioll8 : a) 1t>.,a(E ) iI CODItant, t.e., th e neutroll8 h...... a pure I/E'l peckum. b) The abeot'ption coefficient p. (8 ) of th e foil l ubetanoe may be in t he fort!'

es p~

( I t .!.!)

1/...put and a put that it the tum of Iymmetri c Brtlit.-Wigner line WpN aNinI from the ,,&rio... teeoD&noel. 'I.'hi.I decompoetion ill generally pl*ible when the "*"lAnce .Dtlrp. are luge. i • ., when .1'.. >1"'. For indium (. DllCV of the lint n»on&nOe - 1.'8 e,, ; r =-15 In"), bowen.., thia repreeentation ill inesact.

Le., it may be di'rided into a

... 0) The ,..funotion may be replaced by the ration&! approximation introduced in Bee. 11.1.1, ria.,

'1'0'

,.Cp.6) _ Hip." aDd it may futber be written

1~7'L, "'I'.(iT) Y:;:6+2~- (._~)'-- '

(12.I .3)

, 1+ -/'iii + 2.-'.."

In deriflDc

uu. 1aA tlquaUoo. .... haTe UlWDed t.ha* we ooWd oeg1ect the 1o&1f.

1fYf-

(.moe

Ibieldina of the l/..pen of the Cln* MOQoQ fl- (iT) 6 fa uauaDy < 1 wheD 8> and IeplU"fote the appromute ,..hmctloo into oootributioni from the iDdi:ridaal ~ which an _eel to be well lepal'&ted. With tb.e UlWDptionI, we haTe

Be)

C-... [I.oJ;.('T)!f'{-6!:+ z.. f 1+ (8-'i1~) .-- ~}] , •• ' -~~- + 2~t "

(12.1.4'

;' (i _l.2,3, ...) IpecifiM the fraotioo of the abe0rpti0D8 in the W1 r-;)nanoetha& lead to the actint,. being ooaaidered. For e:umple, in Jn1a. 001,18% of th e capture. in the maiD r-aDOe at 1.48... aDd 66'" of the _pttuM in the r.ona.oce at 3.158 fiT INd to the M-miD aotirit,y oI1D1W; the ftIIIt IMd to &he 1...., aet1Yity. J.. ref__ to tbe l/..al.orptioo. The iDtecn.tioa OYer the l~ 10 Eq. (I U .4.) '- ...u,. CI&I'ried out aDd whea we ~ B-.. _ 00 JieIde CtJo- J J..,z,. . .(U ,

~ 6 _2J..lf>..P- (BC)6!

(11 .1.15)

_ J 4'. . N crl,.= (&0) jj . We h 01':

e apln Introduced

d -6/~

and a-pf!IN and han fmthennore writw.n

.t.cr."'. Tbll oontzibatioD of alli.nglll I'f*)D,&DOO it BiTen by C _ <1>..... ,.:,6 J (8 ~)' ': l+ f'V2 + 2".:..,

I

_ "",,'-.N r..d

wi...

r.. - et'.. Here

1

J1+( ~ ) •

r• • ldeow.J. with the effeotiTe

'8

J\ '

(12.1.6 )

(12.1.7)

T ' +2 N 4 cr'..

reeoa&rIOe iDtegr'U olio

aymmetrio re.ona.ooo

ihat ".. introdooed in Sec. 7.40.2:, u oept that a. iI nplaoed by

2~4 '

Therefore

r.• - Vl+ 1'• .. 2N 417l

(12.1.81.)

1'·- Too'.. {i-.

(lU .8b)

'"

Finally. we obtain

(12.I .Ga)

with

l.a....- 1cr1.:IBo)+ ~

l'l+~;lIh;'.. ·

(12.U h)

We c&n _ fro m Eq . (12.U b) that the acUTatio n of the foil by the I/..put of t he abeorpUoD ia proportiooall.o the foU', TOlume . 00 the othor band, o-.ina: 10 the oonaiderable eelf.-hielding [Eq. (11.I .8o)}. tbe~OM oontrib ute monaod more .eakly to the .ctintioo the thiobr the foO iI. For thiak foila, 2N "~..> l. and the reIODADOe put of Jolt ... iI proportional to ll)'il. i .e., the reeonanoe &etinUon of the foil ia proportioo.al to the rOlX of iw thiekn_ . For " infinitely " thin folla, N da'. .< l . and loa ... approtoeholl l-a ""'2~.a(Jt,)+

.

•

L -\1'..,- JaMll (B) ":

"

(12.1.9c)

•

Here the eecced e:l[pre.ion 00 th e right-hand aid. ill the ex&Ct form of I:;' while th e tint expreaaion hoIdl only in the approrimation that a deoompoeition into .. IIv-put and .. ~ of . yxnmetrio Breit.WigneI' tenDa it ~ble.

"

FrequenUY,it iUJOI1TenienttO introduce the " . pit.hennN le I!·

lhieldlnt: factor " O"' - -I" 1- M- ' (12.1.10) M

<,

\

I S!

Fia:. 12.1.1 Ilho_ B~a·' TaJ.'* t::>' ••

I&tion done by the methodadeveloped in thil lleCltion, and an enet

,

,

-,

of O... (d) for gold (lower eDelJ1

limitEc - O.68 flT i cf. See. 12.2.2). In thia figure we oompare meuured valUeI, an elementary 010100.

~

I' .~

,,"'

"".,."*-1 "**-- .. d-

tf &

em

{f

calculation. In aD en ct calon1&- ..... It.I.L Tho oolf. ........ _ a.. lor GoW of doolr 11I_ _ ; tion. we may no longer neglect. ....... . r._ _ttering &Dd moden.tioD in the ---- ~;--. ... S4.l1t-UI foil. we may DO longer deoompo. the cr't* 1l\lCtioQ .. we did in Eq. (12.1.2). we maal ue a better re~t.ation of th e ".function, and muA tab the Doppler effect into acooun t . In attempl-o ing n eb a c.Iculation oan fall b&ek on the metbodt deye10ped for the ~aala. tioo. of I'N(lD&DC(I abeorption iD heterop:aeou re.ctoR (in thiI oonneetion _ DusllfU or ADLD aDd N OBDIIlIDI). FIg. IU.l tho_ that the elemOllWy calculatioa reproduCN the e1ptlriment&lIy oheen-ed behaYior quite ....n. 8imllar upeJ'imenw on iDdium Ion. haYe been performed by'r.u"T. BL088. . and. EsT..-ROOIl and by BROIll and booa:-. among others.

... It IhoWd be added th&~ TB01IaT tJ at. bue performed .. mu ch more aoeun.t.e calculation of the epithermal eelf-ahiek1iDa: factor of a purely abeorbing roil than we did above. They Itart from (12.1.11.)

l ratMd, of replacing t'P.(p,,6j by ita aimple rational approximation, itA euct form 1- 2E.(p"lJ) is retained. AMwning a IlingiCl reeonence and neglecting th e l tv. part,

one obtaina after integratinj: the denominator and putting

% _!.;';'11

••

O.pl_":"6'-.J I ~ - E. (t.;~ )J d~. "

I,

es~

b OBav d 121. ban Eq. ue.i.n b) yield8

.:_+

.

1-

" 1tf" Tta.Y " •

""'b..

-

• - ~ "'" " • • '" S

~

....

PIt. ILl.!. no. _ - . . _

- --.- --- -

-

"

"

~

0 ... lor

--. _ CakuIa&lool by Taun I.ftaa-..; - - - _ . . . . , _ _ &4. (IU." )

(12.I .ll b)

...

... ..

ahown th at

,...·mv..., 6) - - -

0 .,...... 1+ - ,

for

O.3274 p.,,6

,..,, 6< 1,

• •

for Pao6 > !. Y~6 For intermediate nluM'of p.. 6. th ey h."e performed the inUp"a Uon I:lumerioally. Tbeir ".u1t u. abown in

O... ~ -_· - -o-=-

3yi

Fig. 12.1.2 ~t.ber with our aimple approrimation G.,. _ I /YI + 2p..lJ (d. Eq . (12.1.8&)].

12.1.2. ReIODanee ProbM We ahan now drop the . .umption that the epithermal flu: per unit lethargy dON DOt. depend on energy. Th e epithermal foil activation ifI then given by

a _ Ni l

J•~.!:t(E)~... (E) ~:- + L.,. (E ).t.P.II!.

I..

..

-

......

••

..~

h"

"

... .... ..... •• W

1.6

.r1.6-

aUI alia

uo••

......

U.

--.,

'"' ...6.1'.461 ...

7U

""

(12.1.12)

j

'

. •

...

o.J :;-:~ u

, ,

...

I'lOO

"60 .u "

._lJ:e •

ul::-C~"· _0 .se _ 0.•

-_ o.n us

-....

...

0\

..

'" <,

<,

.'

.-

•

.

.. b ......

.

,

•

[ J'lI, It. U ..

I .._............

'I-

~_ lJcL

....

•

'

,..... 1t.U .........

.J

r--

••

oto-

~--

UI .... U .1

.

W

'

.,

.r

t

It"

'"

'-"

. '

,

VW

I ...............

-

•

",

.-

,

[ -

..... Iu .n

. "

ir

"

~ "

PreqUlllOtly,., probe n.~ hu web .,1up maiD hIIOO&DOe that th e capture thie reeooaooe are reeponIibl e for the maio put of the activation.

~nlll' in

n....

O_Nd,z,epI (B,lJ..P.,m, (12.1.13) Le., the aottntiora ill proportional to the Dill: at a partiouIar rMOnaIlC8 energy. Sucb

I'MOnaDOlI

foll. are frequently

u-t

(under a cadmium. eo..r in orde r to

-wr- thcma1 actiTatloa) too lUke meuurementa OD eplthen:nal neutZ'oD fielda

wJK. -.:r cU.tribuUon- deriate from. 1/1'. Table It .I .1 cont.a.Ln. -ome u.eful data CG ,.,....... detect.on l • Tbe IaR oohm:la gi""N the fractio n of the actintion of aD infiDiteI1 thin foil oaued by capture in the maiD l'NOnanoe ; In all CUM it ill quite nea:r ODe . H Owe'9'er . the aitoation beoomN 1811 f....or-bie when _ ~ider ~ fon. of finite thice-. In th_ foU., .. - - - the l'NOnanOM anllI8lf·ahielded and indeed man Itrongly the higher the l'NODaIlC8. J..P.,JlelJ . . tberelan decreuM with iDoreaaing foil thiobeM, .. can -wy be Ieeft for l18"on1 foil n b-taru3N in

~

'" r-.

I

•

Fic.II.U. For thia reuoa, _ mut make a neooanoe fon... thin .. pl.-ible or apply

~--

),--

oonectioDa, for wboee caloulation ~opI (E)

.

..

.

man at leut be roughly known. The

aandwioh method. whiob we . hall di8cuM In Sec. 12.1.3, oHera a way out of th _ [~ '" difficultiN. R-onuoe deteotora an ftl. 1Ll.le W1awtable for abeolute mMftI'eCDenta beoauee the
,

-"'~

The ICtintioD perturbation in l'NOQIoDoe foil mMltlftlmente ill nry 1JDal1. The naeon for thiI ill that the anne- eDeI'I:1 Ie. of a neutron in .. oolliIion with aD atc:a of the nrTOaDllin& moden.tor ia Iargw than the width of the

n ... neatroo. tbM

-wI,

hal oooe tnnned the foil with the l'NOIWlO8 eDIlIV hal a nry Im&D probability of beingac.ttend back to the foil with nearly \be laDle - e: th. It ClUlDot C&uae a pertarbaQoD by ita al.eDoe. ~.

11.1.1. The Sandw1eh Method We can eyen make l'MOuan08 deteoton out of thiok foill if _ mike a " eeadwioh II of three llyen of the aame material. The two O1lterlIyera predominantJy ..baorb Deutl'oDlW'itb eoerpe. near the main J"e&OIWKle. The inner layer " acti.....ted nearl1 ... NOna:!, .. t.be cater 0DlII by the l/...part. of the ab-pt.ion and the 1J'ardMr ..t . t - \bat. _ --'oaaUJ -S .. - . - dekdon _ CoM(I32 ...), N.· 12l6O ...). ae' I......J, _ , . (1T .......

'"

minor feIOnanoee. The fr&ction of the differenoe in activation between an outer and inn er layer due to the DWn reeonanoe 11 thWl greater than in the cue of .. ling le foil. Fig. 12.1.6 .hOWl the fraction of the aeti ntion difference due to th e m&i.n r--.n0ll ' " MDdwichel of identiollJ. foa. oI."enJ. probe .u.bft&D OM. Note th e COIUiderable impronmell.t oompu-ed to the iDdiriduaJ. fon. (:FIg. 12.1.4). Fig. 12.1.8 U Oln th e 00D8tnacIti00. of .. typicU foil ADdwioh. In order to act.Intion 01 the inner 1011 by " nnahleJded" MUUonI iDc1deDt OD the edeN

."old

,

'.0

...

\".

IU

1\ ........ <,

w -

'"

O

t-

•• OJD ~1t. U.

»

"" _~

oIu..

M

.... _

•

~

4

.-

» ~ ....-... __

~

~

R~m

. . -... _

o~

0 .,.. LL LJ,.

:n.

r--

• '" '" '"

... SO

dof tbo .... _ _ to 1M

~

<,

'" • "_-..4"'*__ " D)

_ _
. . . - . .. . . - .. ta.r ~

of the MDdwieh. .. lbie1ding ring .boat 6 mm thick mut be eeed. The foil iI

irndi6ted UDder .. oadmium. oover. U the foill are 10 thin that the thenna.! a.ctin.tioD 11 homogeneoua, it will drop oat OD IUbtn.ction, and the UDdwioh 0Ul be irradiated without the cadmium. cover. ThU prooedme I. DOt very &OCUlItUl, however•eeee in it we form the amaJl differonoo between two large qUftDtit1ee. Fig . 12.1.7 m o. . the aeti nt.ion ra tio clllec (the ratio of th e activation of an outer foil to th at of an inner foil) fOl' identical foill in .. 1/E4poetrum : thiI ratio '- importaDt in ehClClGng the dimeD8iona of the foil ...oowich. In omtIC' that CD - Co be det.enniDed .. t.OClUlate1y .. pt*ible. CDlCc mut be .. large .. JX-ibJe. n iI cleAr that in gold, indium, ADd tuDgtten it Ji euy to achie,.e large ntiaI. In Ia.nthaDum and Dl&I18ane-e the differen_ are mWl, &Dd CD aDd. Co mut be dlltermJned Tel'Y aoountely.

'"

ACltintioD by Epit.bermal Neutron.

I t. will bel inatructive now, .. well .. in ecme future diecWl6ioM, to ooDllider the theory of activation of • foil sandwich. Let 118 therefore consider a foil of thiekn6ll8 6 and absorption and activation coefficient".. that is surrounded on both eidfJI by cover loila (6" .u.,;,. Th e activation of the inne r foil in a monoenergtltio, ieotropic neutron flux 4> is then given by ../1

,.;,.

.

J'-..... ,

Oc= -:--2

Ei3

H

i

'. r

oc=~[g.
(12.1.14&)

1

"'~_ ~ -~ [•. v.:6'+1'o6 )- •• v.:6'iJ.

;~.J elm?!:,_ $

;)))))))j

j OOll U• •inD.dD. .

(ct. Sec. 11.2.1.) The integration yields

~h'ls ~

~~"lt;7: 1'11:.

-[I - e-

For

(12.1.1" )

~entioal foila of the seme material (p.=iJ~,

6 =6),

4>

11. 1,&. TM """",lnlOI"'- 01 .. tnlo&l -.1....... 400_

Cc = '2 ('l'.(2,u.6)- lJ'.(p. 6)] .

(12.1.1t.i)

We can eaaily obtains th e aotivity of an outer foil if we note that the activation of the entire sandwich of identical foila is giV6D by ~

,•

0C+2 0D= ""29'-(3p.6 ).

-l

I

V • ~

•

•

-e;

(12.1.16)

..."

._ .- · • ... ---· '" W ~

I!-- -

~

,)p , F1a. 11.1.1. Tbo ra'lo o ...u••tIoa af

'"d _ • • "".... foll of

of UaolrUllo~ _ _ CaInIlaIed

val

..

'"

nd.101eh 4etee1or '" I"'"

;. A " 0 0

ot 11.. 1....... foil .. . ",...110Io

_a~

br EnaT

Thon (IU .17)

We O&n DO'" integrate CD and Co over the IJE-lpectnlm by the mllthoda of &0. 12.1.1 and form the ratio C"jCc ' The cnuv_ln FiI. 12.1.7 weee obtained. in thii (with Ec=O.68 fl.) .

,...y

Simult.&Zleoua Thllnll6l ud E pith ermal. Foil Activation

12.2. SlmultaneoUfi Thermal and Epithermal Foil Aetivation 12.2.1. The Cadmlnm DUferenee Method Let us now make the following &8IIumption for the energy diIltribution of the neutron flux : '"

- 6/l'l' E E ()= L\ (cT)i ~

1fI,.

.d(EfcT) + '"epl .-

(12.2.1)

Here and lfIepl are oonstente, T is the neutron temperature, and iJ (EI} T) is the " joining " function introduced in Soc. 10.2.1. Then if we neglect scattering in the foill , its activation is given hy

JIf;:

..•

~

X

f o(.u.(E) 6) .d(E11:T) dE

(12.2.2)

e

II

, , ,•

, In order to eepa.rato the thermal and ,• epithermal activation, we now irradiate • the foil under a oomp1etely -.led oover I •• of from o..s. to US-mm.-thick cadmium . S , The absorption crou aeotion of cadmium t

"! ~ --

=%,+0.".

\

-

\

D'

(of, Fi., 12,2,1). very hiah at loweneraitoDd drop-lharply in the vicinity of 0.6 ev; thus in a rough fint appro:rlmation we can &8IIume that the cadmium cover eapturee aU the thermal neutronli and tnnamit8 all the epith ermal neutrona. Thu under cadmium we only meuure the epithermal&etintion ; more preciaely [ of. Eq. (12.1.14b)], with a cadmium tbiokn_ dOD

(JCD=

~r

1\

, ,• , ,•

,;

\

,..0"IE)60"') I

I

.,. I .,. I

."r

{I'll. I LLI, Tbe .t.rpUoil _ ~ 01 Ca4ml . . . . . flIMtIoIl 01 _ _ eMlllJ

i': [lh~D(E)dOD+p.(E)d) - 9'• ..,...

..

,

I

-

.d(E/iT)

dE

(12.2.3)

Cepl,

l l ' " FOD'

H ere we have introduced a cadmium oorreotion f&otor

(12.2.4)

-

,...

l In additJoIl, we nea:1eot ihe (thumal) aoU.,..tiOIl.

1 lec*IINf'iI'1rb,

If_...,..

~tioll

both beN aDd Ia IS

n.t

ActiYalioD by Epit.benDaJ Neuvau

which taJr.e. iDto &ooOWl~ the ft.ot tha~ c.dmiw:D e&pt~ DeUUoU bet_ n the "cut-off" eDer'lJ' of the epiu-aal apectnun (... 0.1 eT ) and the " e.dmium eut.off " enet'IY that -.ill be defined later. FCD depoodA on the tbicknMa of the -muum 00..... the thiekr- of the foll. the foil lUbatanoe. and the joining funotioD, which may depend 011 the modentor aubdanoe and under oerta.in circumatul.OM &l8o on the neatl'oQ temperature. Figa . 12.2.2 .. nd 3 lhow ClOmlCtion faoton th.t were calcul.ted for J gold and indium foila using J o. HANSSON 'S joining function (d . aleo Sec. 12.2.2). In orde r to obtain the purel y thermal ~ti...t ion from the total

,

I'

'"

---

~,

'7

"'0

..

d_

".. 11..1t. ",.

Ii

' _ "II. "' _

,. "

o..w

V

/

1::::'-

.-/"

---- --

.. .. .

"

d_

(12.2.6)

be taken into a.eoount; howe..ee, fact
tu

,."

",. CAd _lkNl rII.elot tor lDdl "'" toIla I" . . looVopIo _IIoco fIol4 flI""""","'I Il4l• • 1olIWe_ CdIU... ~ O.• _ I _

PlI- I t.U

1t.U. r.se.laUoa 01 '

~ factor :

All tbe oonliderations of t he eleventh ehap Wlr now hold for OlA • Since ,..count ing meuu"" an ~tiYity pro portional to th e actio ...t.ion. in t hiacue all our rtlIult., putiouIa.rly th e oorreeUon faeton ahO WD in Fip. 12.2.2 and 3, ..t.o bold for the acti"rity. In p -oount.ina:. electron aeU.ab-orption ~~I1.t

CdIUW_ l _

V

I"

aeti.... t.ion of .. foil. we muet aub. tract th e epicadm.iu m aeti... t.ion multiplied by the cadmium eoe-

IoIlo ..

'" o..w 1oII 1IlIe_

"

:1.

c- .

IX)

. . - .....

~

. . 1ooUopIe _IIoco

--

CD

Eq. (12.2.4) in neatly th e

M mtl

.ay and in fint appro:dmatlon canoebl out .

lor a Thla . /t'-AbIor'ler

Let the I/....b.wber be .0 thin th.~ ..,u.ahielding in the foil can be neglected. In addition, let all abeorptioa proe- Je.d to acti...tion. We abo. next that the " the rmal tnnIparency" ~D of a mfficien tly thiek cadmium oonr can be neglected. We begin by noting that

c£D_

-

~ !(feep;D(E)6CD+ p-(E)6)_feep;D(E)6CD)]-tT,-·IU'{; 0_

Ul Jp-(.&')d.8,V4'D(.&')&c") ·fr ,-.1'" :~ •

- 'Itl

I

(l2.t.1I)

'" (lIiDoe ,u.(E)d <:: 1). Thus

~

,f V

E

kT

=

1--112' 1:dE"7'

. .-~,..cDIE).cD) lV~-=e -.IU' y;;: !"'Et V--f T

If'T

H ere we set p.(E) =,u.(l:T) V-iT ; ,u.(.l::T) then drope out. If we aasume the BreitWigner form ula for the abllorption Clt'Oll8 eecticn of cadmium. Eq . (12.2.7) can be inwgrated. Fig . 12.2.4 mows T8D all a

function of t he thick n(l88 of the cadmium rover for aneutron temperature T = 3OOcK . WellOOtbatforooverthickn_ > O.limm. cadmium is opaque to thermal neutrons ; t hus the neglect of c£.D in Eq. (12.2.3) is justified . In t he following, we ehell only admit cadmium thiclme&Be8 > O.1i mm. In order to determine Fe.D we U6J:t form theintegraJ

c:3 "",CCD =

dE

.i:T •

•-,,

<,

-e

-,

-,

-,

•• •-.-, • -r

---

.-.-, -

~

The lAu....... lo
,.... 11. 204.

U>otmaI

:---

\

-,

c.d"'''''''OQ_... fll_oICad'''I. .. ~

J , ~

p.(E) 6Et~D (E l dOD)

(fJepI

(12.2 .7)

d(E~J&T)

dE .

(12.2.8)

U the cadmium

CJ"ON B&Ction were an ideal step function, t.e., if it were eztremely high bolow a cut-off oDergy ECD and eero abov e it, thon OClJ would be given by

(12.2 .9)

It bas boona8/1umoo here that the joining function.1 (KilT) ill already equal to one for E ;&i:;:KcD. Now the cadmium Or088 flection only approximates a atep function. NevIlrthol68ll, it ia CUlItomary to uee Eq. (12.2.9 ) to define a cadmi um cut.off energy. In other words, we require

f

E p,,(K) '!g- =

~

1l~D

f p,, (E)EI(iJ~D (E)6CD) ~

.1(Et

T)

ss

(12.2 .IOa)

0

and detennine E CD .. a function of t he cadmium thickn6811 " CD and the joining

function LI(KIlT) . U p. (E) = p,,(kT ) can be carried out , and we obtain E

CD -

If-!:,

the integration on t he left -hand aide

• tlf1t11 ~~ )'. tJ'''~ (K/I:TI 1fI(P~n(K}

(12.2. 10b)

'"

27'

Act.intioa by Epitbennal Neutroml

Tho main oontribution to the integral in the denominator oomeB (for cadmium that wnot too thin) from the energy range above 0.3 flV i in this range, allsuggeet8d variantl of the joining function are very close to one, and ECD therefore depends only alightly on the choice of joining function . Fig. 12.2.4IOOWl E OD All a function of the cadmium thickness calculated using the joining function suggested by J01Ulf880N d at for D.O at T =300 oK. When d =1 mm , E CD-O .68 ev , which we eball frequently uae . . . ltandard val ue. Th on for our I/t/·absorber we simply have OCD - 1J). pl2p. (B cD)6.

In addition, we mutt calculate

-

,f

O.p1 "",lJ).pt

IJ,. {E ) 6

(12.2 .11)

(EIIrT } r<'l :«: dE .

(12.2.12)

The function .1(Bll T ) .110 appro:dmatea Ito etep function, and in thia cue we introduce. cut-off energy E. r defined in the following way :

- .. f-

f

p.(B)T =

••r

14w (E)

""j'n - r: -dE .

too

(12.2.13&)

0

U "'8 .et p.(E) =p.(J:T) ~. we can carry out the Ieft.hand integration aDd obttJ.n

•

• .-

J

•••

••

-

I.' - _.

uf-- · "' l o

V

I.Z

".. 1I.s...

-

---

V

f- - :::;.;.----

/'

' .6

fU

1.0

I.hwfI1.f

df'bo;
1~1Io

-

,r-- -"-- . ._.. --

,/ /J.4

.>

•

. . - . . . . .. ~ . . . "'11; .. . twoodoa

...

•

~fhid1tw8

~

•

lI.I... n. Codmlo>. '"""" tor • LId. 1/..IoU Ill .. IootroDIo _ _ n....... fwooUoDCII 1M

~

CIIIM~al"'~

••

"""'---

Sinoe th e joining funotion only dependl on HilT, we Tariable :tt=8/J:T in the integral and obtain

O&D

introduoo the new (12.2.130)

m aDd the cadmium oorT'eCUon fadol' for .. thin l /e-abeorber ill gi"en by

c.

F CD- 'CCD- =

l /ZelJ

(lU .~6)

YJiTT "

for e:nmple. when the cadmium COTei' ill 1 m.m. thielt and T =293.' "K. li:D ""' 2.76. Thi8 factor, which i8 ahOWD in F5g. 12.2.lli .. .. function of cadaUQIJ1

ThUl.

'acton

thickne68, it quite a bit larger than t he eort"Ction lhoWD. in FigI. 12.2.2 and 3 lor gold and indium . The difference an- from the fact. that in the latter . u blltanooe th e epitherm al aotivation i8 large ly due to the r-nJanoe., whieh all lie at w eb high energiel. that they ars oo1y Ilightly affected by the abt<:qt.lon in cadmium.. Some of the rmulta of tbi8 eect.ion can be applied to thin lou. made of au b.tan~ ...h~ ""* leOtiona oan be deooIopo-ed into .. l/w.pqt. and .. ~oe pan. .. looK .. tho finlt n.<manoo JioI at .. IIlgh -r'J)' where e-dmJ um no Iongtlr .b-orbl Deut.r'OM. Then (12..u8..)

....

oeD _ (f). pt J~I (E) ~

'"

...

f .u J,.. -: -..(8)

Jh=

~

,,:- .

(It.2.1Gb)

.J..

cr. .. lE) -:

= 1

DO

...

+ .':"

(12.2.18 e)

-..

f~ (E) ~:-

wit h E.,. and BCD given nepectinly by Eq•. (12.2.130) and (l2.2.10 h). Eq. (12.2.160) ill .. "ery good approIimation for gold jut .. it i8 for dylprollium . F OJ' indium , on th e ot her hand, ECD and Bu. mUll. be red.mned .moe th ey ars affected by the fint I'MOnAnoe .1. 1.46 n . tl.l!:.3. The cadmlu.m. ILat10 Ul' th e DetenniaAUozr. or th e Epithermal nu Tbe c.d.rD.ium ratio '- defUNd All the ratio of the actl:rity of the b&re loll to ita aetiflty ....bee entirely eneloeed 1D eadIlUam:

BeD- '~D-

t;1)"

41",

•

~ + £...,..t . -~D

(IU.17 a)

·c.

,.,

f.."'A ",V<. 6)

f 7: ["'Uo~D1E) 6CD+ h{Ej61 ~

-

... ~D(E)6CD)] JJIK~tT)

(12.2.11 b)

a

n. For.. thi.a. foil, (12.2.18&)

In putiou1&t, foI' .. thin l /v-ab.orber, 4>,.. ~.pl

ReD -lh -

~

'.f,

VEOD "'T~'

{12.2.18 b l

ThUi we caD determine the epithermal Oux from .. meNuremmt of th e oadmium ratio if ~... fa known. H ....e UIoll .. thia l /Hletect.or. t uch .... BF,oCOuntClr, ""'

.-

.-

"

u

1'\

!."

.

I"

•,

I.

.-

~

ftc. 11.. ..,. 0 . -... _ _ . l4.. ... (lU ll)' " .. ....

.-

••

of

01 _

~

I

~

....

~

.~ D.

_ _ . . . ,...... ",**- 1_

• ........

.- .- I

•

•

.

~

-- ~

_ . I...... nul.IIIort.a.. ~ 01 _ _ 1 _ •••

....... K. , . . - w loot

~

--

_* ..,..0U........... ... lor _ _

_

need

~

_, _

-.lIIo~

DO lldditional data (..ve v&luee of EOD and ~D' which we obtain uaing the method. 01 Sec. 12.2.2) . U we u.e thin foila of another material, we mWit m ow the epicadmium relIODaD
K_

.

~D - JCD}..=L

,,..... r=r"..

. eReD- 'col, _

.lJ~' ( T1

tt%OO lllJ- )

. -'",.. .

~ ~f.ao. 61

J

X •• -

X

.. ..

[~fJocDICD +,.."I_ ..!I' CD,eD )]

- --

-

--..---

•

...J

- --

....(6) . -

-

d (8 /tTj

(12.2.19)

~6

- --

Simull&ll801l' Th~ Uld Epithermal Foil Activation

n,

For. given foil eubstance, K is a function of deD, d, and T and rJao depends on the choice of the joining function . We can al80 define a factor K" which includee the eHecta of p-eelI abaorption. We can measure K by determining the cadmium ratiOll of foilil of variOUIthiclmeseee in the same field, and we OIl.D calculate it by the methoda of Sece. n .2 and 12.2.1. Figs. 12.2.7 and 12.2.8 ehow IIOme meaeured and OIl.lculated valu&e of K and K, for gold and indium foill . Ulling the8& data, we determine q,IJ
1%.2.4. The Two-FoU Method. If we attempt to IIeparate th e thermal and epithermal activation of a foil in a multiplying modium the following difficulty may ariae. The cadmium perturbs the thormal fluz in the neighbourhood of the foil and in 10 doing rJao .ffecte the eceree of f&..llt noutrollll (via. the fiaIion nUl). Thue the epithermal nuz being meuured in the preeenee of cadmium is affected in • way that ia difficult to descri be, OIpeoial1y when a large number of oadmium-oovered foUl are eimultane· ously exposed. The two-foil method offer. • way out of this difficulty. In this method, two foilil of difforent m.terial8 ere faetened together in a p6Ckage and eimultaneously irradiated. Foil 1 is made of • material with a emall f80n.nce integral (Cu, Mn), while foil 2 is made of a material whoee reeonence integral ill larg e compared to ite thermal orou BeOtion (e.g., gold). Then

1

Ol =Gtl
O' ="-1

"'+"-

(12.2.201.)

pt -

Here the /l;A; are oonatante whioh depend on the Cl'OllII eectiollllof the foU .ubstanoee, the foU thi ckneeeee, the neutron temperature, and the joining function_ U ~be /l;A; are known, "'IA and q,ept can be determined. by separately counting 0 1 and 0 . :

"'IA =bll Ol+~. O. .... .....,1=6n01+6..0. J. "ll ""

I

(12.2.20b)

11",

(12.2.2Oc)

"o,lJol- a" ,11,,1 •

etc . Th e ~ and b... can be calculated from the . pecificatio na of the foila, but the recommended procedure is to determine them ezperimentally by m&allurementa with and without cadmium in I. atandatd epeotrum for which
12.2.6. ThlD Fol1ll: Description Aceordlng 1.0 Westcott's Convention The reaction rate 'P [em.... 11&0- 1] in I. feil that is 110 thin that IIeU.ehielding effect4 can be negleoted is given by ~

,, =Nd!"'u .r;.g(TI ~;6" f1.(2200 m/eeo)+f1'.,.! f1. (B)

,

t1(~kT) dB].

(12.2.21)

... Following ..

~

of WDTOO1'T, we ClU1 nlpl'Nellt .io .. oooaiderably sim pler

IoDd more e1epDt way if we make UN of .. eolDowhat diHerent formal de.cription of th e apeotrum thaD ""' hal'll hitherto been uing. Aooording: to WurooTT, 1t'e write

., _N .d ." .O'. ....

(12.2.2b)

(wi th .., _2200 to /tee ). Here" is the tot&1 neutron deoai.ty

,,-J•. ($I)4" _!•~~gl

dE

(12.2.22 b)

•

•

and u. is aD eHect.ive cr.- aeotion defined by

•

•

""',a."" J,, (ttl"(1,(,,)/I1I _ f IP(B )a. (E )dE . To aid u in caleu1l.ting be th e tb ermaJ density

•

a,. Jet. u

(12.2.22 c)

•

intnlduce

"u - ~ - ~

80IDtI

. ''roZu_

alUiliuy qU&nt.it.iee ; let "ta

(12.2.23&)

and Ie&- .... be the epithe rmal dellllit1

(Ct. Eq . (12.2.13c1.] ObviouJy, " -"u. +llepl ' IntnxluciJ1l th e ahhnrriationJl I -f-,.,J" and uaing Eq. 112.2.21), . e find that

f - ~II.

(12.2.26)

Two additional auxiliary quantities will now be introduced. Fint, let. the exoeu reeonance integralJ' be ea peeeeed by 6-

1

a.t",1

21 f T , Vi' V"293.jO 1 •

(12.2.26)

The quantit.y" is dimeneionllllle and depends on the neutron temperatl1Nl. Second, let us introduce a new .pectral index

,.=t~ .

(12.2.27.)

From the definition of I and Eqs. (12.2.23 & and b) it foUowa that

y;i/.ff.

r

(12.2.27 b)

I+ ~ -~~' f.

al.pI

fPuJ4'~pl is ;>1 for " 10ft " spectra, and thUll ,.~(,f).pJtPlA ' Finany, a. oan be expeeeeed in term, of r and , &8 follows:

(12.2 .28) Det&iled tablee of 9 and , &8 funCtio08 of the neutron t.emperature can be found in WJt8TOOTT (el. a1Bo C....lIl'BII:LL and FRUlU.NTLS). For 8. pure 1/t7-&beorber. g =1 and . =0 ; thue u. =O'.(vo)' In a pure thermal neutron field. ,.... 0 and thus O'. = O'.(tl. ).g. For mOlJt nuclldee, the val UeB of " and I ' do not depend on the joining function, for &8 long u there are no re&OnAn0e8 at very low energiM the integrand in Eq . (12.2.26) practioaJIy v&niehee in the region where .d (KIlT) ohanges rapidly. Lutetium and plutonium are noteworthy eXOllptiona to tbi8 rule; we .hall como back to them later (Sec. 16.2). It ia inatructive to rewrite the fonnulaa for the oadm1um ratio in the kaguage

of the WMtoott convention. The activation of a thin I/t1-deteotor under cadmium b given by (of. aoo Sea. 12.2.2), OOD _Nda. (VI)l;...

or after integration

!• ~ ~:-

(12.2.29 a)

(using 4).pt "'" ~ Ill'''') '"

'V"

~D = N'lIo"a.(vo)lI. Vi' r

li -

BCD '

IIU.29bl

llooaullO O=Nd ."a.(".)vo It follow. that

RCD=

~D :zo -} . ~~ V~f '

(12.2.290)

Thm we can determine r from the cadmium ratio of a I/v.absorber jmt ... we can determine tPu/tP.pl {cf. Eq. (12.2.18b)]. The activation of a .ublt.anoe whose Oroll eootion follow. the I/".law at low energilll (glllo'l) but which h... reaonanoee at higher energiee (> I ev) is ginn by (12.2.30a)

... Th.. (llU.30b)

The We-toott. OOIlyontioD ill putioularly weU auit.ed to the C1nluat.ion of neu.tron temperature meuW"ementa with loila (Soc. 16.2 ) and to th e deecription of ftIIODUIOe integral meuun!lme!l.Y by the two-.peotrum method (Soc. 12.3.3). Ita

8"oeraJia;atioD to thiek foil. it problema.tic .

12.3. The Measnnment of Resonuttl Integrals :Many of the qUMtiom dMlt. with in thia chapter have aigDifioanoe far beyond the theory of foLl activation : Ie the approximation in whieh the neutron field iI considered to be made up of .. th ermal Maxwell lpectrum. and an epitherma l l IE-spectrum , .U reaction rates can be expl"ellJ«l. in term.I of average therm.1 croee eectiou and eeeceenoe integrab. no matter whether absorption, fiMion, activation, or _turing is involvod. There t. therefore .. oonaiderable intereet in tbe direct mMlurement of reeonan oe integral. both at infinite dilution and in aituatio..... where eelf-ahielding play. an importAnt role . In thia 8eCtion. we ,hall famllia.ue OunelVN with the mo.t important metbodl of mewuring th e reeonanoe integrab of thin foila. However, fint . . must m&ktl our definition of the reeonanoe integral more preciae . 11.1.1. Preelle Definition .r the !lMonuee Inu-craJ ~noe integnJ of .. I Ub-

Aooordina to GoWllTUlf dol., th e epicadm.ium 8tNloe • ~/.JWJtl by the relation

(12.3.11 where % _4 ma.na abeorption, z _/ meant fiaaion, z_. meatUI IlCattering, and %"",&ct meatUI actJ.vaUon. We must be careful to dietinguilh theee quantitiN from tho quantitiN that we actually _n in a reaction·rato oJ:porlmont with an intinitoly thin, cadmium-covered foil, viz .,

r."

(12.3.2) Heee tb e DOtaUon I (.&') indicatel that th e l pectnun with which the aetual PJ,euurecarried out may deriate from. the idea1I/B.beharior. BCD depeoda on the c.dmham thjClImt., the at-bet IIabetance, abd the a - t r y of the neatron field or the ~t of the irradiation appuatua. W. hu. alrMdy calcalat«l ECDfOl' a \hin foil with a I/~ Mdion in anilotropio Matron field in Sea.12.!.3. The valll. of Fig. It.3.1 apply for II !leaUou beaDl nonnally incident on a foll with a 1/1I~.ect.ioD. For other geometriM _ IULPSIWf d al. In general, BCD will be different from the 0.&5.,... reqaired .bove. The lIpper limiting energy E_ will alIo pneral.ly be different from 2 Mev, bllt AI a rule thil difference bu meDto. &l'8

J83 DO

appreci.t.ble effoet on 1• • In &Dy cue, we m uat DMrly aJ••,.. apply oalcu.lated rMOEianoe iDt.egra1I.

correctioD8 to the meuured

In addition, we can defin e an,~ ruo-~ i7&legTall; [ef. Eq. (12.2 .26)] (12.3.3)

In contrast to the definition of the epicadmium integral, thia definition iI hot uniquelliDoe it oont&ina the joining function, which d epen d. on the neutron field. As we pn:riouaJy ...... for m&ny n b' ltaooe. the exact form oI.::f (Bll T) playa no role. Here. too, wemtlAt dbtinguiAh bet ween the d efined V aDd directly meuured valu es. The . ymbo" 1• .00 1; intro- .., duoed here Ibould not be oonfu-i .. with our e.r!ief notation I:" and - - I1~ . which eepreeented th e eeeo- " I- nanCle abeorption integr&l of a _ u I U U U d_ Iym.metrio Bre.it.-Wigner f'e8C). ..... I Ut. no- ~ ... - - 0 (Ia , ,.". • 1IlIIIo P&Doe with and without eeJf· 1/"""""'" 1oI • ........u--.. r....u01 CII.o c.d_ _ 111'-' _ _ .hielding~ 1"l:llIpecUvlllly .

.

I

.

- -

"

12.3.1. Determ1uUon 01 th e BMouoee lntesral from the Cadmium B.aUo

i

Detennination of th e eedmi um ra tio offen .. aimple " loy of comp.ring ~ oneace integrah. The reaction rata of .. thin foil11tith and without .. c.dm.Ium cover are detennlned, ..00. mill them the oadmium ratio • formed . For a thin foil, we hu e

(12.3.<)

JeD can

now be e&lculaWd lrith Eq . (12.2.1Dc) (with (16 re~ 0'_ ) with the help oI .. preliminary _ timat.e of the l'MODAlIoe integral. U one !Il~ t.he oadmium ratio of a I t.andard IUbet.aDoe for which t.he reIOI1&noe integral i8 a1rMd.y known then ( 12.3.6)

and OPe c&n de temrlne P:' AI Iona: AI the thermal eroeI Ieet.ioD and the g-faotor are known with luffioient. ~. In order to obtain 1. froJll, we !IlUt. calcula te the contribution of the eoel'l1 rana:e betWMlD OM ev aDd. E CD ' In doing we rulWt. tan baek 011 kncnrIedge of the eDel'IY vari&t.ioD of a.(ll) obt.a£ned from diffennt.1al meMureruentl. 8omet.im. a oot'f"e(l'f,io mUit. al80 be made for t.be upper Umiting energy of the lpeot.rum. but. AI a rule Inch a oorreodon • unneoeeaary. Finally, under lOme o1rownetanON deriatiOI1l from the 1/8..pectrum mUlt. be taken into aoooun t. (d . 1UB.D1' d Gl.).

r:' ,

uu..

'" The

Dleuunlment. oa.n be carried out either in a neutron field or in a neutron beam. The reaotion ratooll that are meuum are uually acUvation rates, but by Ule of lIDall fiMion obambcrt fiaaion rates can be meaaured, and by cbeerveucn of the pl'Otl1pt ,..radiation capture ratM can be mea.ured. . The 2.696-day actinty of gold (0'.(8) =0'*,,(E)) uaually lervee .. a .tand&rd in mob m~mente. By oompr.riaon with the cadmium. ratio of .. boron counter, which dON Dot depend on the Cr0B8 section aince a......l fv [d. Eq . (12.2.18b)],

J ~

Jou.ow and

JOIUll'880H

found that.

a. (S)

~ = 1635±40 barn8.

A calculation

u"

b.-:I on very outlfully meaaured Breit-Wigner p&r&meten gave 1. _1666 baI-m, in aatiaf&ctory agr&flmeot with the measured nlue. Numerou. ,.u1Ui of I'MOtwl08 integral experimenta oaD be found in Ap. pendiJ: n . In the older Iiteratore, Eq . (12.3.5) waa frequently writen with JCD equal to

one, Le., the contribution of the neuUona between th e thermal cut-off energy Eu and the cadmium out-off energy BCD to the l'6&Ction rate of the bare foil waa neglected. Thi8 ill inoolTt'Ct. particularly when the cadmium ratio being meuured ill not very large IlOmpared to one . We may only &et lCD equal to one if we defiDe the "effective" thennaJ. crou fleCtion in a suitable w..y (of. H.t.LPERIN doZ.).

12.3.3. Other Methodl of Heanuing &eonaoee IniegralB In the ,Ul!Hptdru", method, the reaction rau of an (infinitely thin) foil is meaeured in two epeotra of different hardnetl/l(l(l. For exa m ple, the rtlI\.Ction rate in .. neutron field with , = 0 it given by

R.-ftgtl,O' (v,lg

(12.3.6..)

while in. fi.ld 'lrith ,+ 0, it it gi...en by (12.3.6b)

If we meaaure nJ"e with .. thin BF,.oounter. we obtain

....

,

-R1"o --- =1+ -"

(12.3.6 0)

from .hich we can find' if, baa already been determined by mea.l1l'ement of the eadJnjum ratio of .. I/".detcctor. With the help of Eq. (12.2.26), 1' can be deeeemined from,. 'I'hi8 method baa the ad ...a ntage of being free of the det&ila of the eadJnjum, ..beorber. It fan. for IUbetanoee th..t h..ve an e:rtremely lID.all value of 8. Many meuorementa of reeonanoe abeorption integrala h.....e been carried out 'lrith the pile OICillator method. &yv.u: hu described .. tra.Dlmi-don method. for the meaal1l'ement of eeeoneace integrala. A neutron beam. penetratel .. thin foil of the reeonanoe ..beorber ..nd entere .. detector .he:- dlllteotJon efficienoy dON not depend on neutron energy. The tranI:m.ieIion (the ratio of the counting rate Z' with the foll to the counting rate Z' withont it) • then gi"'en by

e- =l-N" 75

f

dB =l-NdjNP. O',(B)g

(12.3.7)

... In the arrangement used by SPYVll:. the neutrona _ttored in the foil could.ull be detected i therefore, the tranamiaaion tnM8uremenw yielded the reeonanoe

ab.orption integral

r.-

p

•

Chapter 12: References I

1

ADr.u. F. T., a nd 1.. W. NOIUlBI~ : ~A-377 (1968). Caloulat.ion of the EffectJ.t'e DUluln, L.: &-:lnanae A~ lD Nuolea.r RMotora. a..ona.a. In,_1 Oxford: PNgamoo Pre. 1960. ee _ _• BKotl.ll:.M.1D~tioD. KM1mIhe 1962. Ct. al.oNokJeollikl, 1M (I",). ) M_taDd Bao. ., H., andM. KxOOHI: Vnpublt.hed KuJ.n&bere~ (IIM3). c.JoUlatJoD. of T:.l1UY. D. K., T. V. BWIM", aDd O. H. EnusooJ:: ORNL.284I. Eplt.hermaJ 8eIf. 2l>' (19/19). eo.t.TU. M.. S.: BnJaMt. : Eurat.orn 1961. Neutron 'IUn&-of·Fllght Hethooh. p. 233. J OIUJil8IOK. E " Eo UJuo4. and N . O. 8.J~ : Atld... F,ymk 18, 1I13 (1960).

I

BloH.ul.R.lIl,&QdC.B.P&.UCI :AECLU28 (1961).

ShWdlq J"~ M-..nomeut of the JoIning Flmctioll.

I

~ (EI"7'). Re.l1l&ll0ll

EHa.r. G. : Atompruit 7, 393 (1961). . GoLl1BII:V. V. I., et rJ. : Atomn.y. Eoergiya 11,1122 (1961).. SUW.t.BT. H. a , aDd O. B. OAVllf : Chapter 8m The Phyaiot of Int«. mediate Spectrum Re&oton, Ed. J . SnD, USAEC 1969. D.l.noll'. I . E., and W. Q. P.1Tu8 : Nuc1eoDiCil Ii, No. 12, 86 (UNl7). F&&lNILU, U. I Neutron Doeimetr'y Vol. I. p.1N. VieDn&: InternationaJ Atomio &orgy ApnOY. lliI63. RIODUN. G. D•• and W . B. LaIlO : Nucl . Soi. Eng . It, 623 (196 2).

•

Detecton; the &ndwich

Method •

t.

EffeoWu Cadmium

SroOOB'l'OII. R. W., J . H.I.lounr, Nld K. P. Iomnu: NlIol. Soi. Eng.

Ca~Off EMrgia

1••'1 (196i). 8'l'oOOBTOII'. R. W•• and J.lULn:anc' : Nuel. Sol. Enjj:. Ii, 314 (I~). XaTIK, D. H. : Nuolooniol II. No.3, 62 (18M) (Cadmium 00rrec\1cm J'ao\on). J'dDt7P. a., IU.o Report No.1. 1t6e1 J. NlIol. &era1 It. n, 143 (IHa).) . J'£IJTR11P. B•• and J. OUlD': w.o Repo" No. 43. 1965, d • • 1.0 NWUoD The c.dm! Do..iUltltr'y, Vol I. p. Vielma: lntem&tJonai AtoUlIo En.gy RaUG MId 11m

m.

Apnoy.IM3. GaUMnaJ.!), K. A., R. L Koolln., and It.. It.. Jan'I': NlIoi. BoI.

Ens· e,

U8 (llllI'7). lUNlIU. W., &nd T. 8!'11b"oa: NukleoDlk I, 331 (1IItsll). auu.&LL, G.• aod R. Co J'uUU1n'1o.: AERE, RPJR 2031 (1116'7). Nm.., R . G. : NeuVoD DoUmetry, Vol. I. p.llI. VieDDa: 1nt.em&tionaJ

c.

Atomlo EIItlrIY It.genoy. 1963. SToUOB'l'OII, R. W., and J . H.u..1'PlK : Nuol. &1. Eng . I, 100 (i lltsll). Wanoon,C.H., W.H. W.u.au, aDd T.K.Al.JI:l.UD••' QeDe.... lll68 P/202,

the E lthermal N .2'_~_ tlIIo.cvu ., ......

eon'ffll1t!oDl1

'01' ~

:::m

R.t.e. III

Vol. 16, p. '70. Fiekk WD'rOOTI'. C. H. : AECL HOI (1960). OoIJ:llTZIK. B .• _01 .: EANDC-12, IMI tDefi.D.iUoo. of 1he a-maooe 1ntegnJ.). HaDr, J ., D. K1.mf, Nld G. G. 8Mrra: Nuol. &Ii. ED,. t. 34.1 (1961). IUua, S.P•• C.O.I4111UlLJU.tl.... and G.E.1'tIolUI: Ph,... Re.,.. 'Zt,II (1960). JDlLO'II', K.. aDd Eo JOIUlfUOM: J . NIIel. Eners1 A lI. 101 (1960). J OJrMftOM, P . J., J . ~. and R. W. 8Tot70BTOlI': J . NuoL ~ All, 116 (1960). K.t.OKUlI'. R. 1... and H. S. POMalWI'O., Prop. Nu al. Erwv &r. I. Vol. I,

1711 (111M).

P. &, "Al. : OeM"'. 1066 P/MII. Vol. 6. p.OI, T£ftaUtL, R. B.• _al l J . N\loL &q:y It. II, n (11160).

8P1't'a,

I ct. foomot.e on P. 53.

~

... 13. Threshold Detectors for Fast Neutrons

-.

Certain DUc1e&r reacQolll. aDch .. (ft., 2.) reactiOD8, ineIutio scattering. and lOme (II. p ) and (", ll) re&etioM. 00C1U' only ..hon the neuwna have energi e. above .. pN'ticular thrMhold eneflJ'. 10 many CUM. th_ reactiona 1e&d to .. radio.w,rity which cu be m-m aftM the irndiation. With the help of thnwhold detecton iD the form of fon. , we _ tIM ~ reaeUoD8 to determine the Ou, the e~ .~. or the d~ lu' DeUtrona. SliGh loUt an ol .pecl&1I1anifi. Niiooe _ _ ta mbWli: meo:lM (I"eMton); .... rule, the deteetora conaidered in Cbapter 3 are preferred in mMWremeDW ill free Ileutron beam..

18.1. General Faeta about Threshold Detedors 'l'be -.etintion of .. di8c.ahaped t.hnlebold detector foil that i8 DOt tee Lhiek ia p yen by ... a _ N il I 6"",, (8 )4)(E) d8 . (13.1.1)

•

for th e Cl:r'(WlleCtiona at high energie. are 10 smaU tb at (for reaso nable foil t hick . n_ ) .oa~g ~d Ilelf-ahieJdjnl can be neglecWd. The Ollntral difficulty in aU tJm:.hold detector meuuremen t.l it th. t th e energy depc!ndent'lM of th e f lUt neutron Ow: 4I'(B ) and the er'OlMl lOCtion 1'1ll'. . <E) are u ually quite oompUo.ted. The energy dependenoo of tho aetintion otI f- f--- -

..

.::

1/

L. •

1\

--

..

..

\

pll (Jl,pJSi"

'" • L...Hc!-+-+-++~..J cf" • • • , • ,t.-• • • n.o _ _ ..... no. __ -.... u.. I

.... 11.1.1.

Ior Ibo _ a " I.."lP'· ._.I._ I.. _ "' .."...._

...r-I<>aof _ _ bot_ _, _

rw- ....

~

_

_.to

. " . ILl.t.

p.. c• •,, ).... ..

for

.

~

...... _Uootl

of _ - . J '

IIeCtioD ill known in IIOOMI e-. but with the exceptioo of t.he l pecial ca.e of ~ neu.trona from an aooeIen.tor, the energy dependence of the DIU ill I'IOt bo_ exaetly. ODe mOlt therefore by to obtain .dd.itiona.l infonoatioo about the l pectrnm . We IhalJ. the problem of enluating thnIIhokl detector m6UlU'Mlle nta in det&il in Sea. 13.2, but to begin with ,...e

ert*

w.eu.

IhalJ.

ute

the limple oonoept of t.he "effect ive t.hmhold eDerK1" .

13.1.1. DeDnlUoD or lbe " Eft'ecliTe Tb ,"bol. EDUO''' Th e aotinUoo ert* ~ of UI ideU t.IlreQoJd detector ahould have t.he follo-.ri.o.a; kind of It.ep-funotion beharior. namely . it mould be r.ero below th e threIhold enel'JY E. and equal to 0'1 abo..e E. . The aotintion woWd then be

."

. I

f.~ (8) 4'(1<)""

•

...

(13.1.2)

- aof fP (EldE, B'f'

I~ I ~ ~ I

~

I

_

~

j . l

l

+ +

...

'I'breUoJd Det.eoton for Fut NeutloM

.=

~J1Jf!t,p) CrJ"

1-''''

1/

""

""

<,

V

•

r

I

..

.;, .,

I

, •

I

r".l u ."01 Tho

u..

....._ -..... lor ~

....u",•. In'"(.. ,100" .. "

..-

fi"(",,pt'W"

/

1.,

\

..

[\

II /

tlI I.

-- -

, •

I

I

oo ,i;; ilfA;,)",'n

.,

•

•

." ...

J

V

•

I,

,.,

I

_

loa

e.- _ l o r UIOI

:r."(II." II." .. " t u _

"'

1"'I'\.

.

..._ -.,.

..... 11.1.'. '1M

r-,

1/ /

....r

r-,

, • • • .,...,

I

..... I ' .U . no. -'loa JJ" (.. ,lilt'" 01 _ _-

lor tbI

II "tu...uo.

t••,l'(",pJ Holt

""

"- -,

/

I'"

/

""., I

, '\,"'ttl

•

I

•

1/ I

I

,

•

t,,-

• • ,/'Of.,

.....u.u.

'l'bM _ ... Iloo - " ' - ....(... ')If ~ 01_,"",,_

...

e

:;;

i•

~x •

•

- a. ! NIBldE (13.1.3) r,'

• ,• ~ ;;,

•

- · f N (E) dE .

•

13.1.2. Sublltaneee for Th resbold Deted ors Table 13.1.1 oontai.Jul uaelul data on (ta. p ) thre6h -

old detector. ; tho n'blJt..n ON ue &I'l'&npi aooording to incn-i.na: thrNbold energy ~. The fourth _"""lW1rU. -w_

~

...

" •

... ooluum givee the (true) threshold tlDllrgy at which the proceM being considered become. enorgetioally poaeibJe. Owing to the PI-noe of the Coulomb barrier, E, is neuly al,..&)'1 very muoh amaJler than JJ.f' . The value. of 0'0 In the fUth column

::

(\

/'

!"'"

.. IJ

•

t

J'lc. U .t.7. " , . . . ~I"(

.. . , IlIl.... .. "

•

•

r

-I"'"

, [.-

IJ • , , • £,.•• • " n...",.

1'11. 13.1.• .

Mo •

3

-

..ur. for _ u..o _ ~a t-.,

I

-..

--

~

Cu...(.. 1II)Ov." . . . Iu~ '" _

""

'.

""'"

r ... 13.1.10 . Tho _ ~ "'" to. u.. Uoll

_ '

I--

C4JJJ(n,rnJ Cu"

!""....

/

• • • • •[.-

_

,')l,,'''... . tIlMUi>a d. _ _ ......,. t,;;, •"'"

~· (". enJ Dl..

,I

'M S

"., 1:t.I.t. Tbt _ _ &kMI lot

0 lor U>e...._

.» I I

II

I"

inTI,,, ?InN m 3

_

Al" (....u l..... " hLDolIool. "' ..... lrOD e....,.

ru...1.IoG of

, , ,

Ja'U (...

i... r tn,-.J'ti

'" ""

V

r-

/

"l:r,

V

f/hJID(,..,n')flhr»m

1/ 1/ 1/

V

n

• • • '.•- •

J'lI, 11.1.11. ",. cOWI..tIoa 'or Iho _

" •

.....

CU.. (.....lOU" . . . fullCUoa of _Ltmlo eDlll'J)'

were takcm from the curves .hown in Figs. 13.1.1_61 . With these values and th e values of 11 in the eeventh oolumn, which were taken from a tabulation of BUUGt, E'! "'ae calculated with the help of Eq . (13.1.3). Eq. (2.6.1) "'at used for the m.ion .peotrum . In addition, the table oontainl data on thMe minor a.ctiriti&a I O'(K} for Lbe r'llIldlon Fe"(•• p) Mll '" ill not lhown ,inOll m_ured yaluM are anilablo ocl.y.' 211lld ., ),In. Go bu been Mkeft too be an, M,'t') iD Table 13.1.1.

.., that ooour after ilTadiation of the natunJ element ..be. half.IiTN are not nea:ligibJy lUJla1l oomp&f':ld to t.4at of the main aeti rity. Tbe foiJl are irnldiated under cadralum ooYert in order to IUPF- act.\....Uon by tbermaJ noutronl. BeYenl foil I Ub.t&nCN han pu'tklularly Ioog haU.IiTM (FeN!_, , IKnM, NIM(., plCo") anti for thl. muon are _II euleed to th o time Inuaration of t he flux no-..ry in de\.(lrminatione 01 the neutron dcee, Table 13.1.2 containa the corresponding data for additional tbrNhold d eteetonl that are ba.ed on (" . ,,'), 1- , ~I . and lit. 211,) J"O&Ctiora. Theee deteetore are comple meDta.ry to t he (Il. p ) detectort in80far .. the effecti.,.., t.hl'eahold energy it eeeceeeed bec&uee they allow m_uremenw with eith er very much larger or very mu ch .maller thn.hoJd• . Fip. 13.1.7-11 mow th e ero. I.,oti<ml of the relevant ru.ctione AI functiorul of the neutron energy. The val uee of lJ again

oome from th e tabulation of But1oi ; E:" and a, were obtained in the u ml! way .. for thcl (II . p ) detectors. Neith er the energy-depende nt Cl'a.ll&Ctiou in Figa.13.1.7-II nor the yaltlelll in T .blN 13.1.1 &bd 2 ani p&rticuIarlY IWlCUl&te ; in lOIII0 ~ indiridual m_ uremeot.l donate from one .DOther by .. much .. 60 %. The ....luN of it for th e S" (JI,p)pat and p:II(JI,plSiQ re&t'ltiolUl aJ'fI probably among th e most tru stworth y . Measurements and tabulations ofcl·valuee can be found in M ELLISH , in RooIfUN, in PUlISLL and HS.TH, and in BSAuoi. LI8K I.1f and PAUL4&If, B... YHVllBT and PaJ:81'WOOD, and BJ:Auoi give information on th e energy· de pendent cr"088 M'cliom . BnRLY hN tabWat.ed th e propertie. of many other IUbetaDCel that can be wed .. t.hreabold d etec:tor..

""rage

13.1.3. Flaalon ChamlHl" AIThmhold neuelon A. we laW in the discu.sai.on 01 liMion Cf'OIM IleCtione in t he first and third ohapte". ffilUly 01 tb eee ClOIII lMlOtiou have a very marked etep Itructure with threeboidl in th e neighborhood of 0.3- 1.3 Mev ; thiI wggeete that we T~ 11.1.3. F"" ~ .. T~ lJOodoo. UII!I fie.ion deteoto" similarl, to the ("" p ), ("' , « ). (",. 2",), and ("'. ",') actl· vation deteetcre for the mouurement ~ O> 0.62 of IMt neutron f1 uxee. Table 13.1 .3 '000 D.ll7 '200 ....0.4 1100 '000 giVI!II ecme 1I1181ul data on fieeion ~ Q7 ,.40 del:ectorl. The UOII ~ as funGo ~U ,.60 ~U Lions 01 energy hav e alrMdy been Ih own in FiB. 3.3.1. Jut. rule, we determine th e fiNion rate in a lilwIion chamber. For thiI purpoee, partio ularly Imall chambel'l which can eaeil y be introduced into a neutron field have been developed. Multiple fiMlon chambel'l which make it poesible to elm ultaneotllly detelDl.ine the lieeion rate in several IUbetanOCl have been built . In th e oonatruction of IUth chamben, eattl mud be te.ken that the fiMionahle material be ~t in extrem el, pure form. For en mple, il a U" depoait oontaine only 0.7 % 01 lJ»I. upon irradiation in • typical the rmal re&Ctor about 99 % 01 a ll fiIIiona oocur in U-. One can also determine tbe liMiOR ra te from tbe aothi ty of th e fialOD prod. UN. To do eo, one can eit her count the fiaaion Ion. directly after iITadiaUoa or to·

...... ""

....

."....

.., . urround them during ilTadiation with eo-celled "catcher low " and count th e fiMion product activity in the "ttor alter irndit.tion (cf. K OHLER and RolUll' oa).

13.2. EnluatioD or ThrMbold Detector MeasunmeDti There are three difforent grouP' of methodA for ClnJuating thteehoJd me&llurementa. '!'bere are " maLbem&UoaI" method. in 'W hich one triM to determine an unknown oeutroo . peetn un froIIl JD~nf.l, with Ie't'o,..] detee torl (See.13.2.I). There are " lemiem.piri~ " mMhodl. in whioh Lbe . pectrum y aJao determined but with the beJ:p of additional . .umptio na ,bout itl fonn . Fin&lly. t here are the CUM in which th e Deutron apeet.rum ia already known from. calculaUoa. I n tbeee~. mouuromenta with threlhold dtlteelon Ie"O to verify th e calculated l pectral dUtribulion and even t ually to IiI t ho abttolute va lue of th e l u I. n Ull (d. K ORUK). Frequently in the inVMtigation of '&Ilt rea ewr .)'.tem. U80 iK made of thlwbold detecton and Npecially of th e spoetral indicaron discussed in Sec. 13.1.3 ; '"' . ban nut go into thia application of thrMhold detectors here . The metboda of detennlning unknown .pecka that we plan to explain in IOmtl detail in what folknn b....e only been carefull y wewked out. in .. few CUM and beeaWll'l of th e large uneertaintte. in tb e Cl'OllB IleCtiOlll probably only give en d nlAJt.. in f....ora ble _ . lUI. .. lIathematkal" MetboclJ

-

(a) The Multigroup Method . Th e a.etlYaUon of a tbre.hold detector J: ill

proportioo&l

&0

Aj

"'"

Jqt ~ (K)clE.

• of - v vouP-, t ben

U th e energy ranae ill diOOed into a

At-,-, r•ot lP,

.em

(13.2.1)

wbe", -P, ill the OIU: integrated over the i-tA group and at ill .. auitable average valoe of ~ in tbe ume groop . If we now npo&e M thrMbold detecton. (J:_ I , 2, 3, ..• M ), Eq . (13.2.1) become. a .y.t.em of M linear eq uatioJUJ whicb ca n be IOh "ed for th e unknown til, :

.-. at.

-P,-r." S:A

(13.2.2 )

j •

Here Sf it the invene of the matria: 'l'hiI inven.e n itti, u ill well known, only if the det.rminaat of de- not v..uab, l.e., only if th e ~ aect.ioM IIl'8 linearly independen t of one uother. U th e IIl'8 known - the y c&D be oaIcuiat.ed if we know crA' (BI, but only if _ make lOlDe plagaj,ble . .um ption regarding the energy ,.ariation of the n Ull in the i 4 ,roup - then we aJ.o know the ~ and ca.D t bue oaIeuiate th e beba yior of the fllU: from the Aj • Small erTOI'I in the and the At caD prodooe rather large elTOn in the group nUlle. ; in fact high accuracy ill probably only att&inable by making the number M of threlhold det.eeltonc quite a bit larger than the number N of energy group- hd detormining tbe 1P, fi = 1 .. . N) to bNt reproduoe tbe Aj(J: _I . . • JI) in the leut-Iqu.&ree_. Both FlaoBu. and DJaTRlOB ban tried to apply tbe multigroup metbod i ct. aleo Una.

at

at

b1

Evaluation of '1'hrMhold Detector :MeuUftlmlmti (b) HA.RTMANN 'S Method . In thia method, the flux
.-,


(13.2.3)

In order that 4J(E) be approximated aa well aa possible,

!(
113.2.<)

should hold. If we diHerentiate with respect to a.. and set the l'elIultingexpression equal to zero, we find

f

.-. N


(13.2.6)

]f we now &lIlIume that the auxiliary functions 91..(E) are the activation 0I'0ll8 8eCtiOIlll of the various threshold detectors being used (or a suitable lineee eombiaetion thel't'Of), then it followa with A ..= f \P(E)ao"(E)dE and 91... = Ja"(E)ao"(E)dE that (13.2.6)

and the all can be calculated from the A .. by invef6ion. One respect in which this method i3 unaa.tiBfactory ill that the development of ,ptE) in functions with the energy dependencea of the 0I'0ll8 eecuone haalittle physical Bell8ll i aa DIBTRIOII haa shown, under some circumstancee we may actually obtain approximation functions <1>'(E ) that aaaume negative valuea in places (d. Fig. 13.2.2). 13.2.2. Semiemplrieal Methods All the methods to be diecueeed below eeeve to determine unknown spectra aaswning, however, that something ill already known about their basic form . Without exception, they are more a.ccurate than the purely mathematical methods. (a) The Method of Effective Threshold Energy. ]f we &86ume that the spectrum to be investigated deviates only slightly from a fulaionneutron spectrum, then by uBing the effective threahold energies introduced in Sec. 13.1.1 we can immediately determine the integral spectrum : (13.2.7) If we use many threshold detectors we obtain F (E) and by differentiation \P(lJ,'). GaUNDL and VINER have suggested a useful mcxlification of tho oonoept of effective threshold energy. These authon start from the connection between 0'0 and Ji1l given in Eq. (13.1.2). Whereas in the usual methods, 0', i3 set equal to the plateau value of a (E) and E'f ia calculated from ii, ORUNDL and VSNER reoommend that we detennine 0', and ~ in luch a way that a. varies aalittle &I possible when the spectrum deviatel Ilightiy from a fiae.ion neutron lpectrum i bowever, no change will be made in Eq. (13.1.2). In order to carry out the variations of the neutron spectrum in a simple way. GRUJlDL and USNKR chose the form (13.2 .8)

... uu.

lor N IB) (B in Me1'I. WheQ fJ=O.17. fonD reproduOM th e nperimentaUy ebeerved apectrwn ofthe f_ion neuUoJUlabout .. well .. Eq. (%.6.1). We illlUltrate S1 the method by m N M of Fig . J3.2.1. Bore 0'. - for the reaction AI"(a, p)Mg" - wu ea.leulated .... function of fJ for yariowl u l uN of the paramete r by JDll&na of the relation

r:

.

.

! fI(E)(Ee-'. dE

11. (ftI =

!.....;--:-- -- .- .

... J

u,

(13.2.9)

yg,-'.d6

For e&Ch n lue of E:" ..e obtain .. curve of a,lft>. Out of tm. family . we _k tbat llUI'Ve for which

d.,

7! _ 0 when fJ =O.77; the value of 0',1,8 =0.77) on UJ thi8 CW"VfI i8 the nlueoial being~ught. Table 13.2.1 git"M n l uClt of ~ an d 0', fou nd in thU. way by ns. 11.11. .. Iloo _ Oa ulfDL and VI !" • . T h_ ....luee are more uwul .. ...1 1 ' -......... than thOllO given in the eulWf tablee .moe they "'" .e.ou.. u...boI4 - . , . apply even when the epectrum to be etudied is no -:"IMnl longer Itrietly .. fiMion .poet-nun . (bl U'l'HI'8 Mothod . In U..ilI peceedcee, the neutron l pectrum II; writte n in

~

....

,

__..

,~

(13.li!.IO)

t.e., .. th e product of .. fiaeion neutron apectrum .nd .. polynomi&l in E . The b. are clK.en -0 that the At determined with the thnrehold d etec toB a re bMt approrlmaW in t he leuWquuel eeMe: ~Jf.m..

T.w. Illi.

T.v...wf

~.............,..., 10O atnl'1>L •

0 . . ..

.

•

..., ......, " .7 ."

118.'

...... .... '" .." 7.15

I." t.78

%71.0

.-.f (A l:b.O'd)' t-

_ m inimu m

(13.2.lb)

with

== f O'.(E )N(EjE"dE . (13.2.11b) DiUerenu.t.infl: Eq. (13.2.1b j with re.pect to Neh of the b• ..Dd eetUnt: the re.ulting eJ:preJ0'.1

.JON equal to eeeo, we o btain .. 'pt.em of lineal-

equation. for tb e determination of the b. from th e A• • Since Eq. (13.2 .8) prob ab ly already Clan ",pr'MMt the ' peetrum of the fut neutron. in a reactor with only .. few b. , thiI method ahoWd gi.... good ftllulte if we nee nfficieotJy many t tu.boJd d etocton .. nd. if th e CI"OlIlJ eecUona are well known. (e) DlrnuOR', Method . In th • • peeial caee e f .. w..ter.moderated reactor, we m ..y UBWDe th..t eacb MUtron produced in 688lon IOIMl& eo mu ch . nergy in on. ooUiIioo. that after the ool1WoD it (lUl DO longer contribute to the aoti,...tion of th• U7

........._

.Tbon

( I S.t .l b)

... L'.(.i') -E.+I.s (B).

(13.2.12b)

Hefer,slE) ill the Itrongly energy-depeodNlt IC&ttering Ol'OMleCltion of hYeUoseD; the abeorption ~ eect.ion aDd the inelutio IC&ttering (If(lM IIo!lCtiona of all ot hel' material. preeent in the reactor are oom· ' to bined in II _ In lint approximation, the .tI' energy dependence of I. can be neglected. It lI I n the energy range from 2 to 12 Mev, I .RtE) v&rie. ap proximate ly AI g - 0.76 . ~ Thu the energy depe ndence of the OU:I

,

ill

giVOD

by

,

-

• N (E )

(13.2.13)

(E)- !+llE-6:u

where ar. ill .. parameter t hl.t ean be deter-

mined by

-

,

meu~ment with

•

1\

I"

,

threIhold -.. rz

deteeton. Thia determination oooaiIWi ~ of eaJcuL.tlq; th e Aa with the Dux given ... ., in Eq. (13.2.13) and changing II. until the rel..tiYl denation of the meuured &tid caloulated uluee. iI .. minimum . In the I l-

I

I

,

of Lb. Daniab lwimming pool rMCtor D R-2 , DuTIUCU found th&ta _ll; in hi. determination. the aeti'ritiN meuW"lld with P-, S",Al", FeM(".,),andAln (••«1 W'Me reproduced rathu ...ell (.. 6 %). It ebould be poeeibM to apply thiI method oonl

to other lyatema.

In hia meuurementll in the 00ftl of DR.2, 1>J:&TBJCB made .. compariaoD of yanoUl methodt of evaluation. Fia;. 13.2.2

. ~

thowa the integral n ux F (E ) _

f et>(E) dE

I

I ,

•

, ,

\ r-,

,

r-, 1\ , , ;r;. s • :n.. ....... opeatna'(~-i.~ •• I

(-

J

I

.....1UL III UIoo Iloo.UlI

~ ,.,.,. _ 0'" -..od ...... em- - . ea...1 : .1lIUo IfOCIIl _Ulod tftIuaUool. o.n..: Z\'aluUoa b, Hunl4•••• -ehod. 00"",' : ...-.1..._ b, D1walOln_

. obblined from mea.turementA with the five t.hrwIhold detectors mentioned above. Curve 1 1I'U obttJned with the multigro up metbod, curve 2 with ILuTlW'l"l"s method. ~d ourvs 3 with the Iu t met hod dMOribed (with oX. let equal to 11). ~pter 13: Bererenees 1 Bl'PU, P. R. l 1. ,..... NeutroQ Ph,...., ,*"1. N_ York; Iatencien. h b ~ leeG,OlapterIVQ. Oa nDlo, L, ADd A. U.na: NIlCl. Sci. Ea,." 6i8 (le6O). Houa, D. J .; PUll Neutl'Oll. ~ C&mbrida-: ~·W-&ll1. 11M, ~P'- ", F. . N..atl'Oll. ~ .

-s-&U,

K OIILU,. W. l

A~

' . 11 (1084).

Neu&roD. Ibi...".. PJoc--tinp of. b 180 lbrweII SJIIlpoeium, Vol I ADd I . v . . . .; b~ Atomio Eaerc' A&-o1. IN!. P.... R.L" T. 0 .. ADd R.. L. H un : Nacl SoL Eni. 1',108 (IM I). RocunI. R. 8. ; N~ 11. No.1. M (INeI. ICf.~_p.A

.......,.... .....

-'

"...hold

... B1U110'. ll.: SeatJona Effia&oN Pour 1M Detecte1.U'l de NellboNl P ar AetlVll.tkm' j Eff . ~ par ~ Grou pe &0 Dommel.rie d'E ura.tQm. CE A·Repvn.I I963) . Kn1.D., C. Eo, NlI~ It, No. J,II' tIMI ). ~ RoY, I.C., 1Md I. I. IU.WTOJI' AECL- 1I 11 (11)80). B.&Y av~. B. p.. aDd R.I. harwooD: LA.14a3 11GeO)' j EDeIv DepeadeDoe of Ule H oo..... D. I .. aDd R. B. Sarw.-n: BNI...SU (1t.58lcaw. SeetioM c:A Deted « WIlDII, R .. aDd A. P4 VlUM; EA.''"DC 1E) !8 ( l iMI t ). 8\1brtance...

o::.tl"

a-...

Aoa-H.&.u ..... R., and I. JrL. De OPl.Qlf : ~'f. 19M P/ISM, Vol. 1'. P. olM. D1ucl::I.. Ro: iII Nntron no.imetry, Vol. I , p. 326, Vienna: Intern&tbW Atomkl Ea-s;y A,eney 1963, ef. .-0 CEN·Rapport. R tll8 IIM I). DI~. O. W.. andI. b o...., inPhyax. olF. . -.ndlft~ Applic.Uoa of R-cMn, Vol t , p' m ; V--.: lJIt«QMiouJ Ate-ill En.t1 'I'Iuw.boId. ~ : A&-Y. IM:. E...l-uon,. Fucna. Q. L : NIKll, fW. Ene- 7. 3.56 (llleOl· 1I.anl&JI1I. 8. a , WA OO-TR.-61JS'75 (1867). KOm.a. W. : FRII.Bericbte No. n (10601. No- SO (l lleO), No. It (IM l ), No. U (1M2). No. It! (1963), No U (1963). UTB&, P.14. : WADC-TR-ll7/S (IV&'l' ). -r-roa, I, B. : CF-66-10.140 ( l aM), 0..11 Itidr National La bor at.ol')'. HUUT. O. B.,_ al.: RM. 8d. 1M\..n ,I63 (IMe). Y.-ioD Cba.mben KOKt.D. w., aDd J . Ro.""oe: NukJeoa.ik" 168 (11l63). .. ~ R llIn aD'!', P. w, and F. J . D. ... : HeUt.b Phya. I, 1&0 (1t68) . Det.eeton..

......-

.

I

14. Standardization of Neutron Measurements A great. many inveetigationa in neutron phyaica require preci8e absolute Th_ ab.oluto m eulU'ementa aro of two kinds : the absolute determination of ecuree Itz'engtlul a nd th e abeolute meuurement. of DUM in bulk mtdia or in fftlO neutron bM.mI. Abeolute .ouroe .trength meuuremenu are ~ in all l.a.veetigatlo Da of neutron_producing nuclear reactioaa, eepecially .. aDd q-meuure menta OD fiMionable IIUbetanCN. Absolu te fiul: measurement. are .. neoeesit y in many erou IeCtion meuuremente (d. Chapter 4 ) ; they are abo the buia of neutron doeimetry. particularly in nude&!' reactors. Th e problema of abeolute flux and ecuece at.rongth meuurement lire nat u rally quite intimately oonnected. In 8ecI . 14" and 14.2 we Ihal1 tr-t method" of abeolute flu meuurement, in Sec. 14.3 metboda of determining IOW'('IO atr'en,tM. ud in Sec. 1.... t he important oompuiaon methodl that permit us to oompare arbitrary 80UlCle Itrengtu or neutron flusell wit h atand&nb. me&l~eDta.

14.1. Absolute Meuunlment of Thermal Neutron I1IlXe8 with Probes Neu tron prot - are puticularly well ' uit«i for th e a b-olu te meuuroment of thermal Mutron DuzN . Th e main problem in th eir U8e ia the absolute meeeure ment of their actinty . I n tJu. RCtion we ahall di8cuss ~me .wtablo lDetbcxU for th e ab.ol ute meaauremen t. of acti"it.y. The a pplica bilit.y of th _ method. it na t uraUy not limiW to al-oJ utcl fiu: meaaurementa ; in fact . th ey form t he ba&i. for aU mcuuremen ta of acti...t.ion en- .eetion. (d. Soc. 4.3 ). If we once knoW' th e a b.olutoO valu e of th o t hennal nUl: in a medium, we ca n flMily detclrmlne th e aboKllut.e value of tP.1'I by th o cadmium !'totio metbod. We

can al80 ab.oJutely determine the bat fius from probe me&aUnlmenta . but only under th~ rwtrietioM (explained in detail in 01apteJ' 13) that ariMo from the difficulty of determining th e l pectrum. . 14.1.1. Probe 811b1tanee. lor Akoillte Me&lIllftmenta Let UI I Up poee that the flu x lpectrum in a given medium is that of Eq. (12.2.1). If we determine t he activation of a foil in this field and eliminate the epithermal activation by the cadmium difference method .. explained in Sec. 12.2.1, th en th e rnmaining th ermal activation it given by (cf. Chapter 11 ), 4>,. (~ • • (P. 6') (I+,j f(l + ... 1

C,.=

(14.1 .1&)

Thu. 1Fe obt&in th e th ermal Oux from th e actJvity~ =CufT (T _ time fad.or) UlIing the equ ation ... _ 2.4 u T(I+ x,1 b ~l4 • (lU.l )

(~; ...c,... 61) U+ fl

For thin fOla, }'w k

l and

'We

h....e

·- V::::.t1 ·.

~l4 = -_·__

~L _ _

-r-

,

(14.1.10)

Yi, tTl - .;--. ~1-.l6ll1+') Tb u. ltlu can be ab.olutely determined if .A,.. can be a beolutely counted, if t he activation correction., is known . a nd if }'wIE ) and Pod (B), or in the caee of a th in foil the thermal activation crote ~n a nd th e V-faetor, are known. In addition, the neutron temperature must be known l • We ought to demand from an ideal probe aubetan oe th at ita decay scheme be well eulted to ebeolute counting, that ita activation Ol'OI8l1eCtion be very accurately known and deviate at moee alightly from th e 1/1I.law (g .. I). and th at it have a conveniently large half.life. Of all t he probe l ubetancee di8cu-J in Sec. 11.1.2. gold com". elOllCllt to filling th_ requinlme nta : I ta activation crou tee tion (<7_ =-<7..) for tl l _ 2200 m/8eC (<7_ _ 98.6±O.4 barn) is very well known ; ita v_factor (Fit!:. 11.1.6) is alwa,. clOlle to one ; ita ha1f·life is favora ble. a nd itt limple decay acheme permita precise abeoillte counting, puticu1arly by coincidence methode. Indium h.. oeeaaionally been aUggNted for abeolute meaaW"fIIDenta . but ita activation cru. teetion is 1_ certain and ita oompll:!. decay ecbeme If. auita bll:! for abJ;oJute countinB,. Sodium (<7. . (tlal -=O..63I± O.OO8 bam. V- I . T, - lf.87' h). cobalt . and mangaDflMl hu e oocaaionaIly been used for abJ;oJute meaaurementa. but aI",a,. witb lID. loClCUraCy 1_ than that. attaina ble .lith gold. According to O IUIULoL'f D. th e MXlium. activity may be counted qui te limply by irndiatillg a NII('I11-cryl tal in a neutron nold, lVaiting for t he iodine actiot'it y IT1- 24.09 min) to decay, and t hen counting t ho tcin til lationa of the crptal with )/ k

I The 1_ ,"", "'''' pef&l.lIre nnly an "'U'o..... ly I.h... 1,....1.....1>« ing t he u.m,......w....

th e t.rondo-ity _ _ . aJi thlUUlh , IT} &nd ... . 0 1 , t 7') - 1I !.h1l. _ ... ~iho. wiUKtut

... a photomultiplier. Sinoo-.eh dea.y of •

tion,

DO

IOdium DUcle ua prodUoeII on e .cin tilla. CIOrftlCtionI for at.orption or oountel' ge<:HDetr)' are MOeIIMfY.

14.i.!. The Abeolute DetermlDatioo of Probe Aetlri~ M FOr'tho ab.oluto detoormination 01 foil a.cunUell, W I can fall on th e preeedUJ"N, deYllloped in Duolear pbyw. for the ab.olut.e oalibn.t1on of ndlo.ctlvfI p. and y -emitten. The m08t import&nt prooedUl'N are : (a) -in P-Counting. (Of. Hoon.BMAIOI8 U cd., PAT.. end Y.uu). The foil to be counted Y plaoed with .. few mechanic&l IUpport41 .. poMiblll in the interior of .. 4n-counter consisting of two half .aylin . drioal cha mben (el . Fig . 14.1.1 ). An arrangemeot ia IOUght in whicb all th e p .particle. lea vinj: t he pro be prod uce put -. Moet ~ullntly employed are methane Do.. counters. which are ch&racterized by uoeptional ltability and .. good d etection efficiency l _ l.OO± O.OI aooordinB to p n and Y.un). We ahaJI DOt10 into t b ~blo kind8 of count.er oonatruetion here. The DUQ1ben HI and H. of ennu counted per MOODd in chamben I and 2, ftlIpeetiYely, AI .-el l AI the coincidence nt.e Nil' are reecsded. After ClOITflCl· tioIl. for dead time .nd background. th e quantity

*11:

·.riou.

N, =~ +N. -NII

(14.1 .2)

it equal to the number of part1clee leaving the probe per IeOOnd l (not in. __wr cluding IIOme oorreotJora of hlsber Of· der) . If we wisb to go from N, to th e true probe aotivity, we mUlt acoount for the IlIIf.abeorptlon o r th e lI·radiation in th e pro be (el. Bee. 14..1 .3). (b l The 111'
..... 14.1.1..... _ _ Iill'oqll • lno-.l ••, ~I_~

"·A

\be _1tiQC _ t

ra*-.

M,

l'eClllf'd HI . H• • and N•••~' ill thedet.-iDKlDa of N, . U we two _ _ben br. parallel and witboIat deY., btt_ them, olmDaal, N,.

I It . IIll& - , . to ~ the outpu\A of the

Tbld the true decay rate fa ginn by

'.A _~-; .

(1U.3)

Apparently, the p-aelf-ab&orption doee not appear in thil f'Mult. In order' that Eq. (If. 1.3) hold, tbe following oonditiolUl muat be fulfilled in addition t
I. No clOrrvenion electron. may appear. 2. Th e p.CIOUDter auy not eeepced to )'-ra )' nor the )'.-.nter to p.p&rticlet. 3. The apecific ~thity of the foil mun be hcmogenoua. 4. The aenaitivity for p.radiation may not depend on poaition . 6. Th e aenait.iYit y for ,...~tioo may not depend on poaition. (Only t wo of oondiUona 3 -6 need be fnlfilled .1 AA a rule, th_ conditiolUl and thoae mentioned initially are either' not fulfilled or only fulfilled approximately, and it fa n-.ry to apply cor. rections to Eq. CIU .3). The appearance of th _ .'OrTeCtionl. limit. the a0cura01 .ttainable with the p.,.. coincidence method j enD with goJd. FA can bardly be determined to better than 1%. (0) Th e

..... IU .s.

J,. _

lJ1I'o&I4.'- 7

$

_

Ibr-".

"

_ _

fn/1.,.. Coincidence Method (CAKl'IOlf , Wou). The accuracy of th e

p.,.. ooincidence method can be considerably tneeeeeed and Ita realm of application

-

almultaneoull1y extended to aubet.anoee with CIOmplex decay acbemea if tbe fn·

,..,... I

--

---

counter deaoribed in Ca) fa uaed for th e ,8..counting (d . Pip. IU.2 aod IU.3). Th e 1O\1l'ON of e!TOt' of th e ordlna.ry /1.,.. coincidence method diacu.ed in (b) do not han auoh a atrong effect hen. eapeciall1 lor thin loUl. CJ.K!'tOlf givea • detailed dlaoualon of thia method and the oorreotlona it requiree (&110 d. POluTZI. With thiI method uaing gold .. detennlnation of FA of met&llic fou.. with an aoaur&C1 of 0.3 % fa probably poIIible.

(d ) Aet.intion Me&lIurement with • b P_Liquid Scintillator (Su n :, Bn.. CBn). P.-elf-abeorption can be gotten around by diMoJring th e radioactive IUbet.ulce eed mWng it with • liquid lcintillatol'. It it theD poMibll in principle to oou.nt NCb. decay .

IU.3. E1eetron Self AlMorpUon In 471: Ii-Counting Of th e methoda desen bed in Sec. IU .2, t h.t requiring the lea3t expenditure of 'ppar.tUA and the mOlt convenient in e:leeution it 4n p-counti ng. With the except ion of the difficult liquid lICintill.tor met hod, it it the only one th.t can be

.. :-..,

" i ', " I ~ .....

•

-.

. ;-

.

i'< ~

'""'" ~

-

_

~ r,y Mupr. (o.,cw*rl

.. .. .. .

1WSInd 0,. fIWu

" • '" ..... u .u.

.,

~ QIIf'IIW':

d-

.,

XI

""'"

_'"

n.. _ _...~ l:8d« 4"l ' j ' - tlrIen>tJlJ _Mol 0014 folio

used with ...e&k aetivitie. luch u occur in low neutron OUJ:flI . Ia thU method , however, the IIlU.• baorption of t ho elect.rolUl mUllt be Uken into lOOOunt . Tb il i.e done by introd uction of an effective IIllf_.beorption fa.cklr 8, defined by t he relation I (l4.U)

The ex preeelon "effective " dencte e th e f. ct t bllt S, not only take. into lCOOunt t he p.eelf•• baorption but . 1110 .ny ot her deviation of }I, from the .... w numbor of eleetrolUl leaving the foil d_ per IIeOOnd. S, i.I • function of th e foil thicknetlll; itean beexperimentally r,"".l ..... IU ..l. .... lor a..w. ""10 .. - . . ~ J"f\.m determined by detennining H, for foila ;oi'"I~ of ..... riou. thiek~ In . 4 :11: {l-oounfAor .nd th e troo d ecay F A , for e xa m ple, by the 4:11: fl-y ouinc klenee mothod (for whieb th e I\«'f'a8M)' eot'ft'diona mUlt be caleulated) . Ono can al..o deeee, mlne ,'4, without. coincidence 'pparatu by tho follo1rins extra polatio n method : Ono aeti.....ta • IIet of foila of .,ariou,l th ickne-'1l in the ume neutlon

, '"

..

hi""

I A- __ aho_ in Sec. 11.1.6 , IUlld. t'OlII14ia l' ireu ~ th fl ' r . - t j!-aeti. il.y of , ' oiJcL.pand. 011 itoo onooatoaUoa. No....... th .. i. no4. ~ " " ,$-rounli nc t -........ben both . idN ollhe f"il .,. _ ntfod the "'fed. cJ. the ..... ...", ~ y""iIohN..

'01'

30' field and eaJcula te. their true &0tivity .A_ CIT with the methoda


Chapter 11. One then enra..

~

of

~ <,

pol.tee th eir l pecifiC aetivitiell to u ro foil thickn_. for which H, --~ f-should be one. <, Since tb e aelf.abBorption factor ~ f--. .." . depends on the depth distribution ~CIl.a of the foil activation, H, is differ· ent in th ennal and epithermal ec_. --r- ti'ation, and under eome circum . , tanee. the two valu/llI m\lllt bo IIOparatclJy determined. d_ Fig• • ".1.4 &howl 8, (6) for "-,IU." n. tftocClft . .~ ' - - s:" (' !1of thenna1ly ltCtinted gold. foils . The ........,..". ~ Ia toiII 1- ' - ' ' ' m. .ured "a1llN of MaII'RR were obtained wit h the extrapolation method by \1M of a {J-t»tlnW!r operated in the Geiger region . The ,..,Iu. 01 P O..." , tho,", in ,..". . Fi«. If .1.' ..e~ obtained from. carefully oorTeCted b h eoinei . llJ XI ",/Go ~ dence meuurementa ca.rried ou t dwith .. methane no.. proportional r-.U.l.1. no on-... .11~ lM40r I1' C'11or -.all, _ ... Clo _ (l U ll _>11,1 counter. P61f1T't baa by the ...me method determined the eelf-abeorpt ion factor for epithermal actiVAtion of gold foUs; S"l'IJ1t from hie meuurementa is shewn in Fig . 14.1.45. Fig •. 14.1.6- 8 ahow th e " thermal " 8..(6) of indium, copper. and mangan_ (Mn.Ni ~ foile) found by MIU8T&a with a OJ d_ Geiger count« by t he eJlt.l'apola. ..... 1.. 1.1 n. oft sri') fof o.oIJ J(~ _ _ IfI lMIor foI1I tion method .

I

•

---

I

• ,

..

.,I

'n:

.

" ",-

........

•

"'f'..

!

4n\.'",-

. .... "

., .,

-

-- -

"-- .

., I ., .,

..

_..

~ b'

14.2. Other Methods of Ablolute Flux MeuUlt'ment I n tb e foKowinj: 8urvey of oth er metboda of nUl: meuW"Oment, we -hall be mainly ecnceened wiLh free neutrofl boam8 from ~ and aooelenton. Let 1111 only d eal wiLh monoenergetic neuLrou. i.e.• either let the 8OUl
... .eetion we moat mflU'lU'lI the rMCltioo rate of the deteetor n t.tanoe in .. known fiUJ: ; in order to meuure thi. flw: , we mut know the ~ IeCtion of anoth er detector nblJtanoe, eee. ThUll in making . bAoluto fiUI m~W"l:lmenti we mUlt try

to UMI detector IUbltanoel WhOM Clnl88lMlCtiOM bave been determined by method. that gin abeolute a·nlnN dinJetJ,.. Th e main l ucb method is th e tnnemisaion mothod.

In the eMlJY range froID 100 ke. to 20 MeT. we ean make ab80lute flu mea.W"eIDente by me&M of OfIUtron·prot.on _ttoring, the Cf'08I 8ection for which

• known hop!. tnnamiMior:a nperimenta and .emiempirieal interpo1atioD to about 1 % (d . Fig . "".2). We may umm8 th. t up to about 10 Mev the _ttering in th e C"Y"te m ill iaotropio; at higher energiM t be aniaotropy kno1VD from ItO&tt.ering experiment. mUBt be tak en Into &coount (Gu u l aL). Suitable doviee. for oounting are th e proton recoil ecunte r diaou8llOO In Soc. 3.2.1, ecme of th e oounter teleeoopel diecUllllCld in See. 3.2.2, or nuclear pbotoplates . The accuracy of .ucb Ow: mMiurementl ill about 3-6 %. Sin06 the fi.ion croa IeCtion of U- " Uowu to . bout 4-7 'J1i. in th e energy range from 100 key to 10 MeT - from eompuaon meuurementl wit h the neutron·proton _ttoring Cf'08I .ection - and &inoe f-ion oounting iI Maier to perform than recoil proton countina. fi.ion coun ter. wit h U" are frequently used for ab80lute neutron flul: meuurement. in enugy raJIICI. In the enerzy range 0 -10 key, one can make abeoJut.e flul: meaaurementl by mea.na of the BI"A. «)Li' reaotlon. The croet MOtion hu been meaeUl'ed by tranl' mi&aion (1rit h a oorrection for tb e .eattering Cl'OIIlI Ilect1on. wbcee influence inCrea&elI 1rith inCf'eNing enerzy) , 00 ill known. to a few percent ; within t his accuracy the ~ ~n folio•• a 1/1'.la. . Bit ia used in BF.-counteR . in borc n-coeted oounterl or in beeee-jceded I t -. I n the latter. COI'T'eCttone for neutron IC6to tering may .ametimN be - . r y. When th e neutron energy iI kw.I th an 10 ke.... a IIUfftciently thic k .I ab of BI. will ab.lorb all ncuUOra incide nt on it and we ca n determine the flul: 1ritbout Ir.no...ledge of t he B" (II, «)Lif crose eection if we can determine the ruction rate by a bsolute ooun ting of th e 4?8·kev 'Y -ray of Lin. For tb il detennination. knowledg e of tb e energy .d cpend ent bra nching ratio of th e Bl' (n. «)Lif reaction i.s neee-ry. No good techniqUN ha ve t-n d eveloped ee far for abeolute flIU me... urementl ill th e energy range 10- 100 k. .... T'hough the DI!IUtI'On-proton _tt,ering CI'OllI Iection iI well·bKnrn in t hil energy negion, recoil proton detectonl c.nnot be used IiDClll it iI bardly P*'ibl. to count recoil protoN of . ucb lo. enefJY. Therefore. ODe mut ue detectoR ba-t. on the reactionI B" (II, «)1.l~, He' (• • p)H I or lJ&{. , «)HI, bu t unfortunately the crou ItlCtioni for th_ reactioN aro not very well known in tho eoergy rang e 10-100 b ... owin g to th e difficulty of mcuurillg them in, trall6lDUiion el:periment (0•.':.0.). A t horough m.cUftl:lion of th o problema of flul: and CI'OM lleCtion meuurementl in thia enorgy range can be found in

uu.

B.l.T()H&LOR .

BesidN th e procedUftll diIou.ed here for direct nIU me&llurement, thoro eDt. in man, ea- an indirect poMihllity . ,.h.: if the eDef'IY lpeetrum a nd th o ltrength at a Deatnla 80UfCltl &AI known , thea .. 1ooa &I there . DO b.c"-tterina: the Oux • &leo kDown in the immediatol Deigbborbood of the 1OUnle. We CUI then ClOmpue the known 01l.Jl produoed in to .a'11rith ao Wllc.DOWft by meanl of a .u.itable

detector, e.B., a Iona: counter. UM of the .o.oalJed ataodatd pile, which hall oocuionalJ.y been employed in the abeol.ute determination of thermaJ. aDd epithermal DeUtroa flllIN, can. be counted among the indirect. metbodl. A atandard pile ill a graphite bloek IIOme %x 2 x 3 m along th e edgee in which a ealibrated DeU tron IOUI'Ol'I b.. been pla.oed. The thenoal and epithermal fIllI fJI are then calculated witb th e bclp of age and diHueion theory. An unknown flw: • dete rmined by oomparing the foil acti vity it produoee to that prod uced by the fluJ: in the standard pile. In making the oompariaon, th e foil ecrreetlon mUll. be taken into aeoount .

14.3. Determination 01 the Strength 01 Neutron Sourees Tho following different mothoda for th o abeolute calibration of neutron may be di.tinguiehod: I (a) Tho Method of AMociated Particle.. Hore we make QIIfI oj th e faet tlr.at in lIOII10 neutron·producing reactioD8 a charged particle 18 emitted eithe r promptl y or after lOme delay; if we eeeeeed in counting th_ particlfJI absolutely. _ know the IIOUI'Ol'I stre ngth. (b) Direct Flu Mewuremont. If the spectrum and the angula r distrihution of the emitted neutrolUl iI known, the IIOUl'CfI . trengt h can be determined [rom the OIlI in the neighborhood of the ecuece all long M we are careful to IIUppreu any b6cbcattering. Thia procedure it the invefllfl of the indirect method of determining the neutron Ow: that Wall mentioned earlier. (c) The Method of Total Abeorpt.ion. The IOU rofl it plaoed in a very large absorbing medium. Then all th e neutrons emit ted by the eoueee are elcwed down and fmally abeorbed in the lWt'OuDding medium ; th e ecueee .tnngth folio.. from a . pat.ial. integration of th e ablOrption rate. In See. 14.3.1, we thall deal with th e method of auociated pa rt tcae.. In Seot. 14.3.2 and 14.3.3. we . hall diaeua& v arioua . aye of applying the total abeorptlo n method. The prooedure doeeribed under (b) requil'ft no further n . planation; in fact, It is not particularly accurate and it alao only applicable in special caeee. IIOQI'(l(I&

14.3.1. The Method 01 Auodated PartId..

In th e rea.ct.kln H' ld, ,.) H8' . &.II «-particle appeare llimultaneoualy with th e DeutroD. Ita kineti o energy depend. on tho d.ireoUon of o.m.km of the emitted M atron aDd on the deuteron ~ : for 8 . _ 0, 8 . ... 3.• 9 Me... in aU direetiwMI (d . See. 2.1.2). We can....ny detect the ll·particJfJI emitted in • limited range 01 80hd angle. from a tritium ~t with a ecintillatkln eounte e (prefera bly ZnS ), a proportio nal eeeneee, 01' a ItOlid ltate counter. Fic'. 14.3.1 tho .... a ty pical oounting apparatua UMd by l.aB.ssoN . At low deuteron energiee (E. < 200 ko...). th e angular distribution of th e neutrolll it quite iaotropic in th e C·syetem and approJ:imately ao in the L-llyetem. If LjJJ is the lI01id angle eubte nded by th e e-ccunter, Le., if aU the partielfJI emitted from the target in thia angular region are counted, then the oount rate of tho Il-oounter is llimply given by dD

N.~.;- Q

(14.3.1)

and the~.trengtbe-neuily bedet.ennintd. When higher _ ura.eyi s - . r y .nd partioularly at higherdeut.eron energill8, we muat take th e .ngular d.ietributlon of th e neutrona eed th e ct·partielee into account. Thua to determine the 8Our08 Itl'eogth from N. , the (deutel"on.e nergy-dependent ) angule.r d illtribution of the neut.l'oM iD the C..)"'tem nUlat be emp ioyed : fortunately, it is very e.oouJ"'tely kno~.

Th e reaction H' (ei', II)Rea can . 180 be abllolutely counted by tbill method. However. we must t.e.ke into e.ooount th e f-.ot that when deuterium is bombarded 'tritb deuteron. th e reaction RI (d, p jlll can a180 occur ; at low deu teron energiee t he c:ro. eeetion of thiA reaction • about the lI&me .. tht. t of the UI(d, II)lIea reaction. Th e Hi lI', II)He l reactioD. can aI80 be . t.olutely eeuneed. Binoo th e emillllion of the neutron .. nd the helium nucleua in .. II ~ th_ reaotklDl OOCW'll eimultaneoully, the ct.pe.rtiele or Hea nu eleu. can 1110 be used ... .. tim e m .. rk ; th e ..ttainable precWonis ..bout 6 xlO"' _ . Th e a.oouracy wit h which 8OUI'CfI Itnmgthe m..y be determined in tbill w..y is about 2% in th e CAlle of th e H' (d, n)He l reaction. I...u.s8olf givee more detailt. According to RiCJDIOlfD and O dDUR. the . t.olute Itl'e ngth of .. neutron lIOW'Oe baaed on the photodWntegr&tion of the ..... 11.1.1. ... ,,..... " I ' lor _ . . . . deuteron can be determined by abeolutely . - - __tIDe counting the photoprotou ariaing from th e ~ UI(y. II)Hl. Theee au thors used .. ThC" preparation .. the y..-ut ter 18, =-%.82 Me,,). and placed it between two deuterillDl·filled ioniution cha mbeR. In eac h (y, II) prooeM, one ne utron and OrM! proton with. kinetic energy of 190 kev a.re produced. It 11 quite ouy to detect the proton. The aouroe .trength is very amaU. Varioul neutron.producing nl&Ctioos 1Md. to radioactive ftlIIidual nuclei, and it iI pt*ihle to determine the ~ Itrengtb after irndiation by a t.olutely counting th e radioaetirity. M .. rule. to do tbill it is n~ to deetroy the 1OW'08. AD important UAIlIple wthe (110 II) r'MCtion on F" (Sec. 2.2 .2). which Ieade to the well-known polIikon emitter N..... The J.if (p, II)Iklf reaction leada to an electro n eapture aotivity in Bef (71 - 63.40 d ) that ean beet be detect.ed by meau ol th e ",'8.b,., · y.radiation of u.... Finally, _ can determine the -.lUl"C8 1tl'ength of the reaction Bet(y•• ) BeI by quantitatinly determining the amount of helium. formed by th e ct-d_y of Be'.

U.s.t. ft. Water·Batb Gold Method Thia method iI the mOlt uaua! and. far a nd. .....a y the m<* aocunte for the at.olute detennination of radioacti ve IOUroe etrengtha. The 8OUf'OO wplaood in the mkidle of .. filled with didi.lled ...ater that w10 I&rgeth ..t only a nesligible

,,_I

fraotion of th e neutrow eecape from it.l • Then all t.he neutrona emit.ted. from. the IIOUl'Oe are slowed down and ultimately absorbed in the water. Thua Q ~ffE. IEI
(1....2)

Th e integration e:d ends over t he entire volume including the IIOUl'Oe. E.(E) is eit.h6l' the absorption C1'OM eection of t he water or of the II01U'Oll eubBtauce. H we beee our oonBiderationa on the thermal Dux, take the absorption of neutroWl during moderation into aooount by meall8 of a resonance NOape prcbabilit.y p, and 00l1"6Ct. for the neutron abeorption in the 801U'Oe by a factor KG' thon pQ =KgfI-.A !Pel{r) dr . (14.3.3)

b'"

We have asaumed here that. the 1IOur06 radiates i.eotropioaUy 110 that !Pel will only depend on the diatance r from tho lIOutoe. If thia » not the _ , .... oa.n force the reaulta to agree with thoee for an iaotropio OOur06 by rotat.1n& tho OOIU'Oe during the meaaurement. E... is the absorption croaa aectlon of water for thermal

E

(E... _->;-

neutrona ~;8. o {"" )) ' I. (v,) ilvery aocurately known [0'0 (11'> for hydrogen is 327± 2 mb ame ; the oontribution of the o:l:ygen can be neglected]. Since we can a bsolutel y determine the thermal neutron Dux very &OCutately ( ".,,0.6%) by cadmium difference mea&uremente on gold foils, Eq. (14.3.3) permitlJ a aimple determinati on of the eource strength. In making this determination, however, we must carry out a flux integration over the water volum e. This integration is moet.limply done by me&6Uring !Pel'" a function of r and grapb.ioa1ly integrating the fun ction 4n "'!P (r). HOW6V6l', we can &lao move the foil through the water bath during the irradiation in euch a way that lUi activity at the end of the irradiation directly determinea the flux integraJ. (el. Cu1ma8 or Loa). The oorreotion factor Kg can be oaloulatad with the help of diffuaion theory from the opecifieations of the source; for 0 typical (Ita- Be) aource, KlI .... l.OU. Taking tbe reaonanoo elloape probability into account iI more diHicult. p iI eompoeed. of a factor Pi that de&eribee t he (n, «) proceaa in OlD a bove 3.6 Mev and a1Bo the (n , p) pl'OO6M above 10 Mev, and a factor Pi that acoouuUi for the epithermal absorption in hydrogen. PI can be oaJoulated. using Eq. ('l.3.21), or it can be derived from the epithermal absorption rate using the meaaum cadmium ratio ; it ia about 0.986. P:a depende on the opectrum of the 1I01U'Oll, and in the cue of a (Ra - Be) SOUl"Oe, for en.m.ple, cannot be calculated with auffiaient a oouracy because the oontribution of the neutrons a bove 3.6 Mev to the total aouroe IItrength is not &OOU1'ately kn own (on aooount of the uuoerta.l.ntiee in the epectrum for 8 <1 Mev ; of. Sec. 2.2.1). We oan detormine Pt.uperimentaJly byouooeul.vely introdncing the aame OOUl'Oe into a water and a paraffin bath and meuuring the Dux iDtepaI with a nlIOnanoe probe. Sinoe the p&raffin oontainl no O:l:ygen, the value of P:a iI one there, and the ratio of the reaonanoe Ow: integnJa ahould give Pt.. Since f I. is different in RIO and paraffin, we must make an additional nonnallza· tion meaaumnent uaing a n (Sb - Be) IOUI'Oe , for which h iI alwaY' cae, In thiI way, D.TBonaand T.t.VUNIUfound Pl -0.918for a (Ita-Be) aouroe l; P(I:IIl'rZ IIf a (!\a- Be)

touroe

an the DellWOIII _poe.

111 plaoad in \he center of a ophere wit.h a 1.m. ndiUl, om1 o.G% of

I Th1ll "aJu. 00_ frolll a roe"aJ....tion of !.he 'l'aJue __ 0.876. x..._

~_tD of

D. TaoT" NId T....·

11'1"; the origIlIaJ )lDc~lI$/Wlrta,

~

10

follDd p""",O.082±O.CKK. p" c1ecIreuM rapidly with increu.ina; I01ll'Oe energy bec.uae of the riM of the crOM IClOtionI for the (It, II) and (It, , ) reactione in (}II ; for I..Mev neuWooa. P.t-=0.826 aooonling to I..u8olf. It ill obriouly DO longm.eoeible to uae a ..ater bath with hia:h-energy ~, and it ill much bettor to aM puaffin or another organic liqWd that con taila8 DO osygen . Sinoe with in~ neutron enere the requirement of .. infinjte .D.e" Ieada to eYer greater dim.enaiona of the bath, a pat&ffi.D bath for I4.-Mev neutroc. u quite npeneive. BoW'C81 of error in t he wate r-bath gold method ariae from (i) the ebecl ute determination of 4'", (... 0 .6%), (ii) the flux integration (1Ilil0.6%), (iii) t he correctioa. facton Pl. Pt , and K O (about 0 .3% NOh for .. typiC&1 (Ra - Be ) IIOUroe), and (i..) from the ab.orption cro..-ction of hydroge n (... 0 .6 %) ; the total error ill about Ui % . The -eatter of the ..a.ri0u8 independent source atn mgt h determiDatiooa ill of the MIlle order of mapitude (d . Sec. if.6). I,U .s. Other Total Abeol'JUoD IIMboi.

(a) Tbe Ma.npnNe-Bath Method . Tbe lOurce ill placed in th e middle of a .ufficiently large v - t th.t oontaiJla an aqueoua -olution of MnSO, . A lUBe fraoUon of the moderated neutroc. are tben ca.ptured by the mang.n_, leadin g to the weU·lmolm 2.M-hour activity of )(nM. At t be end of tbe inadiatlon, t be lIOlutio n u carefuny.tirred to uniformly diltribut.e the acltivity, a definite volume ill taken out, and t he acti vity of the MnM it oontain. absolutely counted . If we could make the concentration of lh e manganMe in th e w..te r lIO high tbat ..U th e neuttonl would be abeorbed in th e manganeee , thiI method would ohvioualy yield a result independent of any kDOWI~ of the ere. 1IeCtionI. In praet.ioe only about 60 % of th e DeUtrona are abeorbed in th e manpn_ , eed th e ..bo ~tioo crOM -uooe of Mn and B,O mut be l-t. in the evaluation. In addi tio n, neutron abaorptioo in the l uUur ca.nnot entire ly be oCllJiected. Tbe accura.oy of abaolute determination of lIOurce atrengtha by tWa method 11 .. bout 2%. Thia error la mainly due to the difficluItyof aboolutely oounting the mangan_ activity. Tb e mangan _ bath method ill much more aecurate when used AI a comparieon meth od (d . Sea. 14,.4). (b) The Boron.Bath Method. Here the ablorption of the neutrons emitted from the IOUItle oooun in &II aq ueoua lIO!ution of a boron compound, e.g., borio .cid. U N.(N.) lathe number of boron (hydrogen) atom, p&Eoml , then neglecting r-.ooanoe abeorptioo aDd abeocptioa in the IOUItle we ha"e

.j4'1'"J-

Q=(N.lJ.+N.lJ.)

z= N.a. (1+ ~:';i)

•

fZI (r) dr

I

(14.3.4)

h'" fZI (r) d r •

Now we determ ine 4'(r) with a Ima n boron counter. If V. ia the Illnlitive volume aod 11. t be number of boron atom, per om' of ecunter volume, the oountlng ute ill given by (14.3.6) Z(r) - fZI( r)N. Y,a• •

U we n betitu t.e thi8 in Eq. (1• .3.• I, we obtain

•

Q=<

:;;;(1+ ~:::) J4" r'Z (r ) dr. •

(14.3 .e)

The crotlII IIllcl.ioD.l of boron and hydrogen only eeee e here through th e correction term NgaslN. a• • which for b.lgh boron concentration is 1ID..u oompared to one. By variation of N.1N., we can make the method entire ly independent of th e CI'OlIlII aeetiol1ll. The achievable aoeuracy is about 2%. (0) Total AblIorption in .. Liquid Scintillator. Th e neutron lOurotl is placed in the interior of a very lArge liquid IlCintillatot in which a oadmium ....t hal been diMoI.ved.. The neutrons are moderat.ed by oolliaiolUl with the atoma of the lcintillating l uhltanoe and are fin&lIy captured by th e cadmium. Capture by the oadmium initiatea a complicated y-cucade with a tot&I ener'17 of about g MeT. Th e lcint.il1t.tionI produced by th _ y-ra~ are detected by pbotomultiplien OIl th e I1Irla.oe of t be tanlr.. An adnntage of t.hiI method it that then 11 a prompt (~20 .... 180) relpoIIM to the neutrolUl 10 that ooincidence methodI are poeIible. Th e coincidence technique 11 frequ ently ueed, for example, in mea.l1ll'ementl of th e av erage fialiOD multipUoity j . A dieadnntage of thiI method 11 the extreme y.ltlnaiti vity of the tank j OOIIDio radiation alone prcdccee a h4rh background oounting ratoe, and it it ntlOtll8&ry to IUPProu low.energy y·n)" by uae of an int.egraJ diIoriminator. In t.h1I way , however. we 10Ie lOme of the yoraYI ariaing from tnle capture eventl, Le., th e detection efficiency of the appanotne i8 no longer 100%. Th ne we mUit either calibra toe the appant.ua or calculatoe the detection efficiency with t he help of a Montoe Carlo calclllation .

14.4. MethodJ for the Comparilon of Sonne Sb'enrtht If we p - a ltandard neutron lOurotl th at hal been O&1i.lnated by one of th e method. dilwlltld in Sec . 14.3. we can reiN' the Itrenith of an arbitrary lOurotl to it .. long aI th ere 11 lOme luitable oomp&rilOn t.echnique. Such a oomparison il limple when the lOurotl to be oompared. with the ltandard hal the ...me energy lpeotrum ae the latter and when the angular dietributiolll of the emitted nelltrol\l alao &gI't'll. Then all we have to do iI limply to oompare the flw:tllI &rising from th e two II01UCM with an arbitruy detector in a reproducible way. H th e energy l peotn of t he lOut"OeI to be oompued. do not agree. tlIe oompari8on can. be carried out under lOme ciJ"eumatanoea by making flw: m-. aurementl with the b1g oounw diaenaeed in Sec. 3.1.1 ; howeTtl!', in thia procedure, deriat.i008 in the two angular dietributioDa mUlt be carefully taken into acoount by an integration over all anglea. Even more ...tidactory are oomparillOn methode in which not only i8 the detection MDSitirity independent of the neutroD enersY. but the neutrona emitted. from the IOW'Otl are abo eceated in a 4",.geometry. In principl e, all totaJ. abeorption method. fall into this catClgory . The mangantlll& b.th method diIou.-i in Bee. 14.3.3 ill very frequently ufledfor aourotl compariaon . Since the mtJn IOUro& of error - tbe abeolutecounting of the Dl&nganeM aothitydrop. out. th e method ill ert.remely aooun.toe (_ 0.6 %) when aeed far oompariloa meuure.mentl. (The erT'OI' eatimatc dOflll DOt. ino1ude enorI .bioh m&Y..n.e from the ablorption of energetic neutrollol in oS)'l'lln. Howner, for DtlUtIoD - . -

...

...

.

_-

... . -

""1" Ln. .......

... introduced into the Middle of the aphere through .. 10 X10 em O&D&I. lWlu.m oaloulated the let\.lith it,. of tim apparatWi with the belp of age IWMl diffuaoD theory; it ill oonat&Dt to within about 1.. of ita ,..loe between I kIlT and 1 MeT and illOmewbere ira th. neighborhood of 0.036. RonTel', it fa11I oH rapidl1 for

__-~-.Jr ft- . ~JPhn

i!~~h-J-~A.1

i

~ ~-t:~~o . .G

II

.~

L__

: _

I

-,

higher energiM. M..t.cmLIl'l "''' lobi. to ahow uperiment&1ly that the .naitinty doe. Dot d epend on the angu1&r diltriblltion and that the le&kage of DWtl'oM oat of the ce.nt.l left open for the Introduction of th e lOutOe b.. DO appreciable effect 00 the eenaitirity. TlWI apparatUi ill partioula.rly 'uitable for the atudy of neutron· producing nuclear reactJ.ona wit h aooeloraton. (e) Other Apparatua. Vuiou arraogmamt. of BF,-oounten in .ater or p.raffi.n han been eeed. AM &Il e:umple.... Iho... In Fig. a .4.3 a parafflD detector iDnmpted by M.uJOll' d cal. A total of 11 BF,.oouuten are U'!'&Il£'lll

31. in two OODceDtrio rinp iD the middle of whiob • 1ooI.ted • una! for introducln,: the neutron IOVOe (aooelent.ot target). The MnaitJvity of appant-Qa ia about. 0.1 aDd it. ia ~t. withiD .bout. 6% of It. ....Iue in the energy range from 100 key to 2.6 Me,.. Compared to ppbitoe deteeton, we C&D euily attain .. higher IMJOIitirny1rit.b. am.lIer dim-.iou,. but. becaUM of the aborter diHoaion length . . cannot build detect.ora whole IOnaitiTit y plateau re&chtAI into th.low key region. (d) U8e of • 8uberitical or Critieal Roa.ctor. Aooording to EaaLlla and W.i.'lTDBUO , one C6Jl oompartl diHorent neutron aouroea by pl.cing them in •

uu.

aligbUy subcriti w rea.ct.or and t hen oomparing the reeulting t hermal fhu ee. In cedee that the compariaon yield .. reeult independent of t he IOIUOe apectrum, th e re&ctor muet be .. large that diHerenON in the MO&pe probability of the ....riOH faA neutrorw do not. oome into oonaideration. Thi8 method ill hardl y used any more. If onelntroduON. neutron lOuroe into an n&ctJy oritJoAl r-otor, tho lattor ..w _bow an I n _ in po••r tbt. ialineu in tim. and whoM magnitude dependa 011 tbe ese-. neutron production. We e&n ClOIDpellMto the U CMl neutron pro. duction by introduction of • Del_Un 1OUItle, t.e., an ..beoebee. In . teed y ltate . the abeorption rate ia equal to the IOUItle mength. U the ab.orption nte can be determined hy.bIolute oounting of the abeorber activity, the lIOQtee atrength ill abo known . Upon tb_ principlee depends the method of Lrrn.u. The aocuncy of this method ia limited by t he faot that the additional neut f'O M are emitted at high energy but a beorbed at thermal energy. and thul COrrectiOrUl mUlt bel applied for rMOn&nCle ab.orption and leakage. 14Jj. A Comp&rl80D 01 Varlo u, Source-Strength

Mf!u urements

on (Ra-IIe) Soure.. In order to ebeek the .,ariouI prooedutel for determining .aurae Itr'engthe and ~ internationally recognized ttandard of ItOW"Oe Itr'ength.

at.o in order to obta.in

oomJl&ri-ona bu e frequently been made of th e

.ourtlN

M".U.•. l~ CoMJlOri-t M _ _u.

...."" .-..... Ilnllltb ..

"- (

,..}

~

NRC """""

U :I X10"

±U'I.

PT8

:un x 10" ±.....

PrB .........r

t .Nx to"

ABA_

J.U x IO'

t .U8 xIO"

±'"

±' "

......

....... ..........

±, ..

t .08 xIO"

UIlioQ KiDiin

7.88x 10"

7.88 x lO"

±'"

±"

UZxJO"

UM'

UiexlOf

±1.S"4

±'"

s-.

8oou<w .. M-...I by 0""'•

tlkAU

OM

U. x 10"

3.134X10"

J.m x 10"

J.ln'5 x 10"

±.....

H ....

H" U8t xl0'

±'" U5 x l O" ±2.t'l.

±U'I.

Um. BMeI

a.-

(/l4-&)

±.....

UXAEAs..-D

BoJs;=

_

... ..-

Ilnllllll .. "'"

._ ~

abeolutely calibrated by

U I6x lO"

±t.t"4 e.oo x IOf

±.....

l .MO xlO"

±t.?"-

...

~ 1 .;Ref_

..rioua labontorieL Sueh in..ti&atioN have mainly t-o made with (Ra- Be ) lOuroeI . wWeb becauee of tboit' long half ·life and good yield an JlNtieul.,ty .uitable. In tb_ inveatigat.ioM. particular attention mUllt be paid to In· Il~ in time of tbo tIOUl'O(l .trongt.h due to tb e buildup of polonium. 'I.'bU lnonue ia dtllKlribed by

u..

Q(~)=- Q(")

+

1 0. 1.( 1- .-o.ou1/1, +loll 1+0. 1.( 1_.-0:01171,)

(lfJU I

Here Q(tol ill tb e eouree st rength determined at th e Umfl of calibration " ,yflUS) - reckoned from th e t tme of radium fI:rlraotion - and Q (~) 11 thflllOUfOfIltrengtb ~ ye&r'l after calibration. TIW fonoula of ooune only holda whfl1\ t hfl initial inOl'6MO of tbfl -oUJ'Oll Itrengt h (800. 2.2.1) 11 en ded. Tablfllf.6.l oont.t.IM th. reeuIt. of eomfl inte rnational oompoon.on meuurementl (u of Fflbruary 1982). Tho asroomWlt 10 gClnuru.lly IllI.tlaf~'y ; wo oan mOf'flOYl'lr eonelude that aouroe Itrengtb m.,..uremflnt. can be carried out tod..y with an accuraoy of t4. 1.4 %.

Chapter 14: Rererenee8 Oen.ral

_peailI1I,

H uo• •• D. J .: Pile NMitron ~ c..bridp; Addi-oa-W"" 1063; P. 7S1f. : NeutloD 8~ LuMow. It. E.: J . Nil'll. m (1868). NeaVoo Dc.imetly. ~ of tbe 1M2 H-.II Sympooillm. Vol I aDd 2. Vieua ;

r.-v .. In~ A1oIIl.io r.-v At-I.IMS..

R!(lB_o.o. R.,;!'roar. Nllot. Energy. 8er.l. Vol I . p. lalI (11168). W...t T P I _ . A. l

n.

8~tiaD.

of N_Voo I I _ I e . AIm. RaY. NlleL BeL

.. lit (11M3).

I

8podal Oaaxm..o.wo. D.: Ph,... Re.,.. 81. 1137 (11la2) (AbIoh,,,- 11>u: K _t with N.I).

eoa••. R. : AIm. Ph,... (Pvio) XII. 7. 18& (1HZ).

HOtrr....u •• F. 0 .• 1..lln . ... Sc1r1lTuIamn. g . D. H. VnoOPT:

z. Ph,..u.: 1M . I (1'152).

. -eoaz,ter 11

•

PUI . D. D.• -.nd L. YI.J'I" : Carl. J. Chen. U . 16 (19M ).

MOIIUM.E. R.• &D.d W.:M..H.UBPB rt : J . Nuel. Enerc A .t B 14. 26 (1961). ) ~.,. Ooiadden.OII PvT. ..... J. 1..: Bnt. J . Radiol. II. 0&6 (19&0). Method. IUJTu, J . P.; J . Nool. Energy A 10. SI IND). C..urPIOlf. P. J. I Int . J . Appl. &di&tion Jlkltopol4. 232 (1H8{69).} ." fl.,. Coic.eKleDOII WOLl'. 0 .: NukIeon1Ir. l. 2M (IM I). Method. DIlLCIIP. E . H . I J . 8ei. ~.~. 286I IW)' 1 &r1Tlf. J .; ill. Metzology of ~~ . 11 '-SciDtillat.tou. 00watiDa;. p.278; VMnna; Intflrtl&tioaal Atomio Entqy Aa-Y. Ill6O. 1Uwrp, H. : Z. N&.kIrloncb. lI.. m (1t68 ).} SeH.A"'---:- ' • .._.......A.IR_ P
-y-

888 (1M 7). 81'oCOInO•• R. W•• aDd J .lW.naDr: NIMIl tw. WOU'. NukIeoDik e, W (IMI).

a.:

Ene. 1,

en- 8ectioM

100 (18651).

b'1'tw1:tal Neatntoa.

-

At.oIlI"- J1u O...-u., J. 1..: Iza Ilaro. aDd FolfLD, 100. Vol n, a:.pt«V. T. } 1I_ _, .....,.. J . Eo; Iza Ilaro. Md r~ 100. CIllo Vol 1. Qapt« IV. B. wtt.b Recoil

en..

_a . foo&aoM lIIl P. A.

B.I.'IODLOa, It, (ed.l'oz')l JtANDC.J3U (11163) (AbI01l1te 11u.z R&D,. 10--100 bY). Danb, ,T. : CD.Report R 1880 (JHO ) (The Bt.ndwd Pile). B......... ' H. B., L. ~D'. U1d. R. F. T~ : lWr. Ph,.. H. 1 (1162). K. K. : Arkb Py.lk t, m (19M). x..n.. P. c.: Nucl.:rn.t.rum. I4et.I:IoU I. 26 (19li9). Jt.I:OBJlIOlfD. R., ud B• .T. GaD.": AERE.RfR 2OU7 (1967). Laaeoll'. K. E.: ArkIT pPit 7, m (19M).

IleuIuem«lt in the

Enezv

Kod.l Souroe S------

Luuo_.

:~ure:'~~

pvtloJet. ) PL&:n.... aDdP . HD".: Boll'. Phya. Aota 375 (19M) . The W.. t«-Bath Gold ~n:a. A.. II" '" O. C. T.I.,.DIIID: BIIl1. Ac.d. R. Bel(., Cl. .ci.. Method. fWr. 6, to, 1M. CnTlu, L. 1'.: IDWOduotion to NnlZon Phyal_ D. YOD No*aDd CompanJ Ine, (1m). FlD.J: Integn.tloo 1rith &oil, W. : Nuk1eoDik" '13(1960). MechaniO&1l.ylloved Foik. W..I.L:&.JI:Ilo R. 1..: 1IDD0.41f (lVo&e). Ano., E. J ., &Del P.cao... : J . Nuol. Enuc AilS II. 22 (11161). ) Fu.... O. J . : Phya. ReT. 108, lit (1967). The MI.na:...- Bath J _ . J. DB.N1d J .CIIIJI': J.~N ..~ Bur. BtMd. ".311 (19M) . Method. WALIII:D. R. 1..: GOO '14, (UNe). O'NlLI.Lo R. D., ud O. SouuT-GoUl........ : Ph,.. ReT, It, 168 (1D66) (The Botoo B&th

c..

n.

PriDCll toD.')

.......

).

Imu, B. C., ..... e Ph,.. BeY. lot, 1011 (19M). } lJquld Soiotillatol"l for 8ouro&Rllrx... F., filii. : R.T. Sol. Iuw. II, 1061 (liM). 8vqth ~t&. OoLVIlI'. D. W., aDd M. G. 8ow~y : 0 -.. 19as P/O, Vol. re, p. 121. On.VDI, D. W., and 11. O. 8o.... Ed1' : EANDC(UK)·3 (I MO). of 8oUftJOI.rTn.u. D. J .; Proo. Ph)'L Boo. (London) AM, 638 (lUI ). Method. Strezlith Comparieon. lI.t.OIlUlf, R. 1..: Nlltll.lnaWum. M:~ocU I, 33li (IU 1). HdroJl, J. B., .111. : Nuo1. IlllItNm. Methode 8, 291 (1960). W 4tftnDO, A.,1Wld C. EooLD: AECD SOOJ (1960).

15. Investigation of the Energy Distribution of Slow Neutrons In thiI chapter, we &hall OOD8ider experimente for determining the lltationary .pectra. of very IIOW' (8<10 ev) neutroDl in bulk media. The main part of the ohaptel' ia oent.ered around the dieouuion of v.now methode of meuuremenu (See..I5.l r.nd US-2) ; in the Jut Mort IflCtion we lummarize lOme important rewlte for nriolU moderator lUbd&noea and relate them to the generaJ theory of

ChAp'" 10. We mud diatinguiah between dillffeltli4l and il'<gr'cU methode of mfllltllur&meot. Th e former are ba-l on the det.ailed eo6l'81 analysis by time -of.flight or with a orpt&l .pectrometer of a neutron beam extracted from the medium. They immediat.ely yield the lpeotrum 41 (8). In oontraat, we only obtain from lntegreJ meuuremente oertaln averages OTflJ' the spectrum for whoee further evall1&tion we need additional knowledge of the .peotrum. For nample, we O&D determine the neutron temperature if we know that the thermal part of the .peotnun CI&n be appro:rimated by a Ma.:nrell distribution. With regard to the php1callnfonnation they lupply, the differentJal methoda are therefore far nperior to the integral methoda. H owever, they require fu more apJ-r&tlU ud are limited in their applicability to relatively simple sy'8teml .

For example, it iJI UllU.uynot posaible to eztrt.ot from a reaotor lattioe a neutron beam that is repreeent&tive of the entire neutron field. On the other hand, integral probe meuurements can nearly alwa,. be e&rried out. Today, therefore, we use differential methods for experiments on "o1e&n"IpteInll ; these uperi. menu are directed toward revealing the baaic phyaioal ch&ra.cter of the IpfIQtrum IP(E) and toward &iding the development of a general theory. Integral meuure· menta are then used to check the theory for 1pfICIi&1 .~ma thAt ani not alwa,. aooeeeible to the diHerential mMl1mlmenu. Before tbe introduction of .ufficiently reliable differentia.1 methods, integral mfllllnll'flUlflnts yielded the only UMble information on thermalisation in a It&tionary neutron field.

16.1. ObservatioD of the DWerentJal Speetrom. by the Time-of-Flight Method A dilierentiJl,l epootrum can be lltudied. 'lll'itb a cry.t&I epectrometer or by the tune.of.flight method. Although many very Mn:Iful invfllltigatioll8 have been oanied out with crystal .peotrometen, thie method bu not prevailed over the tune.of.flIght method, chiefly becaU18 of the difficulty of oaloulating the energy depeedeeee of the Ipeotrometer'llIlInaitivity. We Ihalllimit OW'lIllIVflll therefore in the following to dilouulng the time.oI-f1lght method. Fint, however, we ,han di8cuaa a prohlem common to aJl differential methodl, viz., the utnotion of a repreeent&tive neutron beam from a medium .

16.1.1, The Problem. of Bum. ErlraetIon In order to obtain a neutron beam for the IIpfIllk&I ULlJ.yaia of a neutron field, we UIfI an utr&ction tl6nal that reaehflll from the ewf_ of the medium to the point r at which the lpootrum iJI to be obeerved. Such a oanal generally h.. the form of a thimble, .. ehown in Fig. 16.1.1. Let us denote by J (E) [om-'ll&O-1 ev- 1} the energy-dependent cur. rent denaity of the beam 110 ebt&ined. What we wish to have iJI the energy .pectn1mIP (r, B) of the nUl: at the point r . We shall now determine the oonneotion between IP(r, E) t.nd J (E) under rtI'. 16.1.1. TnIoaI_ ........ ,......_ ""'""'"
'"

The t.tl« relation talr.8lIa partioularly simple form in an isotropic neutron field. There '(r, n. HI ilIl11i.m.ply 4'(1', glIb and J (8)_4"(I', 8). We Ihould thua take the beam .beDeTtlf pcaible out of .. region of Uotr'opio flu. Le., .. region or Taniahirlg flu gradient. U there fa .. _k dependeooe of <1> (1', E') on poetion. we C&n tAke the OOnneoti.OD between ltl'(r. E ) andF(,., O . E) from diffuaion theory : 1

F(r. n , E)- "ii" [ltl' (r, B ) + 3D( 8)0 - P'(I)(f', E )] .

I

(16.1.2)

6)j.

(1Is.I.3)

''''t 8J _111(,.. BI ll + 3 D(8 ) a~~~". I

J IB)-(I)IF.E)+3D(E)

(16.1 .1)

6)

or approrimately 111(,., B)_J(K) 11- 3D {E )

81a~~.

Here 8/8, Iignili8ll diHerentiation in the dimltion of tbe edraotion oanal (aDd in fact in the direction mediwn-+OIonaJ) at tbe point 1" of ob.erntlon. ThUi we mUll. apply .. oorreotion to themeuul'fld J(8) for .. hOM! calculation D (E ) .nd tho nu.r. in&<Jlllnt mu.t Lu known. Under Lbo . .umption that the flux gradient doeI not depend on energy. it caD be determined hom probomea.sunllDenw ; D (S ) mua' be e-Jculakld approD' mately·. SJW¥nt '1'1ait procedure only giVN ftli.bJe _IUM of IIP(,.. E ) when ""1..1.&£__-- _ ' ................. ...... the 00l'r'6C1J0~are _mall (I.. than III"'_~"" about. 20 %) . A oompletely different procedure iI beeed on the integration of the vector flux over aU ana:1c. by use of a _tterer introduoed into a canal. that completely penetrate. the medium (Fig . 16.1.2). If we nesleot the perturbation of the field by the canal and the Ample and uaume that the _ttoeret' illO . mall that multiple _tt.ering in it plaY' DO role, then we bave for the .pectrnm of tbe neutrolUl elMtieal.lylCattered by the _\.tenIr

~

bJ

~

J (E>-f ,
(l U .f)

fM i8otropio _ttmng

JIEI-E,IEI!F(r,n', E) dD' -E,(EI!l>(r, E). An addiUon&l CIODtribution due to lne1uticaJIy_ttered neutro.rul may &l8O apJMl&f. In an idMl ICI.tt.Clrer, thi8 contribution &bould be nesUgibly &mall; in addition, I For B.0, ..b.. t.b....,. deplDdeolwoft.b.difflMioa oonAul~. part.iouJarI., Nonjj:, ... _ _ lU va)_ 0« _ II (6) and CJj(6) In Pip. 10.U and 10.1.8. 1D ppllite and beryUiUlllo, D Ia DMrtl 00lIft,u, abo... the D\I-.off-.:f"

Ii".

the el&atlo _ ttoering Ihonld be .. iaotropio u JX-ible, the Ilbeorption t:I"C* MICtion &I cn&ll .. p<*ible. aDd the _ttering ere- aeetion .. eDerKJ·iDdepeodent .. p<*ible. Zirooniwn meW approach. tb_ ~tiOM (Bnn. . " Gl.); INd, graphite, and D.O ha ve aJeo beenused (JOJUlfMOlf It cal. ). The que.tion of what perlurllatiolw .. neutron . peotnun auffen owing to the iDtroduct1on of en utr.ctJ:OD canal hu DOi &I yet been theoretloally in...eeUgated in det&il. In thia connection, one mun owefully diatlnguiah betwe&D bomogeoeo ua and het.orogellOOUl .y.tema a.oootding to ...hetber the f1peet.lum WngN alowly or rapidly hODlpoint to point UWdet.he .ytt.em. In .. hoWOgeneoUlapt.ecQ.the inuod uotion of an extraction ca.n&I can perturb the.peotrum to be.tudied very etroD&ly ainoe neuUoIUI from. other regions, Le., DflUtroNl with a n lIotinlly diHenlDt energy distri bution. can penetrate to t he fro nt.urfa.oe of t he oanel. In homogeneoWl'y"

-- -----------,i

C#Iimff".""""

11'1I •

-~ BzlII fill

t

~ ~.

..., .,

N'

I

eIMII'.

1-- -

--~ ~ ~~ ~~------------11! "''' "'"

•. - -------- cr--"'._- .....-

"'~.~. ""='------............. _ . . - . .. ..... 1..1.1. ",. _ l - ".. ~

_ _..

1..Ul

tema, thia doe- DOt. OOCW'. We Ih&ll . .wne t.b.r.t other .pect.rum per1wbl.t.ioM rem.a.in _aU .. long &I the diAmeter of the extr&ction eanaJ. ia ImallllOlDpued to the me&D free path of th e neutrona. Experiment.. on Ia.rge graphite and .ater

'Ylt.em.I (POOLJ:, B U BTJ:R) ahow that t his is indeed the CUll a nd that furthermore the obeerved en~ dmribution J (8) hudly chaDgN when the diameter of di e extncUon e&naJ ill inc::reaeed to about 5 em. One come. to thia ooncluDoD on one ha.od by oompui80n with ' pecUa that han been taken with extremely cnall canaJ diameten (for which then ill _Iy DO pert.urb6tion) and on the other frnm 1D.tegral meuurementa 1D. which one oompal'M th e actI...Uon ratio Lultt /Lalft [Sec, 15.2 .2 ) at th e point J' wit h and wit hout a canal . By m6l.na of Fig . UU .3, we lIe:lt call attention to t he important pro blem of th e eaU,maIar that mlllt be uaed whene ver we eJ:tract a beam. ]t ahould guarantee that only thoee neutrona that ellw the flight tube through the front . urfaoe of the ertn.etion e&n&I oao rMOh the detector. It mIlA be made of a material that IItrongly ablorbl &1011' neutrona. Hob as boron C&l'bkle (B.C), boron met.aJ, or enn better. boron · lO metal, and mu t ha ve a form. th a t is well adapted to the opti cal oonditiona. .

!i.U•• The Chopper Meihod Fig. 15.1 .4 explaiD. the principM of • chopper fOl' the m-.urement of .pectra. The beam the medium. ill chopped into mort pm- by • rotating inter. ru pter ; the pm- traVflf"le .. flight tube at wboee end is .. detector. Th e energy diltributJon of J (B) can be dete rmined from the t1.m.e diatriblltion of detector aignall. providing that (a) th e rMOlring power of the ')'Item iI .uHicientJy high. In other wordl. the nnoert&inty Ltt in the flight time (oauaed both by th e obopper eed the det.eotor)

Iea...m,

... should be mWl oompared to the flight time of the neutrom along a flight tube of length I. W.1ha1l return to the general connection between the energy lpectrum .00. the flight-time diRribUtiOD with finite !'NOlTing power again in Bee. 16.1 .4; lUffioe it new to 1&1 that in pra.otioe.d 'Iri8 moet often around 10 lJoaec/m (thua e.g., ..:U _ liO.,._ and I -lim). (b) the tranllmjM!on function of the ohopper and the efficiency of the detector are aocaratflly known either from. oaloolatiollll or from m6&llurementa.

IJt!tdIt' I/IilJI

r,f6J1irlll' ""

IimI-'YUI'

J!I. 1..1.&. TblIIllopptr -..p . . . br IIwI u4 1II.o'l'£CU l« oIow_ _ ~ _ D .

(0) all perturbinl effectl are either eeppeeeeed or very ee.rmilly taken into aooount. For thia purpoe8, the detector, or even betw, the entire apparatua, mWlt be ahJelded againat atray DeUUoIlll. In addltlon, _ mut avoid. ".hereYer poIeiblt1, IIOatterlng

or abeorption of

neutroDl on their way to the detector. The perturbing air acattering that oocun on lona: flight pathl can be IUpp~ by use of an evacuated flight tnI-. r The deteotor of ohoioe ill the BF.-oounter,InanyofwhiohOf.n be ecaaeeted together in the A~ form of .. oounter bank in order to obtain '" large .. eenaitive area .. pouible (of. Fig .16 .I.4), ~ 'M To incre&ll'J the aellllitirity, "..1..1... .lbpbl_I«..,.·_ _ _ _ lit partieularly at higher neutron energiee, IeveraJ oounter b&nb can be arranged ODe behind the other l , The 8Dergy-dependent det.eotion Ie1UIitivity of .. oount.er bank O&n be calculated from it. dimenaiOM, tho gu preMUI'O, and tho CI'OlIlI IeCtion of tho BlO(ft. Ot)Li' J'MOtion; in doing thia ot.loulation YO moat anrage over tho.urhoe of tho bank. Neutron abeorpt.ion in tho ....an. of tho oonnters mut aIao be taken into aooount.

""'"

~

IlL :roan ~_ • aimp'e eleoRonio delay de'rioe for oom~tq foIo \be ~.

flIP' d i f f _ bel_1ob1l inlUridul bulb in UMI ~. of 10bIl Conf_OIl OIlNllUilon n-.of·J'1i&bt KMboda,:ena-la1E-*oaa IlMll, P. 457.

.

One 0l.Il al.o euily determine the -wtirity eJ:periment&lly by oompu;.on with an extremely thin counter filled wit b na tural boron, whoee aeDBitivity .ocurately lall a. . the 1/".1.",. Fig. H5.1 .6 aho. . & rotor used for the &n&IyBi of DOutron .peotn.. Since the lpectra to be Itudied uau&1ly OOI:ltain epithermal Deut.roo., cadmium 0&DD0t be ued to make the _alii definiq; lobe alit. ; inltead we m ud ute a lubet&noe that ..beorM neutl'Ont aalkongly .. pc»'I lible at .u energies . Plaa tiCB loaded with

boron or suitable &110)'1. loch ... K·Monel (30% Cu. 70% Nil. IU'tl cuatomary. The fonD. of the a1ite iI n ry imporu.nt in determining th e traoamiNion funotion of the .ptem. Let \Y w.ou.. t bia point furthor for tho _ of .. roto r of rad iue R with .. plano allt of wWth .\ (of. FlJI:. 16.1.8). Let th e rotor t um at .. con- toant

•

"'r~::==~t=::::=~r. "'_ _,

":.-- - - :if-- - - -.

,,,.1 1.1,• • Uwlyo""" '" U.

1_

."or"

&ranaoa-..

,

angulu velocityw in the oounteroloek .... direction. At' - 0 (and al80 att =± ~ . 2, 3, •..), let the Ilit be parallel to the direction of incidence. IAt tho pualIel incident beam be trimmed to th e alit width A by &0 entrance collimator. btl < R. Then aooording to 8'roNZ and SLOUOU. tho chopping action of this attaIlIelDent ill the II&mfl . . that of a llit at z _ O with .. timMepeodODt Ilit width l . W(I , ' ). W(o. ' ). the fraction of the uan.uutt.od lle1I.UOD8. i8 in the velocity nnae f Q)Rt/ l :iii ,,:; 00:

.=.,

Let. ,.

W(Il', I) _ O, W(Il', f) _I +

I :;; - Af2w R.

- "' -1,

'.(OR )' 'O( 4:A 7OR + 1)' ,

W(., 1) _1 _ U W(Il', I) _ I _

.-1.

'oR

-

, 'R h.B :ij I S - . ,

OR ::;; '::;;0.

- .

'oR W(v, I ) = 1- - , - ' . W(• • I I - O.

in th e velocit y

~

.

JR

(16.1.6)

O:\iil:\ii -;- ,

•

,

,

:\ii I:;; 2. B'

2.1 :\ii ' .

wR' /l :;,, ::;fwR'/l :

W(". f) - O,

V .. _ lr« V;; +~ .. '

WI••ll -I - U •

-V* + 2::;1::;;0•

W(", l) -

0:;;1::;; lfii"

•' (OR - I)' • •' ('.R l- U +1)' •

.

WI' , .) "",0

(16.1.6)

V;;; _ .!! .. .

.. .

~-

A

2R

- -

" •.

For .
w(•.,

± :"). Yip. 11S.1.7••nd b ahow WI., ' I for t wo typioal ,..1_ of • . W. _ lha t l.p tb. lliniting cue ._ "". W(o. ' I ~ • • lli.mple triangle function .nth • bMe width A/fAJR. At. &maIler DeUtron n locitiee. the finite neutron t.im. of fli&ht. mak. it.elf felt.: neutrona t.hat.enter the.tit.t. . partieul&r alit orientation and th.t. could go all the w.y through if the rotor ....6nI au.tiorw-y in fact collide

- "f'

with the morine rotor ....all. In partioular, Deutrollll with th e velocity 0< -:cannot .ot through the rotoor.t. allainoe the tim. 2Rlv that they need to traverae the rotor it greaw than the maximum time 2A/wR the rotor i.I open . The following ",,","

• .Jl. , . l8.

•

..... lLU . .... ...

•

I

•

~

.... _ _ "'- . IW ....... -...-. •

• /. ;," . <. . ..16 ,. ;>•••

obnou. relation emta bet ween th e hue widt h 2 ·,d, (which ia important in det.erminingthe r->lvmg power). th e J.im.it.ing nlooit.y Vf . ' and t he rotor ndiu R :

111S.1.7) Th. choice of dMigD apecificati0n8 it aharply limited by thill relation. For e:omple, if we wiah to h....e .n=600m/lOO and 2,d, _ IOO",aeo. t ben it follo. . that R _ IS em. ThldlAlw-5 X10-4 em 100end w _ 400 lOO- l for ,. ... 2 mm.. We can.d ecreaetl vf • by choosing curved I1.itl. Calculatiollll for curved I1.itl CIon be found in M..u.s•• OtrDJU and P.lou. More important than the function Well, t) fa the transmission function Tell). which ill defined .. the time.averaged probabilit.y of penetration for a neutron of nlooity II. By integrating Eqa. (16.1.5) and (15.1.6) between the J.im.itl ;S aDd multiplying by w11l. w. find in th e n locity rang e f wR'fA ;S II :;; "" :

-z:

,:; I:

' 11 - -'J -~'" - I

T 1· )-"'A"":"'1<" I"a

eI 6.1 .8&)

in lhe velocity range wJll/A;S . S h, IPfl :

TI.I= -'[! ~.- - 8 ·u +.!!. 1'-;;;1. b R S" • J V.

(11S.l.8bl

Fft!qoenU, ... inttodQOfl the ,..riahle p "' .fJe and form T(P). the traMmj_ion relativ. to t.he tl'aNmi-on

J:

R for neutr'oM 1Vit.h infinite velocity :

, (,8) -l -Iil'.

( IIS. U ..)

,(,8) -lil'- '~ +'" ~,

(15.I.~b l

Fij:. 16.1.8 aho_ T(P) aooording to Eq. (16.U ). nu. fuDotioo eaten directly ... a faotol' in t.he eoeqy dimibutJoo obeened in .. choppel' nporiPlent. and mu\ be _ fully eceeeeted for. The e~n- for T(P ) are more oomplloat.ed in t.he _ of curved. alita. Th e tran1m jMion fu.nctJoo can be determined eJ:periment.ally (d. J OIW'l Il9Ol( d aL). Chopper eJ: perimenta for the determination of neutron lpect.n. are deeoribed in more detail by STOll'. and SLoUClR, by M OBTOVOI, by J OlW'lll8ON, I.£kp.

" u

,

I

f- ""1\. .\

-

I-

..

r\\

I-

I-

-

-

\

'\ ~. -

II

,

I

az

I

u

U

U

II

1I'll. 11.1.1. TbII

_un

~

UlI'IlIIIIl I fI&\ pia'" eIloppw. _ _

~.

(15.1.1); • •• •• - " " " D,

lttoII l ...." 81.01' £c 1&

and SJOlT1UlfD, and by eo.TU . In order to be able to carry out a olean eJ:· periment. wit.h good ft*)lution , we IDWlt. b&1'e an t.hermrJ flu of froin 6 x 101 to 10» em· 1 lleO - 1 .t. the point. of o'-" atioa . Such .. high flu: eJiata only in .. reactor or in a non.mul tiplying or IUbcrltioal &lleIDbly that • fed with neutr0n8 by .. r-ctor.

."era.ge

lU.s. The Method 01 h.1INl 80vcN Fig. 16.1.9 aho," aoo theJ' very uaefuI anangement for the tim e-of.fIigh\. meuurem l nt. of neutron . pectn.. A puIaed -eurco periodioally emit. abort burtf.I; of neuUoJlli into the . pt.ern being Itudiod: the De Utronac le&ring the .,.nem after Mcla pulee tn.._ .. flight t.ube and are flna1]y obeerTed by the detectors. If we take care that. the decay time of &be neutroo field aft.« M(lb pu1M .. abm ClOIDJlU'ed to the DeUtroo flight time, ... caD. obtain the eDeI'I1lpeetnun from the fIiIh t.-time diltributioa. ju\ .. in .. chopper nperiment..

320

In priDoiplo. the MmO oolllliden.tions govern the deeign of the detectors and the flight tube here .. in the chopper method. . In non.multiplying media, to which we Ihall netri.ct oUl'lloelVei here, the decay time in the epithermal range , Le., for energiea above 0 .6 flT, ia of the order of magnitude of the Blowing-down time to the particular energy. On the other hand, in the thermal nnge, the decay is ezponent.ia.I with tho time conatant 1l""VI:; +D.B'. 1/« is uaually much l&rger thaD the I1owing-down time and» therefore dociaivCI for the resolving power. In an infinite ample of pure water, for oJl&mple, 1/« ia 6f the order of magnitude of 2OO..,aeo compared. to I1owing.down times in the epitherm&1 region of a few 1411eC. U wo at.rive again for a reeolving power of .. lea.at 10 IJ-IIOO/m, the length of the

flight path muet be about 20 m in the cue of pure water. The reeolutWu in the epithermal region ia then very much better « ll.l.eeo/m). In graphite, tho slowing-down timo to 0.6 IIlV .. about 26 "'_ i the thennaJ lifetime ClI.n be ... much .. MlveraJ. mjJljleOOoda aooording to me and absorption. Thus flight paths of lIevenJ hundnd metonlare DeoeIMI'J ; loch long flight paths at'O very OOIItly and lead to large neutl'on loeeee. However, we can drutically reduce the thermal lifetime by.trong poieoning of the moderator. Le., by mixing it with an ablJorber. Similar oonaiderat.ioDl hold for D.O and beryllium systems. In multiplying media the reluatioD time is independent of the neutron energy and. for strong multiplication. is very large ; here alao long flight paths are DeOllIlIftr'Y. In practice, flight-path lengtha up to 60 m are used, Le., only 1)'lJtetu with relaxation timlll < 0.6 mteO are oon.tidered; for longer-lived IYIt.etn.. the chopper method iI prolorable. A more preaiIo dileUllion of reeolution oHocta in the two methods folio". in Bee. 16.1.4.

The putioular advantagea of the pu1eed-lOurce method oompared to ~e chopper method are (i) no oorrection for the ohopper tnnUTlj-iOD is n-.ry. and (ii) a muoh anaUer avenge flu is n-.ry in the medium. Let U oompare, for enmple, a chopper and a pu1eed.-ouroe e.aperimeDt with equally long fliBht

p&the and the -.me bN.m eroM eectioo . 10 the chopper uperiment , only .bout 1'" of the nellUoDll leaving the ertnetion cuW are ued; the other W'''' are intercepted by t he cloeed intenuptor. 10 the pu1fled.1IClurCll experiment. ioU the DeUUoU are UlIed ; for the lWne counting rate, the .verage DIU: C&D be amaller by • faotor of 100. Poou and BUSTJ:R tJ al, h.ve Undertaken experiment. with 1arze lineu .ooeleratol'8. RJ:IOH4RDT baa done acme mUIIllrtlmentll using . pulled T (d, " lBe· neutron eouree. 16.1••• The EnUD' Re.olutloQ1D Tlme•• r.FUgh& lI_uremeutil Let J(B) be the energy distribution of the neutron be&m leaving the medium, and let . (B ) be the energy-dependent deteetor MDIIitirity. Let any ot her cor. reetioc., Ineh .. tb
of ldNJ. re.oluUon, Le.• nen there '" fliBht time.....e b.....

Z.(' )=ooon.

DO

uncert&inty.t all in the

J1(B)6 {y1~/_ - ,)a .

(16.1.111.)

B ere the cou tant cont..ine irrelnant facton .ueh .. th. IIOlid angle eed the ftlpetition.lrequ.ncy. Th. 6-function e~ th e fact th.t only ttao. Deutrool contribute to Z. (,) whose Bight tim.

~ _ eqwJ. to ,. Here I '- the length , Ill/_

of the Bight path. which in • chop per experiment _ mUilured from th e middle of th. rotor to th. detector and in • pu1&ed.aouroe experiment from the point of obeerntion to t he detector . From Eq , (HU ,n.) it follo_ that

Z.ltl -oo~t.J (E -T(f)')I~1 )

_""""J (E_ ; (fW:

ns.i.n iii)

Le.• tha t Z, {tl !d' I _ J (B)!dB] . Thillatter OOlIoiuaion _ immedi&tely obrioUi gj) could have been used instead of Eq. (16.1.11.).t the beginning of OW' OOMiderationa. The ue of the 6-funct.ion _ better if we intend ultimately to SO o.er to the more important practical. CMe in which the uneertai.nty in the Bight time C&D DO longer be De@:leeted. Thoa in plaoe of the 6-functioD there .ppean. nonnallzed Lime rMOlutioD function R(l) (,nth _t - R(l )dl - l). R (l)_determinedontheooehaDd by the tim&-depeodent tn-nlmi-ioo. of the cbopper in • ehoppet . speriment or the time depeDdenoe of lZI (r . H,') in. pu1aed..-ource experiment and OD the other haDd b,. the channel width of the time an.alJler oonnected to the deteeton. A. a rule. eoough chanoe18 .,.. a• .n.ble to .now them to be made 'ff!IItJ ILII'1'01t' aod tboreb,. to eliminatoe th e effect of ebann.el 1ridth on tbe re«IIution functioD; t.beret'ore, in - - . . , ...... )1:.-.. ~

!I

the following we ahaIl neglect thia effect . A difficulty now ariaee because .. a rule R(l) dependa on the neuUon energy. In a chopper operiment with plane alitl, the time-depondent trUlUTIj811jon depende on velocity through Eq•. (16.1.6) and (llU.8); in a pulMd-eo\lf"08 experiment, the decay time in the epithermal range depend. on the neutron enetlY. However. for the folloW'ina: .cmewhat qualitative of conaideratiOM we man . .woe that R (l) • energy independent. In the _ the chopper, let R(l) -

~i - (~A1ll ' 111:;;.d I;

R(A) - 0 othenme (cf. Fig. 16.1.IOa) .

(16.1.12a)

Eq. (16.l.12&) repreeentl a triangle function with a hue width 2.d, =A fRw . A-=O COlTelpouda to the zero time-point at which the chopper mtl are partJlel to the incident beam . In the eaee of the pulsed-ecurce uperiment, let

R(l)ClO, R(l) =u- U-+ d

1< -1/« ) l . l>

_I!«.

(Ui.1.l 2 b )

,

•

,

"

.

•

.1

"

•

, •

1

7If. 1..1.10 .. u4 11. 140UtIId rtIOIaUoll fluIoolloDIlor. : mopp. ; 10: palood._blt

(ct. Fig . 16.1.10b.) Here it baa been auwned that after the injeotionof the pw.e, tho noutron f1UJ[ <Jl(r, E, I) dooa,.. purely oponent.ially at all enera:iN; thia • eertainly not tho eaee in a DOn-multiplying medium in the epithermal range . For ftl&8OD8 to become obvioua later. the ~ time-point is not coincident with the injection of tho pul&e, but is.mtted from it by ono average Wetimo 1/«.

We now have (16.1.l3a)

.-

--

Z(,)_

f

Z, (I

+ l)R(A) dA.

(16.1.13 b)

Eq. (15.1.13b) connOCltl tho actually obeerved flight.-time distribution Z(,) with the flight-time di.tribution Z,(') that would be ebeersed in the C&110 of ideal reeoIution. If we develop ZIt') in a Taylor eoriel , it follows because of tho normalisation of the rekllution function that (15.1.1b)

'" We cen ea.s.ily convince oUl'861vee by me&nl of.Eq . (1Ii.1.12&) t.hat for the chopPflr beeaUlie of th e Iymmetry of R (l) all odd moment. of the J'fI8(Ilution function, in putiou1&r th e lint., va.niab. In the cue of the pulled 1OW"Ol!I, the tint moment. vaniahell bec.ue th e eeee tlm&-point ...... Ihifted with reapeet to the time of pu1.Ie injeotJon by one mean lifetime. Tbua when we e&b De(l:leot terma of ordBr higher thaD th e -md. we ha ve

.-

--

Z (II =Z.(' I+Z; (,) . J I IIB(l) IIU .

(16.1.14hl

.r

(16.1 4'0)

In onler to determine an enefJJ1 1pectn1m from Z (, ) we mtat proceed in the following ....y. We calculate Z" ('J/Z (,) from th e mNllUl'ed ZI 'I ' and ueing the eeccad mOintlot of th e neolution function aleutalie Z.(' ). J (B ) then foUo". from Eq . (16.1.11h) and J (E) from Eq . (l lU .10). In order to go from here to the flu d>(r. El , It ill n-.ry under certain circumlt&oONto apply the oorreotiODl di.cuMed in See. IIU.1. U we 1181. (l5.I.Hia, If l ' R(1)cU= t .11

.---

by ...ayof abbreviaUoD. . . obtain from Eq. (Ui.1.l 2a) fon chopper with .. tt"ia.qle retOIutioo function of half·width .1, ",I -

1

T

•• (.1,)1; .i=}'i

(I.U .l 6 b)

aDd for .. pm-! . -mbly with .. ~y time l Ila

.11

- (f ),;

LI _ ~.

(16.1.16c)

F or .. rect&ngu1.r t'MOlution fonotJ.on with the bue width •.U AI _

o

(dr)' •

A

.lit'

• . o-Vi'

(16.I .16d)

If Mlveral l'MOluUoo funoti ona are auperpoeed, then ,dl - .::tf+ ..:i:+ . . .. FOI" eJ:&IDple, if in .. pul8ed.eourc:e eJ:perime o~ we obeerTe t.he tlme-of -Oight. diHribu. tion with ehNmeJ.ofwidth.::tt, th en A -

V{~t + -} (.::t ()I. and we can negiec\ t.be

effect. of finlw. ohanoel width .. long .. .::t t .:5 Jl•• If we kno ... A. we can Nt.imate ...ba~ erT'On .n.e in .. meuuremeni owing to DfJIlIeet. of the l'MOluUoo correotioo. eoo"enely, we can IIt&rt. .nth .. gino . peotrum, require thai th e error d ue to DfJIlIect of the rMOIution oorrecUon be - . .., . thaD 1 'ro . and from thiI req uirement oaIculate the ~ aile of it. Fig . 16.1.11 th o. . the ....Iu.. of AI' (in .... 8eC/m) thai may not be exceeded in order that lla orct. \0 Cl&IoWat.e Z"(I)/Z (,). 1:(1) Dlut be "et'J _teIr bo... In ~ ... fit • IDloot.h OIQ'Ye \0 ~. ~ 1:(1) ba tome a.vro... fUlItl ADd u.ea dltf_lJIw Ule fi«ed OW"I'e. •

".

-e

.I'.: (f} -z;, (11 r.. (,)

be < 10 • i.e ., in order that the ez'T'On in the 1lIlCIOlnlCted lpect.nun, be !ell th"D 1 % (Poou). The . .u.mpt1onl UDderlyiDg the caloulaUoD. wen! bla.ek detect.on, '(E)=-l , T(E) -l, and J(E)_M(E) or J (E)_l/E . iJll natnrallr depeDda on the energy and decreuN with increa&ing enere. The upper end of the Maxwellian regioD iI puticu1&rly criticallli.nce the ,peotrum ch1rlSl:II rapidly there. It a-een th ..t .. iJ/IAtl3-10 paeofm illulliclent to allow rn_uremonta to be caniod out ..t ooorgiee bolow 10 ov to 1"" or beeeee. The talOlution aohieved in lOUIe actual oJ:porimenta ia also indicated in the figure. Additional oonaideratioo of tho reeolutioD ecreeetion can be found in S'l'oNa and SL(Jv.. oaa: .. well ... in the .orb of POOLl: ..nd of BaY8TaR.

...,... .

,

•• u

~

'C_\~ ",!",~. "' ~ '~ --I~' __. '--I ~_~_ \_-~~~~

"'"' " ' - "..LlL _ _ n . ~ < I t 4" . _ fllIbI,.

•. . . .
--

''' '

.if

_

_

1o~1

4 "11II.

•

... _

.. _

.

u~ :

16.2. lnnltlgatlon 01 the Spectrum by Integral Hethodt Int.egral methoda are baaod 01'1 tho m6&luremol'lt of tho effective CTOU _tiol18 of prot- in the neutron field. Th_ meuurement. can eithor be tnl18mlMlon rneuurementa (..ttonuatioD of tho probe aotivity by luporpoeition of filten) or acti...t.ioo meuurementl with probea who.o aoti...tion CI"OIII IeOtiOI18 h.. n acme chan0t.eri8tic energy dependence. Suoh rn_urementa can be intorpreted by comparing the meuured effect.ln ~ I&Otion with .. value obt.t.ined by ......en.ging the kno'tm differential c~ eection over the theoretically caleulated neUU'oD lpocttum. Somewhat mon direct but not alwaY' applicable ill the .nJ.uat.ion of the probe meuurement. in terml of oHeeti.ve oeuuon temperatUl'fll. Aocording to Sec. 10.2.1 . . CNI freq_tJy appro:idmate tb e l pocUum of Ilow neatnml with the form "' (B) """ tP", . ---.!..- , - .IH' + .,. . .d 16{~T) (13.2.1) (12')"

.....

6

The .peetnuD ia thea fully characterised by th e panmeterl "'."JtP", and '1' (01' wben the W..toott. con.....ntion ill a-t by r and '1'; d . Sec. 12.2.6). "'.,J41", (or r) CUI be determiDed &om the oadmium ratio by the method. ~bed in Bee. 11.2; . . thaJl DOW.how tha\ the t.empenwre '1' caD. be deeieed from probe ",_to.

In.~

of the S ~ by Iot.egnJ JIethodlI

16.2.1. Tllmpllwure DllterminaUon by the TraDamisllon Method Fig. 15.2.1 explains the principle of a tnorwniaaion meawrement on a free neutron b6am. A oollimated neutron beam with a Maxwelllan energy distribution

J(E) ......

(I:~ e--/U' ~avenee an absorber foil of thicknll88 d and a l /ffoO«*aection.

E.(E)=E.(kTo)

n V¥.

and then impingee on a detector. Let the

lI~I!~t~~~~~ ~

eecncn of and the detector substance the I/o-law let the detector be soobey thin

• " '.
--------- 0 1"1£/ - 11£1t -r./fJi

lI£I

"H-+-+++-H-+-t~

I{v- dtltcfv'

,

V,

1.11.+UU

,-I" IA·T!· i -

J'Ill.1D.1.1. it. oobo-.$l, _ " 0 1 ....._11_ pon_ _ II b, u.o ~

cto8II

l'

__ eM! IboraMI _ _ UINaIll. lJ--oIMorIlor loll

~ 1O.1.t.n..&r.-IooloaeM!.~

~

that ita IIllnsitivity likewise variee &8 1/" . We then have for the ratio of the oounting rate Z" ~h the ai>eorber to the oounting rate Z' 1OtlJIowI it the following upre88ion :

,

(16.2.2 80)

JJ (E)

or

with~E.(lTo)d =yandir =

~

#

(US,2.2bl

Fig. 16.2.2 now" Z"/ZO &8 a function of y aoootding to Z£HN. The decrease is approIimately exponential, but with a 810wly d&ere&aing decay oonatant. Thia deoreaao ecmee from the hardening of the Maxwell "pectrum by the 1/t1-abeorption in the foil. For thin absorber foils, E. (kT. )d < I, and we can expand the exponential function in Eq . (l6.2.2b) and obtain

z·

p =l-

, y;:y.

(16.2.2c)

For very thick abllotber foill (E. {kT.)d> I) L.L!'oan gives

-Z' ... --=• ( -'

e,

Va

2

)' e-I(,II)" ~ ' I1

+ 17 - -" I. (')' 36 . ' ~

(".'.'d)

..

,

In each CMe Z+/Z-w a unique function of y. 1I1011oWII uniquely from a tJ'anamj"'ion meuurement, and thUi if c:I and 1:.(.1:7''> are known, we c&n find T. Suob meMure· menu have been carried out by FI:IU(I and M..usu..u.x. .. well as by HvaRA, W.A.LL.l.CZ, and HoL'l'DUlfJf. A variant of the tranemieaion method of int.ereet to WJ baa been deeeribed by BJUNCJI and further developed by KOOBLJ:. In thia variant, we U8& an indium foil all det&ctor and eneloee it with two gold loila. A typical sandwich &lTangement is .bOWD in Fig. 16.2.3. Here again a shielding ring ia used to avoid activation of the inner foil by laterally incident neutroll8. According to Sec. 11.2.3, if we neglect _tt.ering in the foil, the activation of the bare inner foil in a Maxwell spectrum i.I given by (16.2.3)

If the foil is oovered on both udee by OOVtlt"l whoee ab80rption coeffioiont ia

••

•• I ••

kA.

~ ,"'." ~

_.

..... U.t.l. A IoU .....twlcl:I1or ....a- ..... pooratue

l; . .

....

.

r-'"

l-

,.-

.

r.. 11.L1. e+/CO lor ... tadham foB ('- eu 11I&I_') 1>, IOId.IoIlo ( ' ~ MO ...I_ ') " or u.. _ _ M1npenohlno . lu _

110_

p; (E) and wboee thiCkn688 it Il, then again neglecting _turingl, it. activation according to Sec . 12.1.3 is ~

O·

=i J(t~ltC!-·lt2' {PZ~f{

ft;(p. (E) l1 +~(Kll1') -

f.tu.: (E)~')]} dE .

(115.2.4)

• Tbul for given Put' p.., ~, 6. and ~. the activation ratio C-/CO is a function only

of the temperatUl1l. FJa:. 115.2.4 IhoWB C·j(JI aooording to KttOHLK for a p&rtioular II&ndwich arrangement. One obtaine C·/CO by counting the covered and bare foiIB, reapectively, after irradiation; we must correct for any differences due to different oounting and irradiation times by meana of the time factor. Tbe eenaiti'rity of the counting apparatUl doe. not matter beceuee one Ia intereated in the ratio of two counting rat.. However. one mUlt be very careful that the obeerved counting rates are proportional to the true acti'rity of the foil (Sea. 11.2.15). One muat therefore either count y·raY' or average the p.oounting rate over both aides of \he foil. Aa a rule, the thermal neutron mldt to be .tudied ..180 contAin epithermal neutro...... We muat then again eeparate thermal .. nd epithermal activation by I The 1nfI_oe of _tlering in tho roTer foilil 011 (7 hu been Itlldied in del&il fora gold abeorber by KtlCllU. It tlUM out 10 be Mlligibly email; for on one hand, .lOme nMtIy normally incident _trolll ~ reach the foil owing to baok-tlering, while on the oUaer haod, lOme obliqlMlly Inddent _tiona ~ would olhenriIe rem.am in tho .beorber are _u.ered into t.bo foil

'"

e-

of the cadmium differen ce method. To do th» , . determine C· and oooe without and 0008 with t.be entire I&Ddwieh tightly enelceed in .. oadmJum libel] (0.6- 1 J:IUD thick) and thereby obtain four nJ.uN 0:. (teo.~. aDd ~CD ' Then we fonn (16.2.6.) 0; - 0;O::D

IIleN1.I

Feu

Gt=~-F&etCD'

(l6.2.6b)

reD

POco and are t he eadmium. oorntet.ion fa.cton (el. Sea. 12.2.1). PO co ia identieaJ. with th e oorTeetion factor for .. hue foil; F"co ia that of the MDdwioh. The mea· . ure menta and ulculat.iora of KttOllLli ehow that for th e foilll ceed by him, F"CD -.l1D i this ooncluaion ia probably valid for &II conceivable praotioal und. -.riches. With J'~D ....Fl!D =FCD ' we can employ the muea given in Fig. 12.2.3. In thi& method, th e activaUon perturbation hu .. Tory .moUi effoot on the temperatwoe meuurement aiDOll .. folIlIW'f'OUDded by aD .beorber ~ .. mnch larg er Dux dePJ-ion than. hue ODO. Thua the meuured ".hle of c"cJc:. m11l\be eeeeeeed. 'I'hi. C&D be done with the help of th e formw.... given in Sea.II .S. Ho... ever, if, for eu mple. we 11Ml gold ab.orben e&Ch with 360 rns/em' and an indium foil of 65 .7 mgJeml , the entire package hM • p..fI of 0.28, and we have overshot the raDgil in which there are reliable meuurementa of the acti...-.Uon perturbation. Th e perturbation ill greater th e Im&1Ier the traupon mMli free J-th in the lour· l'OUDding medium. com.p ued to the foil ~. aDd probably ncludN .. klmperature dot.e~tion in B.O by uu. method. We e&n "Toid the perturbat.ioD.. ho.e...er, by placing the foil pr.ekage in .. lIUfficientJy Wge canty. There, there Y no field perturbation and th e mea.ured ttaruJmilaion ...lllN need not be eoeeected. Th e Il
16.1.1. Temperature De&ermlll1otloa. with [.a&et(um Folb We caD obt.&in information about the neutron .pectrum lVithout tnm.miaaion mea.uremelJu from the aotf....tion of individual foU. ... long ... we 11M a foil . ubatanoe whOle loCti...tioD crou IIl!ICtion dfr,iIot.ee from the I/..law. Whereu the a.eti. ..tion of .. foil with .. l/~ -UOD ia alwaY' proportion.I to the donat y and independ ent of the 1pl!ICtrUm. tb.t of .. foil with an eoergy.irdependont cr..eectiOIJ. for eKloIIlple, ia proportional to the f11Lll ; from the ratio of the ewe, we CIoQ

dete:mine the .....l!Ilafe neutron ...elooJty. i.e a auitable foil.ubetanoe for .uch m&aeurementa. Natural lutetium oonem.. of 97.40% Lut" and 2.60%' Luu-. Lui". atable, .hile Lut" decaye with a hlolf·life of 2.2 . 10M )'Mb. Neutl'OD. c.ptQnl in Lul" IeI.da to an ieomerio ata te of LutN which decaye with a half·life of 3 .584 h (ct. Fig. 16.2.610). Neutron capture In Lu.IN IMde to Lui" .hich decaY' with a half.life of 11.8 d (d . Fig. 16.2.6b for iu dooay ItCheme). Wb erou the a.ott...tion ClIWI .ection of Lui" followa the 1/".1&.. in the thermal range. that of LuIN de't'iatel from it eharply becaUll!l of a .trone rMOrtaDoe lot O.l~ n . Therefore. if we lrndiI.te a lutetium foil in a Dl!IQtroa field aod ~ OOWlt the Lot". and Lot" act1ritiN _P'I'"'td,., .. obtain a .-tal spectn1 Die. from .b.ic:b . . caD det.ennlne tb-

Lutd,"m

DOQtron tempentw'e. We a.n &lao ignore the Luln actintion and instead U8lI a copper foil .. a ltv-ref_ Dee detector. In th.ia CUCI. we do DOt. Deed to d etermine both of the lutetium actiritie. IIllpuat.eIYi we limply wait. long enough after th e inadiation for t he Lu l ,. . activity to decay. The WOl'O of 8omoD and STulOll ... weU .. of BUB1U.T oonta.in numerous practical hinta for working with lutetium. foill. . We -hall abow iD the foUo1ring bow we can determine the ne utron te rnperatUf'e from . ueh rnMlW'eDlenti. FOl' thit ~, _ . hall UMl the form aliam of WasTmang~ 01'

. .', "_

lI'."-,

,•.......

"' 1

~'.", 'fi~

~

...,, {I_IIIV-

.."v-

•• ....

.."'..."

.......

,."",

+'"

,..new

••

J7

11l7",W-

rI

... ••

••

J'lI, lUt.. Tbo

~

. .

_ _ 11III La,

La'''.

'j J'lI, u.a.n. Tbo

..... "" .m

+r• ..... •

~

'" La,n

iDtroduoed in 8eo. 12.2.6 i our oonaideratlone therefore hold only for t hin foU.. Let the M utron .peot:um be ohanoteri&ed by the pararnetenl r and T. Let UI 00Il8ider the acti..tion of a lutetium foil or of a aand1rioh oompoMd of a hrotetiurn foil aDd a It--ndBl''ll!lOl deteotol'. The qu.utitiel , (T) aDd . (T) reler to the uoitation of the La'" aetirity; the quantity p (T) men to the r e I _ detector (J••• • either the 1.ul'" actirity cw the actirity oi the copper cw ~ foil l ) . Tben".. bye 00ft

1""

C-K(g(7')+u(TIl 0 1"_ Kl"l1 + ,""(T)].

I

(16.2.D)

Here K and K"- are oorwtant. that.OODt&in t he neutron deneity, the abeolut.e nine of the acti..tion ~.eotioD, and the foil loading. Let. u oow fOl'1ll th e activation

nUo

IU ~ _ ~,~

C • , (T}+ -.I2') K- , (2')+r.(7') CI/f - :lUi 1+,.1"11') Hrfi1i(:l')' .,.,. ol tM .... fo8owI . . I/Ha. ~ II AO$ tM _ for oopp.r, lIIIUIC-.1Dd

Lai-.

(1' '' ') .

'* an-.-. ~ . _ 0;

Here on the left ia the (meuurable) quantity OJ(JlI'; on t he right are the unknown q uan ti tiel K. ' . and 1'. K can be eliminated by additional meanrementol iq • kno wn Muw n l pectnun. The neutron field in the the rmal oolumn of • rMCtor ill partleularly auitable for thia purpoee. In the thennal oo1umD there ia oerta.inIy an eqnilibrium. ,pectrum. ; then , _0 aDd T ill equal to th e moderatortempenture Tro . ThUl (-~1i)7'O- iiv(Tre) . (16.2.8)

Since 1're ie known. g(1'ro ) can be taken from th e tabl l!l of WaTOOTT (d . aJ.o Fi8:. 16.2.8) and tho K can be determined. Then ~ _)' 1 (2')+" J 21 ( _CS/W - l +ul/W ( f')

(11S.!.i)

whClre th e .tar indicate. th at tho meuured activa. ti on ratio h... been dividCld by the ii.value foun d in th e additional meaeuremClnt.

Eq . (16.2.9) rtill doee not permit. unique deter ruination of l' flinoethe right-hand aide rtill oont.e.m. tbeunkDownquantity'. We m UJl.thereforeml!UUltl thCl cadmium. ratio of the I/..referenoe foil. Acoord· ing t.o Sec. 12.2.6....e hav Cl [Eq. (12.2.30b)) Cl i.

ReD ""'" -c,;z--

1 + , . (7')

.[ • VJ" , . (7') + . , . - y.

I'

~

•

/

I

I, ~

1/

, I

-T_ .... .~

I

(16.2.10)

,

KCD

• .. Il1I ,. l_ Here HOD ill th e cadmium. cut.-off Clnergy', ...hieh T~pendJI on the cadmium. thicknmI (d . Fig. 12.2.6). _ lULnoo-..We no... have two independent eq uatioue for thCl ~.....'" dotonniDation of the tw o p&l&!IIClterl , and l' ...hich can be ecleed graphically. for enmple. Ho...e....... the folknring iteration p rocedure 11 -.impler. WCl Itart with a m.t Mtimate of T and determine .. ....lue of, from thCl meuured cadmium ratio uaing Eq. (lIU.I0); from t hie .....Ine of r and the m - m activation ratio (Ol()l" )· we can ca1cu1&te a DeW' ....a lue of T. with Table l U I . 1''''' ~'-'::~'.e ~ ...hich the "alue of, may be imprond, eee. Fig. 16.2.8 abo. . th Cl g.factor of 1"ultl :":.:":::.'f-'::;~=~+-.c.!.= "­ ... .. ftmction of T. The .... a1UI!l of g were Ud OJl8']'U±o.OOO35 I18.8±OJ calculated with Eq. (6.3.20) ; the ClI'OII .. eo 1.117 o.~ ± O.08 ~on W&II repreeo nted by a IUpt'rpo-.iticn ' .36 0."" ± O,Ofo .. eo

.....

.,-

or Indl'lltl ua l Drult.-Wlgnef

ro.t) lta nON.

IUS 9.11

ue ...,

~ O. I "

± US

•

1lO

eo

Table 16.2.1 giVI!I the parameters used for The ,.facMr iI a.eWa1I1 1ISIkM.... th e firat fow reeonaDOI!I. Fig. 16.2.& aJ.o batI ~ 1_', , iI De11'11 t aDd OODt&ina .cID.Cl g-facton experimen t&1ly !'r. ...r•. dCltermined (by meuuremClOt. on a hMtod tbmnal oolumn) by80maD and 8TnlSOlf. We _ that for T
Cf088 aection. ,(T) for Lu'" depend. IIOmewh.t on the joining function since the firBt resonance liN in the neighborhood of the tranaition range between the thermal and epithermal rqpone. However the differences, even for enreme I aMumptions about the joining function, are not very great. For Cu, Mn, and LulU, g(T) ill always 1. Figs.16.2.1aand b show , (T) for theee three substances. We see from ",' the figure thatl (T) for Lultl ia rather I~ large; moreover, its veluee are only poorly known. Therefore, the deter-

.'

,

I

I

~

•

t

,,"

•

1\ \

I

I,

- I-

\

•

.. . ... . " .

J'la. It- tor.""" to• • (7')

.-

,," r--

•

...

..u-~

V

V

/

\

T-

V

,

.,

1J1I

»J

1G1 "C ill

T_

•

WSft'OOTI')'or ... t.ta_ ...." b. _ ...... •: La'''. CII", lU", to: La .. • ~

v

minat.ion of r from the cadmium ratio of Lu l7l ia Dot at all preferable to use Co 01' Mn as the 1'".reference detector.

-....._ I•

prectee, and it ia

The &OOuracy with which we can determine the neutron temperature with this method is about ± 5°. It ia thus more &OOurate (and . lso IIOmewhat simpler) than th e transmission method of Sec.15.2.1. A disadvantage is th at because of their strong p.aelf.abeorption, Lu foila muat be very tbin. In order to obtain itdequate ttatiatics, we must iJTadiate them. in a nUll: of at least 101 cm - I _ -I (as compared to l()1 cm -t aec - 1 for tb e tranamiaaion metbod ). We sball return to the result. of such measurement. in Sec . 15.3.

u.s.a, O&her Metbod'

01 Determlnlnl the Neutron Temperature lnateadof Lui", we can aJ.o use Eu w for tbe acti vation measurement described in Sec. 15.2.2. Natural europium oonsists of 47.77% Eu w and 52.23% EuUl• Neutron capture in Eul.U leads to a 9.2·b activity (t be decay scheme ia given in Fig . 15.2.8); tbe crose aection 101' tm. activation deviatel strongly from tbe 1/•• 1&.... and in the thermal range has praotically a I/B.behavior . In oontrut with lutetium, the g.factor dec""" Ittongly with temperature. Fig . 15.2.9 ,hoWl I(T) for Eu Ul ; nnfortunateJy, theae .,.In.,.are only poorly known. The evaluation

331

of the meaa ureJnenu l. d one eu ct1,. '-II for lutetium. The work of SPRDfOSB givell more det&ile. We l'eOOaunend the oombination of IeveNI d etector l ub.tanoee, e .g., Lu , Eu, and. Cu. or Lu , Pu, and MD, Iinoe we may then judge bOlt' well the lpectrum ca n

...'".

.".

,...

r--------

.....u'"'

u'"' u.

ur

u'"' u• u'"'

..u._..

~

.~

.-..

h ili -

t. . .

u"'

...w...,

.m

u •

...

.~,

•• • - I ..... u.s.e.. n._,. __ ...... be deeeribed by. MaJ:weU dilItributioo by bow weU the ..rioUll temperature mOUW'tlment. agree .

u

., -

~

j.

I

u

"""1F1IIiItIJ

U

\

• ..... l U I.

" "

\

u

-, <, • .. .

u

<,

no

II

4',

f

1JIoI"-*,,* K....(.. I'I. .-

•

·

...-

.

I

u/

.

•

~

T_

....

1-

... ....

.,mtl"'J ~

T-

..... lU ' t 'V"'- .le. . t'!bo - - . - ,.."1 ........

...

...

Boe.UIIe of . etroDg reetlnulCle at 0.297 ee, the fiuimI crou Ndm of "'~i 2J9 alao Iho_ • eharp deviation from t he 1/11-1&... Pip. 16.2,10-12 ahow geT ) and . (T) for Pu" and UIM. We Me that particularly.t hiah temperatUnll 'IT) for Pu- in~ very much with inoreuing temperature. T hua we can determine

'32

the neutron temperature by limult6neoua fiMioo-rate measurement.. on Pu" and lJ'I'. Th e meuurementa are evaluated in principle exa.ctly aa in the C&&6 of lutetium. lnvMtigationa with fiMion clhambera have been C&J'ried out by c.o.p-

B:n.L, POOLS, and hBDl.lKTLJ:. STnt80lf. Smnnn. and HEINE)Ulfl{ describe a procedure in 'll'hich UW and Pu" are irradiated in & neutron field and their tiaion rate. determined thenafter by counting their fission produot aotivitiee. A p&rtioular advantage of fiaaion-rate mMllurem entl in the study of reactor .peotra it that - quite independently of the . peotnun evaluation procedure we immediately obtain important technic&l data.

, ,

w"/fi,,,.,/

•

,

,/

1\

,

1/

-,., / •

,.,. ffjll/flr/

III

III

'" "', ...

.,

\

.-

'l/#IfI8MfiII

J'lI. lU.1L

""'"

.en n.I_1w h - lIooloa

Fia;. 16.2.13 indioatee how we ean use cadmium to measure the tomperature of .. free neutron beam . Since an O.5-mm·thiek cadmium sheet ill "black" for all thermal ZHlUtl'ODlJ. the capture nte in it is proportion&! to the fiU.J:; in oonUut the captllRl rate in .. I/ll.detector (a thin BF.,oountcJ') ill proporti onal to the den.lity. The averase DlIutron velocity and thu. the neutron ~peraturefollo_ from the ratio of the two Cl&pture ratM. Unfortunately, Routron /1£1 capture in oadmium. does not "., l'-LU. ... ~~, Iw ....._ ~ gI tbt • "*"ll, lead to activation ; however , the captun! rate in cadmium can be determined from the int.cnaity of th e capture y-radiation. In principle. we can use any "black " detector, e.g., a thick :Lt-l CI')'lItal or a high-pI'fJfUIUl'fl BF._ oounter, inItead of cadmium. We mention in thia ooW'loction a prcceduee given by A1(DBR80N and later improved by O...V Ilf for determining the neutron tomperatlml in a rt'l&CtOr. If we put an at.orber 811beta.noe in a reactor, ite multiplication faetor faU. by an amount that a proportional to the number of additional neutron at.orptioIUI per eeoond. By oompuing the effect induced by cadmium and by • l /lI-abeorber, 8110h .. boron, )(n, Of' Cu, we (l&D determine the al'flrage neutron l'fllooity and the neutron temperature jut .. we did above.

16.3. Results 01 Measurements 01 Stationary Spectra 16.3.1. He&8U1'tlments on Water Meaeurementll of neutron Bp6Ctra in waur a nd in aqueous 8OIutiol1fl of Tarlow abeorben haTe been carried out by POOLJ:. by BEYlITBB. and ooworke~. by lbJ:. ORj,RDT (aU by the method of pul.eod 8OU1'ON). and by 8TO... and SLOTj,OD: (by the chopper method). Measurements on water are easily carried out becsuee the

'" L.....

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diHu.ti.on length (2.76 em in pure water. l.IDaller in ab.orbing 101utiona) is l.IDalI compared to tbe rel..ation length A with which the f...t neutron intenaity fall. off. Aa a result, an equilibrium Btate 11 rapidly eetab1iahed in the neia:bborhood of the SOIU"C6; the l pootrum in the adjaoent equilibrium region th en diHen only al4!:htlyfrom that in an infinite medium with homogeneolll1y diltributod SOU1'ON and is &aoIi.ly aOOOMible to theoretical calculation. In Fig. 16.3.1 a re mown the lpeotra observed by POOLJ: at di6tanoel of 2, 6, 10, and 12.6 em from the eoueee. At source diatan06ll > 10 em, the lpootrum no longer ohangee. Fig . 16.3.2 Ih owl lpectra. in bori c acid 8OlutioJ18 of varlow oonoentratioJ18taken by BlCYSTlCR. For oomparison, infinite-medium lpectra caloW-ted with the Nelkin model are 10180 shown. The agreement of theory and experiment is good, except for pure RIO . The Nelkin model al80 describes the mealurementl well in the cue of IAmarium 8OlutioJ18 (of. Fig. 16.3.3). Note the minima in the.peotra caUied hy resonance abBOrption. On the other hand, the Wigner.WilkinI model (neutron IOI.ttering on free protelll) dMcribei the meaeuremenU leN well.

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POOLa and I&w R.ICIUBDT .nd B~T h.....e tried to fNOlve the lpect.r6 meuured in boric acid aolutiooa into • MaxweU diatribution aDd • I/B. part.. FJg. 16..3.... lIbo_ that ftI~tat.ion of the thennallpect.n.l region . . . Marwe1l di8tribuUon ia in fact quite P-ib1a ; Pia:.16..3.6 abo- neuU'on ten1pen.t l;ll"M determined in thia .ay . . . function of the amount of aheatber preeent.. R&lCIUltDT aDd BftUItT hue aI80 meuured neutron t.empemur. with lut.etiUID foU.: th_ meuurementa agree 1rell with the n1~ derived from. diHenlnt.iaJ. meuurementa . Line "/I" in Fig. 16.3.6 ahOWI the neutron temperature to be expeeted .ocording to Eq . (10.2.100) from • free proton gaa (..4 = 1, a, _20 bamJH.• tom ). Line "b" hu been calcuI&t«I with Eq. (l0.2.10c) for a gu of proWoa rigidly bound to u

•

(:

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•

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.....,...... n.,......r- .. _

» ....... _

»

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.............- .

• 1'.-.0&

o~n

atom.l (A _ 18, a, -80buujH..tom l. The mea.ured tempe rature in· lie between the t wo limitt, ..... to be expoeted. Finally, Fig. 16.3.6 ahOW1l the joining funct.iona for pure ••te r and for .. boric acid aolution with an. .beorption croll If'Won of 3.16 barnJH..tom ; theee joining lunctiona were obtained by aubtnotJng the fitt«l )lnweU diltributiona from the meuured lpect.ra. The two joining funotionl are clearly diHeftlot, indioating .. _1meM of the effectil'e neutnln temperature ClODOept. ~

lU.J. MeuartlDenti on Graphite S)"ten1atic Lnnltigat.ion of the lpectrum in graphite ia muoh more difficult than in t he cue of H ,O. Pint of all, it ia not poNible to poilon graphite . )'Ite1M homogeneously. The effoctin abeorptlon CroelJ eect.ion een only be in crNollfld by oouVuct.mg the .,.tem of <em6tins thin graphite aheeta and abeorber platN of ooppel'. boron.wl. eee. FurthCU'mOre, beoauae of the lars. diffuaion and a1ow. ina:-dOWII \engtha Ln graphite .,...u.rn., DO equilibriUID ia ~ed bet ween the primary DeUtroa. IOW'Oe and the thenn&l Dux, i.e•• the ~ in the neighbor. hood of. fNt, IOW'Oe ia &1-11 ttrongIy inDuenced by diHuioa phenomena ; and " mun - if we 1riah to oompue the calcuI&t.ed and meuured .pect.ra - fall back on the lpaoe-depend eDt t.heoIy.

The only . yltematio oompuiMM:l. betW(I(I.D. t.boor)' &od exporimeo.t ..... owried ou t by P£BU, B uena, and WID'D. Fig. US.3.7 IhowI the experimental an&ngoment. A pphite cube eo em. on an edge II ~y poia.:tDod by boIoD HM1 plat.. The thermal at-orption ~ ..otioD pot ouboa atom. II aboat 0.4 bam jXJe X. = O.43). The entire appustut ill inau1&t.ed loUd CllJl be heated to co. liOO "C. Adjaoent f.oo. I n u-J. for DOUtron injootioo. eed t:-m.

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N tnetion. Tho mMlt1l'OlJl(lQt. are dcee by tho puI-J-eouroo _thod with .. flight ~th eo m long . Since tho relu&t.ioo time of tbonn&l nouuona in lUoh aatrougJy pliIooed l ,.um ia of the order of 100 1l1OO. tho ftIIOIution. .. .clequate. Fip. 14.3.81-(1 abo ... the l peotn mtlUlU'ed It nrioUi tem.perature& in oompariaon with OIJculaUona done with tbe Parka model (Sea. 10.1.3) aDd with a free g... model (using m... 12). In the ca.1culat.ioDl. diffuaion effeota 1t'eA taken into aooount by UlIfI of In effeoti"e ablorption Oft)IIlOOtioa (Sea. 10.2.3) ; the 10GIl 011% OW'Ylture ""... determined from foU m_urement.. P.ulKlI' model nproduoel

eneru

the m_ured .pect.ra quite ...ell. The gu model II bad at 10... temperaturee, but beoom. moob better' at higher If"IPhlte temperaturea llince tbe ohemiOlJ bindins .. then Joel DOtioeable.

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... Neutron temperature meuurement. have been curled out in graphite .ystem.e by Co.t..TU, by KOCB:LK, and by C&JO'BSLL, Poou, and FRnIUHTLll. QU.TES detenDined tho Ipeotrwn in & luboritioal aaeembly of graphite and natural uranium ,I. vario1ll tempen.turee, and reeoleed it aooonling to Eq. (I O.2.l0d) into a Maxwell distribution and an epithermal. part. He ueed the relation

1'=7'.

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(Ui.a.l)

for the neutron temperature l , and found ..4. -30 (T.=- 300 OK ), ..4",_26 (1',-430 · X), ..4__ 23 (T._li20 ~). and A _IS (1', z: 690 "K). Fig . 1l~.3.9 aho. . the Joining funoUon derh'ed from h1.l meuuremenu i the meuW"ementa ..t diHereot graphite temperatUlM lead to the aame fllRction. KOem.:. detennbled the neutron temperature in the neighborhood of .. (Ra- Be) IOW"Oe in graphite by the tAnemiNion method. He found that T- 1', ill large near tho Mlut06 and f.na off with inoreuing diltanoe from the 1Out06. T - T. ia about proportional to the flux ratio IJ)ept/lfU. whioh Wall detonnined in the same eel. of experiment. by the c.dmium differenOll method. U one ellminatee the diffusion oooling effect with the help of Eq. (10.2.20) and fit. the data to Eq. (16.3.1), ODe findI that A.tr =oW±lO, • reeuJ.t that agreee with the work of CoADS within experimental eITOI'. 16.3.3. Other .oderaton

and SlO8T&.&lfD have mado 't'erY oueful chopper mea.uremente of tho apoottum from tho central oanal of a D.O.natural.uramum reactor. The m8&lW"lld apootrum u wu 1'MOlved into a thonnal and ,/ an epithermal put aocording --' -,-- to Eq. (IO.2.IOd). Fig. 16.3.10 I abo_ the joining function 80 obtained; it reeoemblell that found for graphite .yltoml by Co.&TU. '!'here are additional indio ridual rMU1tA on variou other modoratol'l, m&in1y hydrogonolU IUbetanooI, nob ... zirconium M"*'" IimI " f/: It ! ! 'I i ! ,v , u hydride and polyethylene, but -f &lao D.O and beryllium oDde JOJUB880:N. L.KP.&,

t I' ---

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(Bu STO).

Chapter 16: References Geoenl

Poou. Jrl. J ., JI. 8. Naux,

aDd R. 8. 8'1'0. .: The Keuurement. and Theory of Re&Q~

8peotra. Prosr- NuoJ. Ea.gy s.. I, Vol I, p. 81 (11168). BUftD, J . R., .. 01.: IA...... NeatloD ~ QA.!6U (INl). BlJTlJTD, J. R. : 8tatu of Neu!nlll 8pemn ~ k , BNI,718, p. RC I (1862).

·4l 8peelal 1 n"'11RTS, K. R ., and W. RI.IOlUAIlT: in Neutzoa Timo-of.Flight~. Geaen1 II'MU p. 239; BJ'uuD18: Euratom 1961. about the U., BJlY'ftD, J . It, et aJ.: Nuol. Soi. EDg. to 168 (1961 ). of the Tim. ~.aUIJUl.j,. M., and G. P... m.r.: Nuc1.I.Mtnun. Methoo. 4, I~ (19Ml). of.Flight M08T'Ovolo V. I ., M. I. PIvo-u. and A. P. TarrovICB: Geue.. l9M P/6o&O, Met.bod to VoU" r-12. Detoennine Poou, M. J . : Neutron Tiln&-ol·Fligbt Hethoda.p.UI; Bruuela: El1t&to1n 1961 Neutron 81'OlfJI. R. 8., and R. E. S1.ovACn:: KAPL-loi99 11U6j. 8 STou, R . S., and R . E. Swv.&OD: : Nuel, Sci. Eng. to 466 (19Ml1. pectft. 8TvluI. W. J . : Ph,.. Rev. '11, 757 {IN T)' l Heuurernent of the Neutron Speotrum with. T.t.n.oB., B. T.: AERE.N/R 1006 (IH 2). Cry.t.l8pootrometer ZIII'III'. W. H. : Ph,... Rev. 71, TIl2(11147). •

'It'l 'remr

V_Ill, E.,J. M.usuu..andLHa8Il4LL: Ph,.. Rev. 19S (19471. HuoD_, D. J . R . Wu.uol. and R.H. HOLTZMJ.MM : Ph)". ReT. 7', 1277 (1948).

s;

Xeuurem t b;:

t

ureBe.

;;...n:t:"'~ ~ IIIIOD.

•

I

na.U'IOII, 0 .: MDDC-7f7 (1e.6). } Temperature MlIlU1Uement by the 1'rf.nI. Kttom.., rd. : Nliol. Sci. Eog. I, 87 (19li7). ~ J4ethod lrith Voila. LaoIlTJ:. 0 . : Ph,.. Rev. II, 7t (193'J ). } c.Jcula~i~ of Thermal Neutron ~. C. T.: Pb,.. Rev. i!, 67 (1937). 'I'raaam_ _ tmough At.or''*''. B~T. K., ond W. RUCllaDT : BNL-719, 318 (1962). BUBURT, K . : Diploma TheU, Kadanahe 1962. T CBwLIY, B. G., It. B. Tvulu, end C. B. BloIUlt: Nuol. Sci. Eng. ){empe~turet with

I

II, 39 (1963).

L:::'m'FoU..

8oBIIID, 1.. c., end W. P. SmcS<)lI' : Nuel. Sol. Eng . 7,477 (1960); HW·66319 (1960); HW-&HTlS (1961); HW-M866 (1960). BIOB~, C. B., It. B. TvaKn, MId B. G. CBwLIT: Nuel. Sci. Eng. II, 8lS (1963). Tempereture C~paa.L, C. 0 ., R. G. Faa"lAl'TU, foIld J4. J. POOLII: Gene..,. 1868 HeMuremeDt PliO Vol. 22, p. 233. by Ft.ioD. Bate SmclOlf, W. P., L. C. 80m0:D, ud It. E. BaIlUIUlf: Nuol. So!. Eng. M_llnlment. 7,436 (1960). K olU'Il1lll, P., K. RDlZ, and T . SPIUlfO_: BNL-719,401 (196.2) (T_p.ature Meuwement wit.h Europium FoU.). Neutron Temperetare MeMunmont ' All'DD&Olf, H. 1.., lilt aI.: Ph,.. Re't'. 71, III (IMT). } by Comparina RoaottYity E:Ueot. of G4VIlf, G. B. : Nuel. 80!. Rna· I. I (19!i7). V&rioue A'-orbcn.

I

Bn-n... J . R., lilt aI.: Nuol. 80!. Eng . t, lOS (1961). Studis In Hs0 aDd. B~, E., end W. R~ : BNL-719, 318 (1962). A q _ 8olatioDe of PooLJI, lI. J, : J, Nuol. Eaerv1,125 (IN7). Abeorbcn. STon, R. 8., aDd R. E. SLonou: : Nuel. BoI. Eng. I, 466 (IHe). C4KPB.u.r..,C. G., R. O. J'qU-4lfTt,B, and Jl. J. PooJ,.ll: 0-.,.. 1868 PliO, Vol. 22, p. 233. eo.. .T Y, M. 8. : In NeulroD Time-of.F'liiht lIethoda. p. 233;~ : Euratom 1961. StucUe. In G~phite. KltOllLI, M.: liuo! . Sci. Ens· I , 87 (1967). P.uJ[8, D. E., J . R. BUST'" and N. F. WID": Nuol. 801. Eng. 11, 306 (1962). J OH.f..IfSlOIf, Eo, E . L.t..Jn>4, and N. G. 8J~ : ArkI... F,.Ik IS. (113 (J960). Studlet In D.O. JOB.f..RSION, &., and E. JOlflJlOlI': Nuol, & 1. Bag. II, 2M, (1M2)•

1

• Ct. footnotlJ 011 p. 63.

Part. tv

The Determination of Neutron Transport Parameters 16. Slowing-Down Parameters In t.h1I chapter, we CIOnUler It&tionary and non.ltationary experimentl for th • .tud1 of neutron modtlration and dilluaion at. energiN abo.. the thermal regkJa. By far the meet important. of theM espenmentl are t.he ltaUOnary one. for- det.erm..inina the msn -.uared 1lowina;-dowD diatanoo (&0. UI.I) Iinoe t.hia QQ&lltit.1 ent.ers direetl1 into rMOtoI' MIcu1ationa. In addiUon. compwiJon of m-.ured aDd oalculat.ed msn -.UoU'elI Ilowina:-downdiatanc. aUowa u to draw ClODclaaicM:lA about the Yalidit.1 of OW' UowiDc-down thOOl'J' and t.he oon'eC!t.DeM of oar DudM.r data. In 8eo. 18.2, welh&D familiariz.e ounelTel with _ e m_ lWJ'eIQen'" of the DtItltroo dinributioo at. '"Cf large diat&_ from 1I011I1lM j theM m~entl are partieu1arly important. for lhielding probleJ:u. Finally. in See, 18.3. we Ih&D CODaider lIOIDe esperimenta on the time dependen oe of th e .towing-dowo ~

16.1. Dtlklrmln.UOD or the Mean Squared SlowinK-Down D1stanee 1&.1.1. Baale Faria aJIoat the TethDJque. or HNIlU"eIDfIIlt The m...n .quared 11owina:-dowo cU.tl.noe can be determined from foU m...• llUf'll'lDentti of the neutron diatribotion around a point eouroe in a moderatJng medium . The medium MOuld be -0 lafse t.hat. t.he neutron lou hy I...kage is unimportant.. For practical. p~, t.he meet int.ereeting datum is the mean Iquared e1owing-dowo diatanoe to thermal energy ~ ... 8 T'l ' However. acoording to &0. 8.4.1, a IDeMUnlment of the neutron distribntion 'With thtltmal probee doee not gi.. ~ but rather th. total msn .qured. dilltanoe 6M1 _6(T",+P). In other wordI. the diHUflion of thennal neutroPl that OOCIUl"I l ubeequent to th e a1cnrinB-dowo ~ aHeetti th. nperimental r'Mll1ttI. In order to determine

;X-4"q,. we need 10 mt1UW'8 the neutron diIItribotion with reeonanoe probe. at a m-.te -.:r immediately aboTe the thermal ranee. It. baa beoom., outomary to tab ... a

bMia for the determination of the m-.n MJ,ual'Od l1owinc-down the _troD. cU.trihutioa at 1.4e ..... which caD be m-.ured with indium fcilla. AA..., b.a.. Men in Sec. lU. in a thin indium foil « 10 mg C1m--> tmder c.d.m.ium. ~% of theM-min roctJ'f'ity ariMefrom neutron l:.ptw'e In the 1.f6.e... ~. Tbu..., C&D write foI' ~e aot.i1'&tioD of a thin. c.dmiwnoOOTered ~

......... 1011 (16.1.1)

DetenninatioD of the X - Sq~ Blo1ring·Dmm Diatoanoe

•

'r'COD (rj41'1r'4r I ""

,COD(r)"''' 4r

""' ~'" = 6 'f~6ee.'

(16.1.2)

•

Note that the me&8\ll'&ment givee not the age but the flux age at 1.46 ev eince the Dux at 1.46 ev and not the 8l.0wiDg-d0WIl density ifl determined.. The age ifl Bm&ller by an am ount Lh ·= 'f!.u ....- 'fu e ... . This diHerenoe can only be taken

into aooount by calculation. Aooording to GoLDSTEIN del., .1'f· =O.43oml in HIO, 3.7 em l in DIO. and 1.7 oml in graphite. AB we shall eee later, the eIperiment.l errore in the meaaured values of the age are of the same order of magnitude &B theee oorrections. The integration in Eq. (16.1.2) u· tendl to r _OCl, where&8 in the pr&Otloal 0&iIB the flux can only be meaeured oyer a finjt;.e domain. Fortunately, in mOlt 0&8N flux me&lllll'e1D.ent over a lirn.ited range of dietanON lJuHioe. beceuee we can determine the flux vari A atwn for larger dietancee accurately "'.1'-'- -}- -i;-- -i;-----;;;;-,! enough by extrapolation. We explain this by m6&l1IJ of Fig. 16.1.1, which tOl' ...tml\l.-..-I. IDclIIlIII eboW8 r1CCD(r ) for indium foila in the J'II, 18.1.1. ,. .()'CD(.) ool CIl& _ 1lo>1_ neighborhood of a (Ra- Be) source in water. At large ecaree diatanoee, we r1OCD (r)-.o/l-r, (16.1.3) clearly have D(r)-.o/l-r' or CC /r1, and we need only emnd the meaeuremente far enough to determine the decay ooutant E. For larger I01U'OB diet&noee, we extrapolate aooording to Eq . (16.1.3 ). The explanation of this behavior liee in the lintoollieion nature of the elcwing-dcwn pI'OOelIB at large eource distanoes that was repeatedly mentioned. in Chapter 8. We IJhall return to thifl point in Sec. 16.2. We now polI8BllB a complete " recipe " for the determination of 'fr."....: Fint we determine CCD(r) for thin indium foile in the neighborhood of a point ecuree. By plotting r1OCD(r) on aemilogarithmio paper and extrapolating it linearly, we obtain OCD(r) for all r. Then we plot ""OCD(r) and r1CCD(r) on linear paper and graphically integrate them. from r _O to r =oo. The quotient of th_ two numbeee giVtvl ~ .... In the practical U8B of thIa procedure, we mUlJt take into acoount a number of eourcee of enor. In the first plaeo, for re&&OllIJ of intensity one cannot ordinarily uee eJ:tremely thin indium foi1l, and RU'faoe deneitiel of 100 mg om-I are uaual. Aooording to Fig . 12.1.4, for woh thick foilIJ about 16% of the activation oomM from neutron capture in the higher reeonanoee beoauee the 1.46-0'" reeonanoe ill already quite IItrongly lBlf-ebielde(J.I. ThiIJ activation by neutrolUlof higher energy l Fijj:. 12.1,4, boW. 101' a 1/8·lJaz.peotnuu. N_ the 801U'O&, the 8peotnLm wiJI. 00I1i&iD

"',-,-,.--,--,

I

I

•

folll"ll~

,-

•

1Il._.

more higb ....ergy neu.tI'onI t.ha.D a I/&·Ipectnun, fond the oontribution of tM h41her IWOnID08I

to tM aotin.tioa will be eqa 1aIgw.

... mut either be elimin&ted by oaJoulation or by using th e u.ndwich method of meuurlDs f1.1l][M m.ou..ed in &0. 12.1.3. Additional elTOn can ari8e from the finlte.tIM of ~b. neutron 1Otmle : neutroM ClaD be abeorbed or Inclutloa.lIy IKl&ttcmd In the lOaro&. Fin&1Iy. domtiOlUl from the ideallphodoal geometry can CAU8Cl errors. 1D media with largo Ilowing-down lengthl, .ucb &II DIO or graphite, it is ooouionally diffioult to ooutruot .. aya.em large enough for age mea.eurement.e . A. Ron hal ahOWD - d. aI.o WaIlfBDO and NODnB& - the INkage out of .. finite Iptem that .. beiDa; u-t. for an as" meaaurem.ent can be taken into aeooont byoaloulation. For to PMJX-, the material. to be inveetigt.ted i8 made into .. pualleIepiped of length G and b along the adea. The height of the parallel. epiped mUlt be very large compared to the Blowing.down length . A neutron IIOt1I'Oll II placed at the middle of the parallelepiped, and tho diatributiOD of the rNODanOO OWl: OD the am of the parallelepiped is measured .... function (If the IOurce diR&ooe e. Then we form the momenta iii of the .J:i&l Du distribution ~ to the formul.

-

J.CCD (.)••••• 7'~~'._.-

__

t CCD (*h

•

The age may be det.ennined from theee moment. with the relation

~

.+1

(..

" )';Jm

•

..:\ (2.+2) 1 1I1 + "p" ...

,-

=,;~;-;,---.,.( ..- ;--: ..,.-).;r. .~ (~b)l 7+11'·

which h.. been derived by WZDlUao and NOD.UR.

16.1.2.lleu&U"emeni or the )[eaa 8,uand 810wiDg.Down DlAanee 01 J'INioIi NeuWonalD Waler enmple of the method developed in Sec. Htl.l, let ua diacWJe the 11,.." for fiMion lIeUUoDI in water. ThU meulU'tlment h.. reoe.ntly been oarried out rePMtedly beo&ue of a l,.nematio diaorepanoy between WJry oareIully oaJ.oulatod valUeI (JIll 26 om l) and e&rlier experimental valuN (....30 oml ). Recently, eqeoially oarefoJ. meuurementl have been e&rried out by DoDl'u d Ill. and by LoKBAllD and BLUfOlUlLD, among others. Fig. 18.1.2 ucnn M1hematioally the apparatua used by LoIlllBARD and BLllI_ mwm. Thermal neutroDa from the thermal column of a lwimming pool reactor irradiate the muon eonree, a UMl'ly &quare (ClI. 8 x 8 om) I_om·thiok Illb of ura.nium-drooDium alloy OOIl.tainiDg .bout I.'" X IOU U. atom. per om'. Indium. foUl (100 mg om"" covered by I·DUD-thick cadmium. are fixed in a devioe hung from • bridge that permitl prooiM and reproducible poeiti0nin8. The entire apo paratu ia immerm in the reactor pool. Fig. IU .3 ahOWI the epicadmium. actrnty of the Indium folll .. a function of their distance from the aoU1'Cll plate. Thia Utribution".. meuured with. diAance of 1.7 om between the end of the

Aa

aD

meuurem.en~ of

thermal oohunn I.lld the IOW'Oe plate. An iDcreue of thiI diItI.Doe de. DOt affeot the relaU", diNibutioo. 01 tb. _tIoM. from whiob ooe CI&Il OOMhJd. that eJowtnl-down In the graphJte of tho thennrJ oolum n d ON not not.ioeably d loct tho Oll:perimontal reaulte. The vrJU&lI Ih own were cbte.inod from a diU_nco meaetlf'elDoot in

r....,

~ a­

which ceD WII mOllIlrod onco with and cece without th . -.:>1U"Oll plate. In t.hie way tb.r.t fractio n of tho foil ecti,..tion duo to tho diroct nUll: from the tbormaJ column 11'11 elimina ted . Th e lWluvation by photoneutrone that are produced by the reaction of ooergetio rlrli«ll"ftl ' " y·ra~ from th e reactor with the deuterium pr-ont in tho water aleo diaappoan &om the cliff_noe. In oontl'Ut, energetie y-ra)'l from "'-I..a n. _I"' ~1.olraaD IlooI'M bllM fa _ _~ ' ' 11M fi.ion in the ~W'Oe plate can peedeee photoDOUtrol1ll iD the _ter and faJaily the neutron ~bution. H owenr, meunrementa iD whieb tho y_ra,. leaving tho eoueee plate wore e.hielded by a l.26- cm·thiek biamuth 8llb ahowod no lIuoh oHoct.

,U''''''''

..

lu order to oet.imate the effect. of foil aetintioo. doe to neutron capture in the hi@:her indium reeonanON, meuurementa were c:aniod out iD which th e indium folla were ooYWed on both r. aide. by cadmium ow in"'*"'1 dlum ooven. It turned out .. that even under indium .. oo"en th e fon. ahowed a oooaiderable aeti....uon ; lb. neu tron. of energy

.,,\

-,

•

other t.h.o 1.000e" oootrib.. ute heavily to tb e &ct.i"a-

I',.

'tion. Howeeee, th o l patill.l ,..riatio n of the aoti vity obt&lned with tho indium oo"en ill th o MmO II that obtained withouUbem,i.e., Fig . 16.1.3 hold. rlsoroualy for 1.0i6.o" noutrone . Ia order to _ what offect the finite thiolmeM of the lKlW'Oe plate hll on the .... ' _ .... ~ .. l..a-~ . . IlooI'M_ ..... • • , . . . .. neutron diltribution. tho ftl. l ..U n. _ mOM1mllIloota O-e' tho lKl1U"Oll reI-ted with lIOuroe plate tbia - of 2 and 3 em . n tumod out that tho nUll: (normalized. to ita value at a IOtlf'OCl diltanoe of 10 em) decreued

r-,

•

r-,

,

,

•

•

•

•

• ••

w_

with iDorouing thiclmeu of tho 101U'OO plate for IOUf'Oe diet&nON < 7 em. Thia deaeue » duo to the _beorptlon of epithermal DOUtrooa in tho plate. ThiI perturbing effect. wu eliminated by utn.pol&ting to sero IOUf'Oe plato tbickDea

1D cwJouIatina the ap from the Ou diatribution "e mlat DOte that aooording to ita definit.ioo. the . . . detenniDed by the neutl'on diNibution tP(e) around • point 1OUt'Oe. whereu what"' heeD m - m ia the distributWD F(~ l in th e oeiI:hborbood 01. • IIOQf'Oe plate. Bet_n th_ t.o quaotiti.,.. the following ftliatioo exiltl:

H_ 8("') ia the aoUlCe IItrengtb pet unit area of the platCIi the integration ia o...er the entire area of the fiaaiOD piaU!;. 8(t") was detAlnnined by meMure· ment with thermal folla at "Ilfiow; plll.OfllJ OD the ecuree plate. 1D order to ebtain tP((l) from F{I). the abon integral ... carried. out with SDf1'lIO)l:" rule . From the .... Iu.,. of tP(e) eo determined. Lo.B~ and BUIfOB..t.1\D obt4ined. 1'T.".. - 28.8 ± O.9 cm' from Eq. (18.1.2). 'I'ha ...elue ia an ....uage of the ag.,. obtained by p-oouDtinB the ~..-ity of the froDt eed b.ck IidfIlJ of the fou. (rela ti.. to their orieataUoo to the _ plate during the irftdiatiOD). A"waging ia ~ .hen ""' oount a /J.acti..-ity becauee otherwi8e as we ... in See. 11.2.6 - t:urTN1t effecte inOoeIlOll the I"fIIIUItof a foil meuurement. EnD after to a...en.ging. the acti tion of a thiclI: iodium foil ia not strictly proportional to the Ow: at 1.016 e.... AI. Ibown in Sec. 12.2.1, all the e1'flDcoefficitllltl in the upaaaioD of the 1'ector Ou in Legendre polynomia18 oootribute to th e acti.... tioD of a foil. SiDoe the P' (I') are s--aJIy cnaD for ' ii:2 and IIinoe the ClCII'I"NpORdi ~ t'lCJlw6) banlmall nJlIN for th.iD folll. the oootributiou of the hi8'her tennI in the erp&D8ion caD Io!J:Qod alwa,.. be neglected. For loo.mgJcm.' indium folll . hO_...... Pw6>1 in the ~DaIlOll, and thu 1hCp.6J....1 and , ,
In t.hi. ClUe, C-r.(t')(I+~ ~~~). and e...en a amaDP..put in the "ector OUI: would ha.. an effect on the foil actJ..... tioD. It tUlDl out that beeeuee of thY

P.-&.otion TaW. 18.1.1. JI_ _ Vol... 01 lAc Flu AV' ..... N ~ .. W...

a

OOmtetioD

of

01

0.6 em' muet be applied to the f1ul: age. The l'fIlIult of Lo.JUJU) and BU.1fOHoUlO is then Ineeeeeed to 32,3 AWDDlIIOW. Yuan, Md N.OL. IOU 21.3±0.9cm'. HJu., & ..."- and Frro:a " .8 In Table HU.I, ricue ...03 Hoov.. and A..unu Ww. 11.0 mfllllured v_JUeI of ft are M.T B1.oeau aad Tann summarized. The earlier in. t7.3±O.O Lod.u.D IoDd 8L&JtCIaAaD .fIItig_tiou .a.. rather high P~. t7.o±O.O t7.M ±O.1 Do......... ....IIIN owing in part to teeuf6cient IlOOOWltiDg for the effect of aouroe aize on the flu diatribution. In oont.rut, the l'fIlIUIti obt&ined. in the 1aA fe. yean agree with 008 aDOther rather well, The theoretioal ...aloe ia 20.0 om'. lnfonn..tJon on the higher' IDOtnentA of 1.44-e. neutrona in water ii, etG.) iI gino. by DosaJllu.. ol.. (of. a1Io Hn.L, Roaun. and FIToa ).

,... ,"" , ,",,... ...

Tt.....-

..

""

(ra.

tt:... .

I"U. RehhI for - 8 ~~...... for VarioUI Jledla and Soureee'. Wdler. Table UI.I..! abowa m-m ...1. . of 1'~ .... for ...riou neutroD aoo.roeI in H,O. The ... a1Qe1 for the radioaoti.,. (<<" _) IOUl'ON are DOt ...ecy well IlIoAaTn ..... aod PuauJq ..... botJr, (iva - . _pIeCe tab. 01. _ _tL

'" defined lIinoethe apeotn of nob ~ depend on their oompolition. whicb may "'1'1 aooording to the ~ of manufacture. In addiUoo, there hM been .. _ _ of invMtiptJon. of the ar i.D homogmeooe minw'eI of _ter with aluminum.. t.iroonium, iron, bialllutb . and lead. Reeultll are given by MVlOC and POll'UOOltVO, by RID.a, O.liNSIUIR and HJU,J:NII, U well as by Hn.L. RoBERTS, and Frroa.

--

..... I

IBbo- Be)

.-

. :..... 1_ ·)

5.48 ±O.15 B&al:ov.lfAL\al lf• ..ad

_.

t&1n UDOllI'f'II!IOIte enon • F ot enmple.. the finite liul of t.be eouroe ia not t&ken. into &0-

M~

",""

IN_ B-,

1.8 II,"

11"14."18"

....u

Sn.... OU .

'I1lickT.....

.c.. _ !50 bY

.... ...

13.8 ± o.J

3U± U

....

(Ra.-Be)

6 Mev

(Po-Be)

6 Me"

6'7.3±!.O

H"jd, .)~

1'.1 X...

160±e

"".~

-

'I'bo. _ t o .ill _ .

Annp"a1.. _Ule-uoa diRributioa - m in dif• f _ t dirolatioM froD:l t.be ~

Duoo.u.dCll V.u.al'lT. IoIld BtrlU'l'11If

.",.,. Tur<

CUW'a..t."'"

I . _ 160 m

In Fig . 18 .1.4,

TU .... for pure H IO is plotted .... function of t he average of the DllUUon 1OUI'Oe. In the figure, '"' oompr.re approximate oalculationa done with the Belcngut-Goertz.eJ. method (Sec. 8.3.1), exact.1 oaJeulationl,.Dd t.be meMUl'ed nlue.. Whereu the eDOt caleu11.tionl .ceun.t.ely reprodu ce the fl rHl rgy

...•

"

- ,.,.

f•

n·.

j

', /

•

~'It,.tl ",-

~:~h-kj ~'k " N1(i"IIJIII'

•

rJUiltl

'-",),

,v ..

( J'lI. ILU. ........ b' ... ..o ue r-doaol _ _ ...,.. _ . . . . . . . . . - . , oa.-II " ...; - - - •• , . -.. ~ IIr u. ........."... ~ ....uoo4: • ~ ftl.- . . ..no.

-

I A~1, the oNoWMioM __ pwformed b1 tM lIonte Carlo method tU in.l lDto _ , ~pio ~ ill OZ)'pll.

... meuared nJ.11_, the SeJengut-Goerthl method a1.... ~ give. too Iarse a nlue fIX the . .. 8PaDfQU bu eagg8IIted Ul iDterMting applioa.tion of the fact that the age in litO in_ _ ahuply with the llIl.etgy of the 1OW"Oe. ObTioua1y,we can determine the a"enr el'Ull'JY of the IOUf'Ile DeUuorw qwte aeclI1ntely by meuuri.og the age. U ..e ' unound the lOuroe with a concentric , pbericai annwu of a hea"Y 'ut.tanoe, IU th e ."eragtl enllf8Yof th e neutrou llII reaching th e moderator will be '" amaUer alnoe eDMgy IoeBee d ue to iDelut.i.o .eattering ocour. One can d etermine th e d OCl'eMe in th e a"Mage enerv "elY aocuntely by an age meuuremeut and thOl'eby • obtAin information about the energy lou in an inelutio 8C&ttering in tbe teeteubstanoe. Such "&Ill"..' ''' • H n - Dt/l ,__. ", meuurementol han been done with (Ra- Be), U I (4, " I He', &lid ... r - . til l1oi ,...., . . . . HI (4••)Ho' DOU.trona. HfIllf1J W~ . Table 18.1.$ men,. .ame meuared flu: age. in DIO. The meuund ,..!oea apply to a DIO oonoentntion of 99.8 %. W...n baa iDYeatigated the flw: 'Ie of fieaion neu.tronIu a functloo of th e D,O eoncentration ; Fig. 16.1.15

•

I '" .-•• ••

N.

r-- "'-t

r--

I"-

•

.,.. 1...1...

~""'1I

• • •......._

• •

" _ .._

_

~ _

TabM 18 1.3. TJtu .4,.10 l*,i_

_

•

unm.- " " H'(' . jH"

n,icIr. TUJtIl K,,_Z60 kev

.e...I-

I)'"

... 11....

·)

Bul7IM_ ill

DoD (to.8"') ee--

.... . -

'1'befmaZ r of UA........ v-.J _&b-_troa

IOHS llU ±U

4*"ktioN nrio:IaIl

- . - d ia boaI tM

d~

~

. '"

,

" " I ... t.t..

~

u,d ...... _

t-

• ~.

--

1.:"'"

/

V

11.-.0- ,u ll

,

t-

,

ewo.-... _ til l1oi_ .. _ .. .....,w.... ....,. ... . ......

,

lin' f _

aho... hie ftIIultoi. I n the eoecen. tntion ra.nge , tudied , the age decreue. abou.t .4 em' per 1 % mer- in the ligM _ter con.
A Monte-Carlo caloulation of the flUI age Tt.. ... of fiation ee utronain99.8% D I Ogivea l 12.2om' (OoLDBTaIl' d tIl.l,and &IIapprou. mate calcWatioa by the OreuliDg. Goert&elmet.hod IOSoml (L-...nr. d al.).

Orapl*- Table la.I ..400ntama

coeaaured valu. for the f1w: age to iDdiwn rlllJCI(WI(lB of ,..riOUU0 11f(l(lll in graphite of deMity I.8 glomi. Fig. 16.1.6 aha... calou1ated valuea of Tt.,," u .. fun otion of the lOtll'OIIenergy. In the figure. a

~oIu..»-

M'

8q-.d .......·Dowu ~

Pre<Me uJoulatioa (GoLDBnIX d aL) ill compared .nth the limple appro:dmatlOD of age theory [Eq. (8.2.11b». The uJoulated nloe of Tt...... for &.ioa DeUtron. ia 3M om'. Table 18.U.

.,tu... i ll ~ Gf .Dnftl.U ,/eM'

.....

(Bb-Be)

UraniWD Fittion

2 Mel' 5Me1' 5 Mel'

(Ra- Be) (Ptt-Be)

RILL, RoaDTe, IIDd Mc:C.t..lOIoa HaRm. til al.

,." 142 SU:.5 ±O.5 _300

.....

8om_

"",418

Sinoe • theory holdl nry aocuntely in pphite, one CUI. re~t the 1lo'triD&.doWD denaity in the neighborhood of a IOlmlfI a.nalytioally. For e1Ullple, t.he nu Utribution at 1.46 ev in the neighborhood of a monoenergetic point eouroe iDa fIery large medium iaginn by

-

flI(l .46 e", r)_,-..-;:....

(' 0""1

Tt.....

..hen ia the D u . detenni.ned from the me&Q aquared t1o'triD&~'tr'D dinaDoe. Eq. (18.U) DO longer hoJdI for a MnlI'08 .nth a broAd energy lpeotrum. We lhowthialnFig.18.1.7,wben the Ou dilJLribuUoo meuured DeU' a (Re - Be) 1O\1I'08 in graphite it oompared .nth a oaIoulation bued. on Eq . (18.1....) .nth T-380 om'. Better a.greement iI obta1ned _hen the lpeot.nun of the 1Oun;J8 iI approximated by I8nral ase group': then

"""

•

I •• ,

"

"'

•"

1"1.'.

1"11. 'not ~ _ , ( "- .l _ til .....,.... I Iil-..d MIcoIJaMd Iol'

.: _

-*'110"" lor

Mn1Z'ON CUI.

po

-

-,.: u ,

0.16 130 em'

"'\ . ,•-_I, _.. 410_....

" . -;

flI(l.46 ev, r) _ fg~. Aooordiq to hRXI, (Re- Be) ase groupa :

-,

~

(18.1.6)

« ,

be repretented in pphite b1 three

0."

"'-'

0.18 816 em'.

Slishtly diHllrent ..Iaee ""u1t from meuunmente of the au thon: g, TlLM.. ""

Th e

nUl:

O.lI 175cm'

0.08

:uo em'

0.21 860

em'.

diarlbution OOlTeIpondioa: to th.....1_ iI ...

..here it ia . .n to reproduoe the meuured val\MII quite well.

gi't'lIlO

iD Fig. 18.1.7,

0tMr Jlodut!Ior, . Table 16 .1.6 oont.&ina valUei of 1'~."n for an .dditional neutrona 1O\U"OeI . The valUei . tem. in part from older meaauremeatl and are not alwap reliable. In lIOIDe plaoeI, calculated valUei are aJ.o gi",eu. Tben are.uJ.I not maD1 ",liable 1'-.meuul"ementl in the metU hy. dridel and iD. orpnic modenton.

.ne. of moderaton and

-

Do

-.

-....... .......... ...... ..... ..... .•

•

'1-. 1- '1

....±...

1.8511_ ' V _

.- --

No...... Ul4 W........

Do

1.78 J1om'

(Ra-B.)

1~±13

ODdn Alll Al.

BoO

U SrJ-'

Un.zailllll

83... ± ".'7

0o‫ס‬I>,;, 0• • lIllCl Y OI1lfO

Dlpt..nyl

C. " .ll85x U..... IVJII H I 3.s211 x

>
O...... 1ILl. and

49.l2 81.016

Lvnrr and 0_

100000 aw-! om'

Z.U

t~ z. ,

".8,1_' 5.° 11_'

Ui J1_' .... 1/-"

.......

UI1IDIIlQl

2M...

.~

..... .....

"""""... V~ _

11.1.4. n. All tor 810wtnc 0.1111 from 1.4& "' to nermal EDMV Let u defiDe th. quantity J1' -1'...-r~..... .

(' 6.1.61

U J r '- boYD, . . CUI c.J.ou.lat.e the age for moderation to thermal eDeI'IY from the uperimeuta.lly determined flu age to indium fMOn&Doe eaergy. According to the elementary theory , I .... .

..11'11II

( I I)

Sf 1- ••

<>4

J~ (E) ~ . • •

( 16.1.'1)

r

Here llnIlll3.6kTi. the thermal eut-off energy diIou.Med in Sec. 12.2.2. T able 16.1.6 oont.am..ome ",al Uei of.d1' determined in uu. ",.y. T1WI kind of c&l.oulation i. not partJoola.r!, aocurate aince the eHect of chemiO&1 binding on the I1owing-doYD ~ '- not l.alr.eo into &oooGJ)t. HoweYN', we C&Q eu.iJ.y determine ..1 r uperi. meawUy. To do thiI we note tba&the foUowing u preMion holda for the migration ...., .hlcb CI&D be detenni.ned from the mean MJ,1lW'ed di.tADoeto ab.orpt.ion of tbenDal DeUt.roM :

(16.l.8) (16.l .8)

aDd we CI&D detennine J l' if .. bo", JI' , Y , aDd rt...... . We CUI meMure JI' eed with the methodt of thilIIIOt.ioD ; in th e Den chapter, we Iha1J beoome familiar with methodt for det.erminina; Y . A difficlIlty In .uah mM· .u.rem.eDtI .. th e ~ that .. . rule J l' il lIIIIIrJJ. ClOIDpued to Jl1 and Y. Thu we mut form. the IIIIIIrJJ. diHerenoe of WJe numbers and oolllllquently obtain. !eM

'Jr......

36' uact reeult . For thi.I reuon. it iI better to uae &D (Sb- Be) ~forlUClb me&eurementl (in order to make Tt6ll.... anall) and to poison th e moderator with an ebeorber (in order to make Y amall). Some meuured valu ee of L1'1" obu.inod in this way Ill'e given in Tabl e 18.1.6•

8,0 . . . . . ~O

(98.8 '1.) •• Be (1M 110m' ) . Graph ite (I .e gJom')

... ~

"ro

18.2. The Bebavlor of the Neutron Flus at Large Distance. from 8 Point Seuree or Fast Neutrons A luge number of YWf canfaJ ItudieI of the .prea.ding out of thenna,] and

1.46-eyneut.rolUl at l.uzediltanoee from a point IOW'CO in -w h.,.e I-n c:uried out . Fip.llU .l. 2, and 3 ll:unraome of th e l'e'ultl found (d . aleo Fig.IB.I.I). Either the quaotityr'4>IM'. (r) or the quantity r'lP", (r) baa Men plotted on aemilog paper. In tint appro:rlmatioD, .u the curvee mow

.. lin6U' decreue at largeklUf'Ofl diat&nOM. In other worda tP(r)_c-z,/,a. The thermal and epithermal

•,

•.r-.. t ",

~

Dux.,. behne in very Iimilar mannen. 'The liSp1&- t .,r naQoq of thiI behaTiot' ill very limple if we DOte :: that the IC&ttering onw ~ of bydrog en falb .. • oft very ah&I'ply with energy (&0. U .2): U .. neu· ~-(- ' troll. emitted from the IOUl'Ol!I oooe oollid.,. and u -

perienoea .. large eneraY . , the

otOIIlI

eootion for

:11 1

l/J , II ., " II IlI'c. /l4 additional oolllaiOnB • large . Thu the proba.billty of the neutron', beinfl: moderated and finally lob. ..... 1&.11. .. . ec-(l) lor -sml_ ecrbed Deal' th e ute of ite first oolliaion i8 large . ...... . . . . . ....... h1~ ., ~ .tobI .r- _ The thertn&1 and epithermal DUN UioWd then .1a_Cb.-1 11 n. _ ___ beha...e like the Du of prima.ry neutrona. The 1_ .. 'I'M l.tter '- proportJonal too hez-e E '- the -w. II LOll _ "".IlIK, total Cl'OM 8lICtioD at th e IIOUl'OI!I - V. U tIM IIOClnllI emita neutroM 'With a broad energy . pread, a hard· ening eUect 000Ul"I. Since the Clr'C* MOtion for th e lint ool1iai.on increa8N .nth dooreaaing energy. th e more energ etio neUUOnB in th e . peotrum of primarice Iml attenuated 1_ . With increuing di8tan ce &om the 1IOUl'OI!I, th_ energetio neuuona become more and more import&nt in determining th e . pnw.ding out of th e DeUtrooa. TbNe facti exp1&in the .tight flattening of the d_ y curve for fiMion ne ut.r0n8 in 1'1&:. 18.2.1. Han careful ICrUtiny of th e decay curvN dete rmined with m~ IOUJ'ON (of. I1p. 1802.! aDd 3) likewi8e Uio... den.Uoaa from RnJaht-line beha rior. In u-e OUlII, tho deoay beoomeIlOCDe..hat .telIpeI' with lncreuin&

....

,-%",., ..

O
..............

_ , In' _

m.tanoe from.the «Mll'Oe. 'I'hI. originate. iD t.he fa.ot that t.be picture of th e ' proW-

ing-out prooeII denJoped abo.., D&lDely, that a ~ radiation compone nt (t.hoee neut.rolul which have had at lean one collWon) iI iD equilibrium with a primary component (the UIloollided lOuroe neutrolll), iI not.quite rigbt. We mutt note thd DtlQUOD. _ttering by hydrogen iI .trongly an»ouopio in the forward directioo. iD the laboratory .,...... TlunIan, . MUboD ClaD with an appreciable

."" .... . . . .. . r-. ,

0 0 0 0

°

,I

•

"l.

.-

°

I~ \, ~~

,,~

\ r,0\

.

\

.., • ..... II.t.L ..

~,)

............

\ •

,-

" - .. _

;\

IlI'_ _ a (lfa • • ) _ _

IIo_CMet-b,_1Ill).

pobability lRlffer only a amalI change iD angle aDd elMll'gJ' iD i~ first oolliaion. Tbeo. aIt.r tM ooUWoa it ItJl1 be10ngI to the .. prim&rieI" and the cU.t.i.noUon bet_ the primary and MOOoduy radiation compooen~ il Io. lharply defined. With inoreuln&: dilt.aDoo from the 101U'OCl, the prim&ry component will contain man and mon DClIlt.rou that; ha .... DIede aome Im&ll-a.nale oolliliom and i~ average -ru will link lOIJIe.bat. ThiI uplaina th e tteeper et-y of the oeu.tI'oa Ou at large ~ U the IOQf'OCl INIll.Uooa have a broad eDeI'IY .pectru.m , the lut effect deeoribed O&D pe.rt.ly off.et the hardening effeot deeoribed above . The theoretl.oaJ treatment of the .prw.ding-out pl'OOelll at ~ diatanoel .lI difficult. A oomplete deeoriptioo of the neut.roo diltribution oan be obtained with the lDOmeo~ method (cf. 8&0. 8.1.3) or by Monte Carlo oaJoulatioD8 ; ncb orJoalatioM inYOl.... oooait:Wabie DDJIlerioallabor . WICK. and B OLft hue gi..en lOme analytio appros:imau OIl method.. In .pite of the denatioDe from limp&. beha"riol' jut m.o.-I, it II ounomary to appro:dmat.e the meewred D8Qtzon diatribatioo iD the neit:hborhood of a

-

..... ,.,.".

Table 1602.1. RlLsm#c/ll. ~ l IE lor Yarioul N .wnn. BotIfTAI ill WaW

•

~-

l/Z tom l

(N. -Be)

0.81 MeT

(Ra-Be)

'M"

FiINoD.

9.ft ( ) Utl (I .ft . ,,)

6 MeT H"(oI. . )He" J'-! MeT ThiclI: Tupt lPG- Be)

16- 300m It - oW om

UV(~

... .

u.......

3.3 (I.ft eTl

1ll..3± O.7 (thermal) 1-& (\benD&I )

10- oW_

HJu.. Ronan. aocI me.

10- oW II" 0&0-120_

~WlIU. •

....

.z.

1\"-

-

.'...

E4- l liOm

point source by • law of the fonn

Rug

NIit.

~

. -E·/~·lnwater thiaia,&lwehave

_0 in Figa. 16.2.1- 3, reaeooably

acourate, at IC&llt over limited intervals. T.ble 16.2.1 m owe uperiment&l n l ue8 of l iE for 'f'ariou neutron II()Ul'(lN in w.ter. A1Io l pecified there ill the range ov er which the Ulouured flux diatri· bution .... fitted with an eJ:pr'M!Pon of the form. .- E·/~. Tb e :& m ouW'l!lmenta were made pvtJy ~ with indium l'MOQ&.DOO foile and partly with thermal foila. Any differen OM between mMSW'l!lmenta eeed e with th e t wo kinde of foile are probably du e to uperimental

.'

r-

-,

•

'\ I If/1:::::;"-"

..

\

\

•

\

,

,

errol'll.

I n moderatol'll in whioh t he eoattering 0l'0M eeotion .how. no eleu deorMee wit h inCf'eMina: u eu tron energy, the concept of nodi&..... IUI. ....(.)Ior ta...a .... ...._ tion equilibrium it eTeD _ applO- ... .. • priate than in ...etee. NenrtheI-, C~ -&.) here abo the neu.troII. diatribotlon it ooeuionany approsim.t.t.ed by " upl'eMion of th e form ._ z·/~. For eumpM. W£DJ: found III_a.' em. in 99.8% D.Q in the range from 30 to rocm from a n.ion neu.tron eource ueing indium reeonanoe detocton. It ahould be ooted that in IIGme GUN. e.g.• in pun D.Q or in graphite, the primariOll do ftoi: dotel'mine the . preading out of the neutroDl &t luge di.IUDooe from the eouroe. lJutead. in l ueb _ . k.ly abeorbiDj: media there it a pure th ennal at large distanOeli from th e IOUl'OO . neutron field th at decaY' like Detailed uperlmental aDd numerioa1 dat4 on the I preading out and 110wing down of fait neotronl at large IOUl'OO diata nOOl O&n be found in GoW 8TJ:m 'lIbook.

,

•

.,

• ",-

_ l-.....

~

oI_r.-.

14.1

..

' -"£1,.

...._

IWlr\l, X... IfOCl

1'1>,..

13

-,

'"

18.3. The TIme Dependence of the 8Iowi0I'-DoWD Procesl 11.S.1. 8Iowlq DoWII to lD4huD alli Ca4m11lDl 8MoDanee

ID H,UopD01lJ .04ent.on We ClaD. atndr tho time dopeDdeooe of the alowing-down ~ by injoeting Ihcri pm- of DeUUOIl.l from. a JMll-l eouroe into .. lDedium and then meMUring the time dependence of the capture rate in rMOnanoe detecton. In this way , we obt.al.D the time- aDd lpaoe.dependel1t. nux 41(7', Ell ' ') or , if tho medium is largo

, -8Mm

J'lI, I.... L

-:::=::-" . .._Ill

no. _ _ ... ~ ..........

enough and • lpat.ial integration of th o capture ra te ill performed, tho time · dependent OUI: 4)(ER • ,), which ('aD be oomPAred with tho calculationa of Chapter9. Reliable m8allUl'Cmenu of this kind have all yet only been carried out in bydrosonou moderaton. Fia:. 18.3.1 . ho_ t ho apparatus EI'IOKUUlflf uaed for .tudying moderation to indium rMoD&llOO energy in hydrog"n. Th e actual lDodera tor w.. lead acetl.te with • proton densit y of l.Ol xlO" clQ·'. Thill proton denlity nn timM Iowortha n .~ " I-. that in ater, ... cho.en in ord er to make th e mea n free - - l' path and th orefore the time _ Ie of the . Iowing -down proc • _ larger. The elowlng-down ---r-- - tim e ill tho large compared to t.he duration of th e primuy per" neutron pulM . A 31.MfIY beta• 1ric. Iu.t. no. _......-- '" _ . . . - .. u. .... tron eeeeed ... th e neutron .!o --.. - ;- w _ a. C" U I IOUI'Oe j It produced 2'1l- _ long put- of 2·Mev neutrora by meane of th e (y, It) ~ in lead. The neutronll were detected by m_ of the captllnl y.radi..tion of aD indinm foil, which w... covered on all .idee by aD O.l6- i/cm l.thiclr. boron layer to lupprea capture of thermal neu trona. Thi. indinm foil, tosether with a n orswo .cintillator cr')'8ta l th at detected the capt ure y.radiation, w... housed in the mod erator volume. Th e crystal ill abo lMlnaitive to y'l'&)'I ariaing from the eeptuee of thermal neut.rona by the surrounding protons and to breDllltnh!ung from tho boutJ'On. Thus thonl ia a Itrong background preMDt, and tho meuurementl mUtt be eerried out ... indium difference meuuremonti, Le., mouunmentl done once with and once without th e indium foils. Fig . 16.3.2 uowe tho behavior of the indium r-.onaoCll capture rate obtained in

..

..

·tf •

.L!

:¥,,

,

"

>,J-

•

•

3M

luch a difference meeaurement , A oorrection that taketinto aooount t he leak ago of neutrona during the slowing-down pl'OOCllIII haa been applied to the meuured n IueI. For thia ~n. the meaaurem ente are ooIDp&ra.ble with the infinite. medium dilltributioDll calculated in Sec. 9.1. Th e curve in th e figure hu been calculated with Eq. (9.1.6) for free prot..on. of the d enlity occurring here. The caJeul6ted diatri. . 1'0 ,~ bu tion ~ with th e mMlured pointe • within experiment.&! error, from whic h w. ean conclude th at binding eHecU play DO role in - t moderation . 00,, 0 1.66 l Vi, Thi.I oonchaion ~ to be upe
..

II I

I--..

...•

.I

••

,

~

•,

MOu.:aaand 8JOS'l'lUlfD have atudied the time behavior of moderation to the ead.m.ium am en Ut IJ4I I., .Va: cut-off energy in water. For thY purpoee they £ahot neutroD bunt. from ......n de Oraaff .J'lI. 1' 0lUJI. '"'" toIal .. . _ _ """'_ ............._ '" generator in to th e middle of .. I ·m- tank of "",ater and followed tho time depeedeeee of the capture ".radiation from a am.u vo111U11 filled with &q1leOWl IIOlutiona of eith er cad mium , u1fato or gadolinium. nitntoO. Th e OIMl rg)' depeedee ee of the capture ~ ~n of cadmium. and gadolinium ia l oeb that .. tuitable mixture of both a beorben b... • Cltt*

IO
•

OM

- ""

1'---' l bo. D in F1g. 16.3.3, viz., lor';'" it i8 zen) above 0.2 I V &nd u .' behaVe! like II" below 0.2 r ev. Thenlfore, by properly U ; mixing the time-dependent reaction ratM meaaured ~ 'lriUa cadmium and gado. linium IIOlutiona, we obtain , a quantity proportional to t the neuUon deneity below s 0.2 ev . Thia quantity atl U 4. ,...balllkJa t&i.na a COnitant value aft« fIl fll&bo " ,r,-. u. . .. . _ a l ufficienUy Ioni time (if '\Ife dilrqj:ani the fact that t.he neutron deMit.y decaY' 'lrith time due to abeorption p~). By diridinl t.hrough by this conatant.value, we can obtain / (') ,the ftact.ion of t.he neutron deMitI below 0.2 ev It tim e ,. Fig. 16.3.4- Iho..... / (f) meuured in thU '\Ifay bI MOu.. . and s,,09TtU1fD. Theee authOf'l defined a dowinB-dolfD tim e to the cadmium

I

.

. .. ,

~

•

.-

•

" ,- "

1 We thaDl1 Dr. Elfoauu,.. for pointinK Ollt all enor in hit orifin.tJ paper, _here \he m.teo.oe of I ItroDg biDdiol effect . .. ClllDClNded on the baIiiI of all inoorrectJ1 ~ dbtribuUon.

23'

... out-<>ff olW'gJ' by tho relation ~

',-I(I-/('l)", e

(16.3. 1)

Uaing the It') in Fig. 16.3.• , ~d with eomo minor COlnMltiona whioh we ehan not cli80uM here. we find. '. =2.6±O.6Iueo. In order to oompare '. with the theory,

ca.nno'

we reeort to Eq . (9.l.7b) emce this gives the 8lowing-down time for a Ihuply defined energy, "hereu we w~t the average time IpeDt above a partio.. ul&r energy. Obviowly. ~

JcP{".JI ·4v~· 1- 1(') """ ~~ '::---­

(16.3.2)

f
•

bolda forthe fra.cti.on of the Deutrondenaity above the cadmium cut-off energy: if we inae:rt 4'1(11. II for hydrogen from Eq. (9.1.6), we find 1-/(1) _(1 + E' ''OD ,) . - r~ ..',

Th~

2

',- -

r

"CD-'

(16.3.3)

aooording to Eq. (16 .3.1). Introducing tlCD corresponding to ECD=O.2 flV and tho _tWlring croI8 lIeOtion oorreeponding to the proton density in water, we find that '.=2.f2l1teC. The experiment and the caJoulatiOll thua agree within the limite of experimental ereee, and we O&n oonoludo that the ooemioaJ

binding doe8 DOt influenoe the time toale of a1Gwing down to 0.2 flV in water. JUUl( and later GB088KOO have shown that the average tim e which. neutron 1pend8 above the oad.m.ium out-oU energy may be obtained from simple ltationary meuurementl. In ~ infinite medium. containing & homogeneoualy distributed IIOuroe of atrength S [em'" sec -t] and a l /t:t.abeorber, the total neutron density ,,[em--] toquala S'le, where ,- =I/t:t 'L'.(Il) islhe average lifetime against oaptUl'&. The counting rate of a l fll-detector, e.g., a boron counter. it given by Z.-.,,-Sle . U we furthermore meaeUl'& the counting rate zeD of the Iftl-detootor under a cadmium cover, we then take into aooount only that part of the derndty ~ that 0001. from neutronJI above the cadmium eut-off energy. Now ".pI "" and therefore

n.

8.'.

zeD

'"

. --~.

(1··...1

ThUll we CUI obtain '. from a meaeurement of the cadmium ratio if the abeorpUon ClI"l* aeot.IOD of the medium is knOWD. One can obtain the nIu. of Z ~d that characteri&e the infinite medium with homogenoouaIy distributed eourcee by integrating over alIlpace the valulli Z(r) and zeD (r) determined in the neighborhood of a point source in a fairly extensive medium. Using tw. method, DJ: JUllJI:N obt&lned '._l.M±O.l3!'-Me for water; thill reenlt W&ll later corroborated by GBOlllIKOO, who found '._l.66±O.IO !'-1eO. On first view, theee ValUM I188m to be in dilagreement with the ,..wt of M6u.:u and 8.l'0n'Lum. However. while the OIdmium oukdf eneru WM 0.2 ev in their experiment (ct. Fig. 16.3.3), it i:I uualJy mnch higher Ice det.eoton eno1oMd in thick O&dmium coverl (et.

zeD

,., Sec. 12.2.2); for th e eIperimenta of D E JUBD' and of Gaouaoo• ..,. can uaume "",0.3 .Vi for tWa energy Eq. (16.3.3) yioldl ',=l.68,....eo in good agreement.

BCD

with the expe rimental valuee. T here are t hlll four independent uperimental reaultl that ahow chemical binding only Ilightly aHecU the time beha'tior of moden.t.Ion in ..ater above cadmium re8OnanOll energy and that therefore corroborate the emple tbeory of Sec. 9.1. In oootnat, CRoUCH find. uperimentally &Itl'oog influence of chemical biDding OIl th e time -Ie of moderation to oadmiwn eut..oft' ~ ('. ....6Ilaee). The I'eUOO for lhi8 di8c:RpaOQ1 ia unknown..

HI.a.!. Slowing Down in H ea'f)' Moderators ; TheSlowiDg.Down_TlmeSpeetrometel' BJ:ROMAlI d al . have et udled the time dependence of ne utron mod_ tioD 0Vtl\' • wide range of energies in It.ad. For thie purpoeo. t hey injeoted puIae. ol l4..:rdev neutrons [from the St(d, a )R. ' reaction) .bout l 1l8eO long into a lead (lube about 2 met.en on .. aide (I.e., about 90 ton. olle&d) aDd then lJIouured the capture rate in 't'arioua l'MOD&nOe .beorben .... function of the time, They determined the c..pture rata by a1 e&na of the capture ,..radi&tion, which they detected with .. proportional count«. FiB. 16.3.6 Ibo.... th e oouuWtg rate obtained from allilver foil .. . fun ction of tim e. Th e eurve hu • .tern. of masima that eeear when the Detluona pus t he energiM of the Ilil "er reIOn&ncell during moderation. Sinoe the reIOn&nCle energiM are well known from ehoppe r meuutemoots, we can get I. rela tion between t he elcwiag-down time and the neutron energy from auch mea · .urements. Fig. 16.3.6 IlhoVt'1l luch a relation i nlnee meuured with Cu, Mo, &tid Zn are plotted. The amooth curve repreeentl the theoretJoal relation l (18.3.6)

which we obtlin from Eq . (9.1.19&) with A_207 and I . _O.M6 em·'. A .pecial feature of the elowing-down proco. in I. hea,..,. moderator ill that the averiloge enorgy 11 . hllrply defined at all timet. Aooordin8 to Eq . (9.U 9 b), yJ'J'IIE - y8/3A-II.4'" for lead. In praotioe, we mtut tab into &OOOuot t wo ot hO!' eUectI that .moat out tho energy diltribution. Noutro rw with enorg1M above 0.67 Mev can be inolaat1orJly .c..tt.enld. in load and can thereby _ large amountl of enorgy. Owing to th 1l inelaItio _ ttering. the energy m.tribntion of the fut neutron. it broadened oonaiderably daring their fint few oolIiDona. However, th1I initial bn».deninB b6oom.. llmallO!' .. moderation oontinuM beoaue the Iloww neutron. make fewer oollisiou per 1IDit time than the r..t. MUtroM. According to BUGJII.- ual.., the energy definition il lItlll about M% at 10 kev and only reaoh.. the val ue of 11.4% Ilt about 1 keTo At Tery Ilmallenergioe. the therm al motion of th e leadato m. makllllit..olf felt and leada to Iln additional broad . ening of tho energy m.trihution. In epite of th_ eHectA, the l'eIOI.uUon ia nJ. ficient in many ouea to permit the determination of reactl.on 0I'0IIl l8Ctiona .. funoUoDa of energy with t.hiI "Ilowing-down.t.ime apeotrometer". To do thiI one inU'Odnoea the nt.tanoe to be Itwiied into the lead block aDd m~ the I

A_

- - . . nWbl. .J[knJ-

{'~~J1).I' ia "-h far.::So.a ...... Ul.edifl'_

bM.- '"' Nld Eq. IIe.Uj" anlmporta.a.L

'" rate of neutron O&pture .... fanotion of time. Since one can always make .. flux meuurement with the help of .. BF.,counter, ODe can determine t he energy variation of the 01'088 lIllotion direet1y. Additiona l normalization meuurement& are neceeeary for an abeolute detonninatioD. In tbi8 way one obtaina Cl'08I &eetiona which beoauae of the limited reeolving power are avenged over an energy band 5IJO wheee rm.tin width is about 12-30%. For man y nuclei, twreeolving power is aufficient to Bepatatethe re8Onancea; the reecnenee JllU'ameters ca n be determined

,

"

by methods Bimilarto those described in

• At tlplW't rift

Sec.4.1.3. BERGMANdal., KAsBUKaJCV, •

~d/f'Ol/flf

,.'"•

A! ! U

-V " f.-- AI IUC~

l~/

f-

. 1\

I

IV

"

/

\

I~ Zll ll.lrl' 1'\0

•• • mol • "

'" "

" ,,

I

1~5Q7W

IIJ

"

• J

Z

-[

r ill. II.Lr., n..

--

-[

tJ~l.pl.ilnono&. hi 80_ - " ' " wlU:l" ................ '1.... . ~IM'

..... 111..... ~D

.

u........1/Y. I"" ...... ta.·

dowJl41"" II*IW_

POPOV I and SlUPlKO, ~d rooently MlTZlI:L and PLKNDL have determined a numbor of capture crou leCtiona and reecnence porametcn in the energy rsage from 10 liT to 10 kelv with I nch a Ilowing-down.time spectrometer,

Chapter 16: References General U&LllI, Eo: Ioc. cit., Npeciall1lJeCtioa IV , p. 306ft. GoLDnEm, R ., J . O. SVU.I'U.lf j r., R. R. OoVaYDl1. W. E . .lCr!nfaY, and R. R. B"TIl : c.JcWaU0n8 01 Neutron Age In R .O end 0theI' MeterV.J., ORNL-2639 (1961). 'hJuoulf, 1.. J . (ed.) : Re.ctor Phr-!c. Co~t&lIta, ANL-&800, Second Edition (1963) ; NpecWly BeclionS .4: Slowing·Down P....mcteD in W.U·ModeRted Media. lloAaTRaY, A. Eo, P . J. PEMLUQ. B. r. Snn.w, and 1.. J .1'DlPuw: Neutron Reeoneuoc Int.cgrel and Ap Deta, Argotme Reector Phyeice Couetante Cfm.1M N..... letw 1 (196 1). Puaun, P. J ., J . J . K.4o.4l1ova, and A. Eo MoAJIT:8aY : N.ul.roo Re.meoce Integral .od Ap Date. ArJoone Reectot PbyWoe Conetellte Ceoter N.1VoIctter 10 (1963).

8peelaJi Roe.. K. E., ~d A. K. WUlfnRO : lIon·pm.

AgeII --..remente in Wanao, A. II ., Ad 10. C. NOOUD t AECD-S41 1, I1I.C\4 (1961). } Finite 000meUy. l

Cf. footDote

OD

p. 63.

Chapter 16: Ref_ _

'"

Al'IDIUL8OJf. R. L. E. FUM I, and D. NJ.o toI: CP· I331 (I" '). BL088D, T. V.• and D. K . TaunT: ORNL-2M2. lOll (19M). urem ent DolunlD, R . c , d aL, Nuol. Sol. Eng. t , 221 (1961). of the Ase of R ILL, J . E .• L D. RonaT9, t.nd T.E. FrroJI : ORNL- 181 (1948). F'-ion Neutronl J . Appl. P hY8. U , 1013 (19M). Hoo n a, J . I ., ..nd T. I. Aan1Tl : ORNIrMI (1960). in K.O. Lolllll~. D. B....nd C. H. BUJfCJWlD: N uol. Bel, Eng. 7. +l8 (1960). Prrrull, W. G.: BAW-I0I6 (1960). QoLDllTJ:III, H., P . F . Zw~ a nd D. O. F08T:U : ) o-va 1958 P/237 r.. Vol. 16. p- 379. Ct.lculati~ of t be Ag" of FY.ion WII.J:.IIIII , J . E., R. L KKLLJ:!f8. .. nd P. F.ZwIW'EL : Neutrona m W..ter. Geneva 19M P/697, Vol IS, p.62. 8J,l1.1tov, LM., V.K.lUuJwI. aDd K.N.Mu~ : J . Nu cl. EQIIrgy 4, M (191S7). Cuwnz., R . S., d aL: Nucl. Sci. Eng. t, 143 (1957). Age M__ DvOOJ.L,V.P.• S.M.PuRl, and K .S.R.ui : OeDeva 1968 P/ I640 . VQI. 13, p.SIS W ~~ Fo.,.... D. G. : Nuc l. Sol. EIlI' 8. Ita (1960). 1M HIOIKJII, C. A., .. nd T. SraJlloJ:a, Nuol. &1. Eng . 10. IISI (1961). V arioWl MOIfIl. A. M., .. nd B. PulfTlOOavo: Can . J . ReI. A Ii. HI7 (19'7). R u D, M., F. OIlIlIlIUI1l'. and R. L KJ:LLl:N': N ucl. Sol. Eng . .., I (1968). """inH.O. SPlIIOJ:I., V•• D. W. OLlvn, .nd R . S. CA.II1IfJ:LL: Nucl. Sci. Eng . 4. 646 (1968). SI'I.lJfou, T . : Nukleonik I. oil (1M3). V .u.Jt1l1:~ F. A., and R. E. SOLLIVJ.l'l : Nucl. Sci. Eng. i, 162 ( 1~9). LIVIlfJ:,M:.M:., d al. : Nucl . Sci. Eng. 7. 14 (1960). ) SPlIIOJ:I., V., ..nd A. 0. B. R IClUllOSOJf: Null!. ....._ A •__ V- " ._ • DO Sci. Eng . 10, 11 (1961). ..no ge....- "..oue ......un- In • • WJ.DJ:, J .: DP_I63 (1966). DUJ:T, W. G.• dal.: AERE R/R 2Ml (1958). DuooJ.L, v . P ., t.nd J. M.ul TJ:l.LY : 0-.,... 19M PJ358. Vol . IS. p. 28. FU»I, E . (Ed . G. BICKJ:ILIT ): AECD·2t\lW (1961). The Age for Varioue Hlll naJ" J. M.• d al .: GeDe.... 1968 P/OOI. Vol. 12. p. 696. 8ouI'CtW in Graph.ite. HILL, J . E.• L D, RouKT9, and O. M oC.uo!OK : ORNL.J87 (19'9). S1'1l0ll1lf. C. U.: TID.I661fo (1960). CUPBJ:LL, R. W., and R. K . P... lICJl.,I,.L,L : Trana. Am. NUN. NOBLU, R., and J. W.u.t.AClI: ANL-4076. 10 (19'7). GoLDllTlIl1l', H. : " Fu nda ment&l. .AIIpeot. of Reaotor Shieldlni". \ RMding : Addilloo-WlIIIl&y Publwhing Compa.ny. 19159. The Spread ing Ollt of HCLTI . G. : Arkiv Fy1dk i . m (1950); I . 209 (1951). N~trou .. t Large VIIRo.. lL. ...d G.C. WIOll:: P~. Rov . 71,862 (1947). ~ .... WICII , G. e.: Ph)'ll. Re .... 76.138 (19019). ~""B, W. : J . Appl. Phya. !~, 1236 (19M), \ C4t...... J:LL,R.S.• dal. lloo.mt. _. . , .._ . _. _ ,,_ . ._~, UnlmeD 0 ..... ~_II ...... 11. ... ~ • 0lITU, D• 0 . .. HW ...... (1-' ""'" . of lI'u t NeuttoM. RUIIR , J . H . : P hY8. Re .... 71, 271 (1948). W"'DlI, J . W. : 100. lilt. CBoO"OlI, M. 11'. : Nucl. Sol. ED3. t , 631 ( 1~1l · DII J o allll. J . A. : Nuo l. Soi. Eng. t . 408 (1961). EIIOr:LMA.J(K, P. : NukJeonilt r, 12lS (1968). The TIme Dependent'JII of GKOS8.IIOo. G,: Cblmen TekJnillb Hllpkolt., GOt.eborg.

ar.....

1

1

lleport ClH.RF....(1963). M6u.u, E. ••nd N. G. 8.J6IITa.I.1ID: BNI,719. 966 (1962); AD Atomenergi Report RF X U8 (1963).

IIoderatiOilm W..ter.

BDOIlU. A. A.... ..L: o-.-.llr66 P/&U. Vol ... p. 1M. Jsuoy. A. L. Tv. P. Pol'O't'.NId r. L SIIUDO : Semel. Ph,-c.

JJ..T.:P. 11, 111(lieO). Yv. P.Porot'. U>d

r. L SIUJ'PO : J . Nuol. Eueru A 14"18 (IMl). lIJTul., F .• aad. B. 8. l'un:>L: Ullpa llu.htd Karllnabe Report. 19M. Pol'O't'. Y17. P., aad. r . L SIlA!'mO: ~ Pb~ J .E-T.P. I" 1132 (lM I); 15, 883 {11le2}. x.u~.N. T..

Tbt SJowma-DoW1l·

TIme 8~.

17. Investigation ofthe DitTusion of Thermal Neutro ns by Statio nary Methods lD thl. chapter, ""' Ihall beoome t..mlliar with .t..tionary method. 17f deter· JQ.in.ing the dlffuaioa paramet.en of thennaJ. DflQt.roDa. FirIt •• Iball oonaidfIr in Sea. 17.1 the cIa.lo.l mfltbod. of meuuriq the clifhM:ion length. 8fI
17.1. Meuurtlmeni of th e Dltrullion Length 17.1.1. Poln* Soarce In au tnllnUe Bedlum All methode 01 directly meuuriDg tho diffu.eion Iengt.h are bued on obeernng tho .,..riation of the tbormeJ. DflUtl'oD OUl: in the .auroe.free part. ol a meditun. Th. linear dimeneioDl of tho medium. mut be large. of the order of magnitude of

.,.enJ. diffUIion lenstha at.. ..

The ei.t.uation ill Iimp1elt. in AD effocti"ely infinite mediUI:Q. i .... a medium . hoee dimenei.OQ.l are -0 J.arp that. praotioally all the neutrora eMit ted from th e -oW'OCl are abeorhed. ThiI bppens when t he diameter of t he medium. ia about 30 diHuaion lengthaj in water thia diameter ia 80 om, in gra phite 16 In (cone. eponding to a oube of graphite we1ching about liOOOtonsl). Thu. we can only reallzo AD infinlte medium. of water or .ame other hydrogenoue moderator with a ebort diffuei.on Ieqt.h. When we ue a point. thenDeJ 1OUI'Oe, It. eufficee to meuun the f1w: ,.a.riation a1oQa: a radiu with folla . Aooordina: to Boo. 15.1.3 •• ha,.e in thia e-e Q

. -rtl.

'lP(r) - bD ' - .-

(8 .1.13)

and we obtain L by plotting los (r . tP(r» agaiDat r. H oweyer,.moe moo neutron emit fNt. neutroM. the cue juet diKuMed cannot be realized immodiately. Ne ....rt.he_, one can create a nesatiee thenDai eouroe through eadmium dif· ference meuuremenu . One meuurtlB the neutron flux near the ltOuree onoe with a nd once without t be IOUl'OO ciOMlly . urrounded by a abell of material, preferably cadmium. th at. capture. thermal neutrobl -trongly. The difference of th e t wo meuured nJ. . 1. cau-J. by the tharmaI beutl'Ollolabeorbed in the abell ; it lhoWd MtiatJ the fonnala gino. a bon rigoroa.ely. BKCKuaTS a.nd KJ.O_KA, amoog othen. 1tODr(lM

361 have eieeaaeed the diffusion length in light water by this method. Only moderate accuracy is attainable since the Dux values underlying the evaluation, being the dillerenoes of two large numbers, oont&in sizeable unoertaintiee. The mea.aurement of the flux distribution in the neighborhood of a (Sb- Be) source yie1dll ccneldere bly mo", &OC1U'&te :reaulta. Bea&WMl of the low neutron energy (Et> 20 om. The latter alternative ill ha.rdly feMible since for reeeoneof intenaity precise DUl: me&8uremenUi for 1'> 200m are very difficult. According to DB JUREN, the necesaary ecrrecucee can be calculated lUI follows. The source term appearing in the diHwlion equation

Dfl tJ:I (r)- 1'. tJ:I (1')- 9a (1') = 0

(17.1.1)

ill given empirically by the el:preaaion Xc- E,

91'(1') = - , for diatancea from an (Sh- & ) source greater than 12 om in water. Heft! 111:= 1.68±0.02 em. The solution of Eq. (17. 1.1) with this source term ill

(') ~*

.-;/L !O- E.(lE- ±J.)+" .'LE,([E+ ±HI.

(17.1.2)

Heft! a iB a coDlta nt l • A eceeeucn F (r) factor follows immediately from Eq. (17. 1.2): (17.1.3) Now tJ) (1') .I'(r)_.-,/LI1'; thu by multipUcation of the me&lJured valuel of. Aa (r) by 1'(1') the influence of the source DeotroDll ia eliminated.. In ordlll' to calculate F(r), mut be experimentally determined. ObvioUlly,

a

:~;)

. . '*

l~-i)' [O_EI ([1:- i Jr) + .1"LE1 ([X+±Jr)!. (17.l.4)

If for constant I' we determine f1>{9u. (from the cadmium ratio - d . Sec. 12.2.3) and if wo know E, D, and L, we can calculate 0 uaing Eq. (17.1.4). However, linoo L and D are not known initially, we moat proceed iteratively. Fint we make I To ..kn1Jat.e C. we mlUt know fl' (') all the ••y to . _0.

Mtimate of Land D. then . . determine 0 and F{, ). and then we oorreet t he m_1lJ'eli data aDd det«mioe &Q impoved 'Value for L. etc. Raa:a and Da J URKlf find that O - O.oeoe fOl' watel' at 23 "C. and obtain the 'Value. dOwn in Fii:.17 .1.2 for F(, ). Th e value L =-2.776±O.009 em fono," from th eir eoeeeeed da ta. Later in &C. 11.1.... we 8h&ll learn of additional e:zperimental reeulte fOl' watel' ud other hydtogeooUi modera ton• including IIOme at higher temperatures.

&Q

...... '

••• ..

I- -

:-..

",

1\

,

f

~

•

\

_.

\

•

'"

1\

• If • , II II II''' lJ,d,tw ",. SHI ..,." r ".. 17.1.1. 'hot - . . I _ _ f 1 u _ ...... . ~,

:

~

'''. • ,-

,

_. (1'-") --. _ . - w .... - - . _. _

\

,

II .. .

""*"

".. 17.1.1. Tho . . . . -eJoa lor . ( Il:>- h ) _ 110 H.o M tI "O

17.U. FiDJte He4Ja; the Sigma POt

Th e method of mfl&lluring diffuaion lengthl jUlt diee useed it oat ap plicable to IUt.u.noee wit h large diffuaion lengthe. l ueh as D.O. graphite. and beryllium . In thia eaee we moo build an ueembly that it finite in compariaon with tb e dif· fwion lengt h and take the neutron leabie throUih t be .urfaOll into AOOOunt . The ltandal'd arnngement for tbe mNlunlment of diffusion lengthl in thit CIUfI it tbe IO-Oaned &i&ma pile. A . a pile it a column of tbe material boinl In .....usated witb a eylindri eal or .quare el'Oll& IIIfICtion that is fed throUib one end with Deutrol\l, The diHnaion length foUo_ from an analytw of th e Ow:: dirtribu. tioo in the pile. Diffusion length meaaurementl in a ligma pile have been carried. out 00 graphite by HaazwuD dal.• by C.f.RLBLOY. eed by BUDRla d al.; on borylliwn by O".f.Sl:V.f.. dol. and by B OOH U ; on beryllium owe by K OItCBUIf dill.; aDd on D.O by S.uoalfT dol. and by M..J:UB and Ltrn. In addition. a ",riot of inveatigatiorul in 1fater and ot her hydrogenoUl moden.to ra b ve been canied out in thia JtlODIetI'y.

1"Ic.11.U'" a 'JPieal appu-aw.1or _ Y Oft gra phite..

Tbe _ ~ I to •• d.iff-ioa \nMl of _ bola ~ eRhet " ndiooMt,I.... -.rM flit 14 ~ "'rt. _ 1Udl ill looMed Oft the _ bal u it 01 \hi pile . . . to iy ...s. ~ MRit fM\ _u-. ..hicr.b. apia Ieeda to OOIIlp&t.Uocw In the

6".... . (..-aU,. 1_" ....1Ie'"u..__ t to S. aod Ieac\bI- 'na __ • lor u.. '-'c\h , pile. One _ ... "

m-

onhat.iorl of \hi - m

~Jy

au

dil&n"'_ NIlI _

11I_

pr'OYIde IoqlIIJ-fIB'

rot e-dmIlllII

-...-......g..

d.ifi_ For tho. ~ _ IMy pnlfi.t&bJy _ a CllIdmlum plate tha., ClOftn \be ..tire ~ MC\ioa 01 die pile .Del that _ euily be '-ted a.ad ftIUIOnd. ODe _ aroid \be ~ c.~ bylut. _\rom by f~ weU·thermeJiud _t.rou from u.. tbolnn.J oo!umn 01 a .. adeN .-ct« iDt4 tbe lIigma pik 'I'boI ~ ..... "'PPM' nrf_ of the aigma pile IboWd be c.nIu.lI.y connd. 'lrith -mulllD (or NlOt.her DeIIWn~) in onIer to provide a oJe.o. bowld.ry oondit.ioD for the t.hermal DelItrorlB. When a f . DeIItnm IOIIIOe 1& being uaed, neub'onll with ~ above the e.dmium CIIt-oHenel1Y can IMve the pile =hindllf'eld. _u...r on \he 8t tltts Ii /lalt f,ih;r". z ....n. of tbe room .. ov. _ other ~ . IlIId &pin .."'" til. ~ ; \Ail Md. t4 a d ..\or\ioa of die Ou:

U ;.J

""~ , n-

_ "OI!""

di.trib'lltioQ .-.. \be .arl-. 0... ll'l 1.-t \berefore avoid plaeing NOy 'Will ftIf1eclton neu a alima pu.. Th6 DIU dl,tributlon hi ~ wllh rem. tJong \he oentnl a:ll. of tbe pile &Dei -..111 tJoDa: \be mid. Ii_ of MlvenJ _ ~ a' ftriou& . .~ from \he alQJ'W" Devicm an ~ kl u-t t be too. ill pree ....y reprodueiblll poaio ..... 1T.La. J. ..... pOI fOI' or.- lo,..clI ••••_ _ t1on&. 8inOll ,be diHuaioA pvr.mtIten I' """'1&00 find thWl th e nU:I diBtribution depend on the modenw te mpentur&, the room temperature ,hou!d be kep t fN«lR&bly oonItant 1±2 "C). Special b-tinl de vi.,. an - . . y tor ,....""'men .. at hillher temporaturM-

,

_.

,,- -!'I-'------,-- ::::::::- - ~-~-~.,.,:,/'

According to Sec. 6.2.4. if only thermal neutrona are preeent th e OUJ: dis tri but ion .. given by 4t (z, y. z) =

t;. A,. 'inh( C~:)ain (~_!) llin( "':--'-).

(17.l.6)

A,.

Here tbe are source-depende nt COnBtantll that are unim portant ror our purpoee : the relaxation lengtht~. a.re given by (17.1.6)

a. II, and e are the effective edge lengtht, i.e.• the actual edge lengtha augmented by twioe the e:lI:trapolation length. Sufficiently rar rrom the ecut ce, the contribution of th e higher Fourier component. of the Ou.. will be Imall, Ind we Ihall have

~(z. y• • )-

einh(!i1-)

117.1.1)

on t he central axil or th e pile (z = l a, Y=lb). Ueing Eq. (17.1.71. one can immediaUily obtainL rrom the decrease or the DUJ: alongtbe central am in thie region. and then hy means of Eq. (17.1.6) one can determine th e difrusion length L . In practice, one prooooda in the rollowing way . Finlt one determ inea at what aource dist&nce the cont rihutio n or the higher Fourier eeespeeente may be neglected . FJuS" mM.lure menw along th e midlin. or t he v ariOllA em. lIfICtion.IltfIrve for th w purpoee . Pis. 17.1.4 abo_ l uch a distribution meaaured in th e pile Fig. 17.1.3; we eee that it can very &OClurately be fit hyaline function. rrom which we may conclude the ebeence of &I1y higher Fourier oomponentll.

lhown in

- ---Th, pboklnltlltrona that .....jec\ed trom Be or D by _I'llec.io ),oray. from the I

.... ~ ~

~

MeUnlinMed ID oadmJam diH_ _ LL

IOIIl'IlI

In addition , the latenJ meuurernentl abo yield information on the extra· polakd eDdpoinL If we &1'0 oert&iD. that a mngle line function lin (n Z'/G) 0Qm. pletely deeeribel the !las dimibution , we ean det.ermine 4 by the method of leut tqlW'ell eed thea determine th e enrapolakd endpoint d from.the re1&Uon 4 =- actU&1 ed.go 1engtb +t.... It turna out that the relati on d -0.71lc, is not al.... ys u a.ctJy fulfilled : ~ P-ible C&WJN for tb e deviationl were diacUl80d in Sec. 10.3.4. T..ble 17.1.1 oontainl 80me directl:y dew,rmined valuee of 4. Since G is usually :> 4, the deviatioM from the limple O.71 l,.-law C&WItI no difficulti ee in the calcul..tion of the diHuaion lengtb with Eq. (17.1.6). In other wom., it ., Qually aufficlont to u determine the quantity 4 from the limple ox~ a =- actual edge length + 2 .(O.71 lv). Next, tbe fin deereaeo along th e central am is dew,rmined in tbat region wbore the C1'OM eee, , .1 I r/f Uonal m....uroroen... Ibow that 1'II-17.U. n.1I.WaI_u-llqdloUlbuUn lu olp>a pUt• tbero are DO higher Fourier com, .)1'-'" 1Iu: - - . (_l_ol• ••• ponent. (et. Fig. 17.1.15). E2;oept

"

T.w. 11.1.1. Z1tJ* ' m_

..-

O,..pb.it. (1.11i_ !) D,O I (1Kl.41 'ro) .

,&,

.-

41_ 1

U7±O.Clll U4±O.oe

_ ... .........

Yal_ of ~ &-..palGtetI Z_ poiu

HS1fllam til • •

AuoD, :M1JJI'1f. and

....

... " <-I

...

""-)1-I- I

•." •., 2.31

U

Polf'nlOOal'O

. ..

11,0 . . UI o.m 0." I A __ ..,aNatJOa. 01 t.II.liI uperi_t by JUu. aad WOODa '10'" . .. 1.81-. . /0.11_ 1.611 0lIl.

for the rogioo dc.o to I =-C. the Ou decaya expooonUall:r, aDd it .. cu.tomaryto inboduoetbo.o-o.Uod " oodpoiDt OOl'T'OCtion" ln order to olilI1inate tho inOuonC8 of the 1'III'faoo. whoM ~ l.d.I to dorlatiooa from the &imple upooontial 1&_. Therefore Jet, Q.l not.e that Eq. (17.1.7) caa abo be written in th e form. fll (I)_.-el""

(1_.- !i;;-) . 1

(17.1.8)

,-, If we divide the meuured vaJ.QM b:r 1_ 1- 1 ,;;;- _ whicb we e.....luate uaing a gu. . for ~l th..t will be imprond it.eratJ:vel:rl&ter - the eoeeeeeed vrJuee ehould follow tho llimple uponeati&llaw. On. IMlli10c plot 'we then got a Rraigbt line wbe:- 110pe p.N ~l (d. F1g. 17.1.5, oune U ). Natura1Iy , lhJe entire prooedure caa be programmed for a oompaUna machlno.

... h will be inatruetiVfI to _timal.e the quantity of material ~ for an accurate meMwtlmetlt of t be diUuaioD 1eIlgt.h in to Iigm& pile. According to Eq . (17.1.6), wo h.... (17.1.9) where we ha ve IIet fJ' '''' n' (~i" -~) by way of abbreviation. 11 ." ...ume that the lltTO f in the de tennin& tio D of fJ' is negligibl e, then the relative error it LIL UI giveR by

+

tJLL _ tJt :!. (l + P'V).

, • (17.1.10) ":i

When the crou Metional
1/

Ion, ... mentioned above, the Cl'OlIII aecti.on&1 dimenaiollll mUllt be . t leut 2 to 3, and even better 3 to 4, diffusion length.. The length of the pile followl from the require. ment that th ere mut be a region : of at Ieut four relasation lengtha in which the decay along the a:ria it pwoJy UPODCIDt.ial. A. .. rule, we c.n acme n thiI by ch~ the height to be .bout La tim. t he crwe .ect.ionaJ. dim onaiOc.. Finally. let ua mention agm that thOM oonaid erationl hold only for the Dux peodu ced by thermal M)W'. ON outaide the Ngion of meuure·

r

tn ent, ThUi we mUll either UN .. thennal column .. the neutron IOW"Oe or eliminatethe .o1U'CN lytn, bWd e the reg ion of mequrement by

/

V

r

I

•

,

II

1/ / '

'11111., $# " t·l -

••

h .17 .U. ".. IU1ool. _ _ flu 4IolIIboI_ III plio 1 11m- 11I11_ I). ~ 1:

-, .... ,.

or.r... U : wtIIo .................

the c.dmium diffenooe method. In media in which age theory holda (graphite, Be, Dea). we ea.o Wo eliminate the effect of u-e latter MJW"ON tbeor'et.icaJJ., (ct. Huzw4JU) d all. 11.1.3. Detenn1naUoD. 01 0 . Dl.lhuloa .Leqtb.hom Oe nu Dldrlh.Uou. In • Medlwn 8QmtUJuied ., • 8urfatt Souee

One needI . maller quantitiee of material than required for. aigma pile when • met-bod .uggllllted by HZllIDlBZM is used. The . u.b.tan0l'l to be Inveetigated 11 placed in tbe interior of a .pbere tb.t 11 . unounded by • reflecto r of pt.taffi.D or water. In the oenter of the . pbere tbllre 11 • .ouroe of f..t DllUtIoDI, e.g.,·a (&a - Be) 1OUl'Oll . One fint meu~ the flu diatribu.tloo ~ (r) along. ndilll nctorj thllD ODe IIIlrTOUDdI the . phllre with a cadmium abeD but the

1M,..

366

1D~ of

the DifhWoo. of Thermal NnW- b1 ~ lIethooh

reflector unchanged and again meuuree the Dull. 4)" (r). 11>(,) - 4Y(r)- ctt ' (r ) then ill the BIIJ: doe to .. thermal aurfaoe .atU"08. In the interior of the Ip here (/) '- the aphenc.n' l)'D1metric eolution of e1Maent&ry diffuaionlbeory "'(r) _ :;; aiD (' IL ).

( 17.1.11)

Th ere is a condition on th o prac tical a pplicability of this method, namely, that th e radius of th e I phere be luge enough to allow d>(r) to cha nge .ufficientJy beeween the eenter a nd th e , urfaee . Tw. method waa used during the 1940'. for the determination of th e difflaion Illngthl of heavy water (U J:18J:NB1t&O .nd D6P&L), beryllium (BarB. and FoHns). a nd graphite (J J:MSEN and BoTHs ). It hu the diMdnntage of being Umited to . pbef'M. and in the cue of graphite, for lIump1e, roquu- ooMidetabLe machining of the lndiridu&l graphite lIuga. 'l'hi8 .uggeeta &hat it may be worthwhile to generaliul th e method to eubee. &.rr.c 011 eM SlIT/Gte of II Cwbt. The IU t.t&nOll to be Rudied b.. th e form of .. cube IUrT'OUDded by. refleotol'of pu-affin or_t« (cf. Fig. 17.1.6). The IRa - Be) .aw"Cl8 ia located at the eouter of the eebe, Th e Du dWtribution is meNllfed 0000 with and ceee without .. cadmium covering around the cube. Th e diHerence repr-enta th o effect. of .. . urfaoe ~ on the eube . The dilfueion equation in the interior of th e cube muet be eclved hy approxima . tion. The sym met ry of th e system around th e center . uggcetll that we tako .. our solution a form oonta.ining only even fun otion. of the coordinatee (the origin ill at th e (ll!Inw of the eube) : lJ>(Z', r. , )_4>,S, + lJ>,S, + lJ>. 8. +4>.. 8.. + lJ>. S. + lJ>.. 8.. + + lJ>.., S..,+ 4>,s,+4>.. 8.. 4>.. 8.. + 4>... 8. ...

+

I

(11.1.12)

it ill not DClO&M&rY to take into account t.rm. of orde r h.ightr t.hao the eilhth.) llet"tl th e 8 &I'll th e elementary lJIDIDetrio funotiona

(It 0U1 be ebOW'1l that

S,_. ;

8, _:r8 +y' +,,; S. _z' + ... + at;

y'+,,:,+,lzoI; S .. = .z'y'+.z','+ y',' +'" zoI +"zo'+" y'; 8... = zo' y',' ; S.._ z' y'+ zoIzI + Y' zo' + r'r,I +z' r +z' r j 8•• _ z'Y' + zo''' + Y'''i 8.,. - "y'zI+,.zo','+a'zo' y' S, -z' +Y' +~ ;

S, _z' +yI +~ ;

S.. _ Z"

(11.1.13)

a nd t he (1J &I'll OOfIItanti (free. at tint). 'I'hi8 geoeral form muet aatillfy th e diffuaion eqQ&tioa. Application of th e Laplacian operator to the 1'&rioUll 8 ·lun e· tio... the following rela tion. :

gi,..

VI 8,_O

Y·s._e P·S. _12S, P'S. =30S. VlS. _MS.

P'S.. _ 4 B, V IS., -U S..+4S. P'lS., _30S..+ 4S.

P'Su -12 B..

(17.1." 1

VlS...- 38S... +28.,.

The introduction of the form 17.1.12 into the diffuaion equation V1- flJIV =O and WMl of the Eq. (17.I .l f) glVlIIo a IerieI of relatiol\l among t he oocdficienu i thus in place of Eq . (17.1.12) '\II'1l have the lOIDowhat limplilied n-ult

4)".

'01I. LIL+ 'll>t~,B(s, '0II , L )L+ ~ C D +....... (a', '.1, )+ ..... . ( II:, , • • , ) .

'11> - tI>. ,A (lr,

I

(17.1.13)

AIlI:, ,. I, L }

_[6+-1. [8.+,l.o [8..+.;;'1 8 +,.'1' 18...- ~;! Jill] + ~ ( 8;, - 4.5S O(r , ,,1, £ )=-8. -7..5 8.. + 90 8 + 3~ (f B (lr. r ,l,L)= 8.-3 SI l

11

-

D (r, "

I,

+.lr[- s +!r-1l 8.. + 7.6 S

(17.1.16)

- 6 S Il]

L )= 8,-14 8• • + 368" .

t IM' fir

(rflll fir

-

",.,ff",

__ _ _u.._lIf. _

.... If.l.&.

_

bqt ....

. , _ ~_

_.. . .,--__

.... 17.I .f. ....

_

loIJo .... .-IootcIIo _

We obtain the diffuaion lelljfth in the followinl • • Y: Th e .....Iue- 4" of the flu det.enIlinod at .. aeri~ of poinw (Z" ll, are fit to Eq . (17.1.16) with th e method of leaat lIquarel, TbU8 '11'11 obtain 4'•• 4>., 4>•• and ltJ, . Th t8e u leulationa are carried out for .. Ml riee of L.nluN that lie near th e expected val ue of th e diffuaion length. Nen we form the aum of th e IIqIlUN of the reaidll&1s :

Q-

.1:("'- ~ ((z,'4 . L»·.

(17.1.17)

• It ill obviously a funct ion of L and b.. a minimum when L ia equal to the dif·

fnaion length. Th e method CIon be aimpWied by Iineoarizing the L-depend enoe of the queeuuee A, B, and O. This method h&ll been applied by F'ITt and by SoHLUUR to graphite. Th e detailed nume rical work wu done on an electronio computing :ma.chine. Oyli rwlrical Sw/4« IJo.ru. We can combine the principle, deecribed in Sec. 17.1.2. of the aigma pile WhOM .1Zl'f&oe 11 black to therm&l neuUOQ.l with that of lUlf&oe JOllI"llN. Fig. 17.1.7 U10. . an arrangement fou'Ilcb a combination. The aubatanoe to be ItUdied b.. the form 01 a oyliDder and Ia alwaya covered with ca.dmium on top and bottom. The (Ra- Be) lOuroe ia loeated on t be ana 01 the

cylinder halfway up . The neutron diatribution along various lines parallel to the uiI it moaeUJ'tld with and without a oadmium aleeve 8urrounding the cylinder . The differeooe di8tribution it due to a thermal 8urfa.oe source 00 the curved .urface of the cylinder. In view of the boundAry oonditioI1l at 1=0 and 1_4. _have (17 .1.18) ~ - A•.I,()". ·r)1in

":.!. .

L •

(17.1.19)

Here 1.(*) it tho uro-order modified Bessel function of tho tint kind and the A.. ue oonatantl. The flux dietribution along a line parallel to the axis and eepereted from it by a distance r, it given by ~ (I) =

L.. A ..1, ()". " .) 8in .!...a :II,!. .... L B..(r, )8in -!.'!.!. . .. a

The quantitiee

B..(, .) .........1. ()"..'.l can be determined by Fourier inveraion of the flul: meaaured along thill line. Beceuee 1. (0) = 1, we have on the axil of the oylinder ~=

L A.. lin ,~ :II! • •

•

It followa from Fourier invention of the flul: diltribution measured on th e axis that B..(Ol= A ... Th~

1. ()"..,,) = n. (,~

(17.1.20)

8,.(0)

and L can be determined from thill tranacendentalequation. The method can also be used on 8UblrtaDoee in the form. of a priem or a cube. It waa used by F'rrj to detennine the diffU8ion length in graphite powder. 17.1.... BetulUi of Varlold Dmll8Jon

I.e.

Heuurementl

Onf,_'y Wokr. Some recentel:perimental rtlIultll at room temperature are oollected in Table 17.1.2. The valUeI were referred to 22 °0 with the temperature TabI1l17.U. Til DifjwitJ"

~

Qf WaUo' ae22 °0

B_UftINldlCJ..tlua(I958) Cadmium diffemr.0Il 2." ±O.03 in aD. infiD.it.e &fda aDd Kol'UL (1961)

Da JIJUJI".oo.R_ (IM I) Ro.. (1M2) Roollrt ADd

8.o~

(INI)

medium Thermal ooIu.ma aad a tlgma pile

2.'6 ± O.OO6

(Sb-Bel _ ill .. JDfilUt.e medium

2.776±O.006 2.778±O.Oll 2.8311±o.oI8

I

CofT'eQtad for dtttributed .c>Ul'OeII aooordiDa too 8110.17.1.1 No llOmlCltion lor die.

l

tributlld IIOIU'OM

'"

ooefficiont given in Eq . (17.1.21). Th e c1eaneet meuuremcnu to date are probably thoee of Ill: J I1IlI:~ and a.aa and of S't.uur. and. K OPJ'aLj the ....ef&le of their VIllu~ .. L =>2.761±O.008 em

whi ch we ' hall take ~ the host va lue avail.bIo. Fig . 17.1.8 tiliOWI eome meeeueed values of th e diffu sion length in water aa .. function of the te mperature up to T = 250 "C. Meas uremenu a bove 100 "C mwt be carried ou t in .. preeeure tank, and t he introducbon of the neutron llOUl"Ce C6U8N difficu1tiel. .. p " .metal

"

~

uf--+- -+- -t-----:

. .. . i -

". . 17.I .L Tbo ~

- - - Kot- C17.UI): -

(.,-1,.:

..... v t " " ' _ V - IIoH.O _ .....~ ••• • I I I _; - , , : . . . ._ . . . ~ I .. ~ :. ; . . . .1a\84 _ ... c: .-Iealo. ~ . IUl ... C., """" . . lfelkllo-.w

(71% MD, 18% Cu, 10% Ni) and mixed ceramioe eompceed of DyO. and A1.0. have pro ven tb emtelvee I1IIeful .. tempentW'e-rNi.ltant foil material8. Th e temperature depend ent me&llured valu es agree lrith one a nother fairly well. In the vicinit y of room temperature, .. good ap proIimation is

L _2.77+ 0.006 [T -22]

(L in em . T in "C) .

The temperaturo depcndonoo of the dif fu8ion length in B .O in th e following way . To begin with

-

_8__ .-./lr f . L - -- - f "·(8}W·-· -

l)

L' I T} - -

__ _

r.

I

3N'

I -

•

8

be inter preted

dB_ t7'

",.ll') t7'

-

(lUI

(17.1.21)

1ar

dB

(11.1.%%)

tT

The temperature dependence of th e atomic density N ill known from density meaeurementa. Th e te mperature de pend ence of the a b.orption term in the denominator of Eq. 117.1.2.t) follow. &imply from tho l /..le w for a• . In order to cal culate the temperature dependence of the \.nnaport term in the numerator of Eq. (17.1.22) we must mak e eorae Ulumption about the energy dependence of O'u(.&' )' T he following ca.sea have been trMt«l : a) fuDItOWSn's PrMcription. a~(B) ill given by O'. (E ) (1-0086(.&')). 0086 1. equal to 2f3A. for -eattering on free nuclei. R.t.nll:owsn h.. luggeet«l that tha relation be generU&ed to the proton. bound in water by introducing a -......IrU,JII _~ M

mitabJ1 defined eouu-dependent effective ma.. '1'IWI mIMI hi defined in the followiq way. The IOat:toering croM MOtion of a free proton • 20 bam. Aeoording to Eq. (1."'.3). the _tterlng Cl'OM IOCtion of a proton bound ill a molecule of

ID&M

A.., • 0'.-20 (;~+fr bam. We can therefore derive an effective

~

(11.1.23)

from. the meMUl'lld _tUring CI'ON .ection of water (el. Fig . 1.4.8) and th en ca1ou1ate ak (B) from. a.(E) aDd A. (8 ). DaozDOV d al. have improved thiI /11 preecriptlon eomewhat by taking III th e thermal motion ofthe moleou'ffru/rI

-

t. ",

u

.,., 11.1."

".

If

.,.--

~

U

u

Ioq\ll .. _..,. . .Mr _

Bo°_..t

"U

into &OOOUDt. The result of thil caloulation hi shewn ill Fig .17.I .S aa curve A (in this connection el. alao ElKIK) . b) According to DJ:UT8CB we obtain "a good appronm.ation to tbetemperaturedependenceofthe diffusion length if we simply take a'r(B)-

Yz ; thia correaponch to

curve B in Fig . 17.1.8. c) Curve Gin Fig. 17.1.8 haa been calculated on the be.aia of the Nelldn model for waw (Sec. 10.1.3). 0086(8) waa taken from Fig . 10.1.9 and a, {E) from Fig. 10.1.8. (Actually. we ought to calculate 008 6 (E) and a. (E) for each water temperature since the 8t&teof thermal excitation and thu8 the aoat tering propertiee change with iIlcreaaing temperature.) OITtu Hrdrog~ MOtltsakw,. Table 17.1.3 oontainl eome measured valuee of the diffusion length ill variout hydrogonout moderatorB j the meaaurementl refer in put to room temperature and ill part to higher temperaturee. TabMlI7.1.3. Tla Di/juioa LelIgI.\ i.. Yllriol+l Byd~ Modtrakw•

........

Dowt.berDl A (20.81';' d1phenyt. 73.1"';' dlphenyloxide)

n;...."

........

Lulllte(CJl.o.,l.181!_->

........... ,'<,

... 36

92

... ,." ..'""

192

.......

Ll"]

6.216± 0.006 ' .N1±O.OO2 U3o&±0.006 6.245±0.004 4.12 ±0.07 3.1. ±0.03 2.751±O.063 2.807± O.0G6 2.833±0.052

I.HI± MilO

3..l20±O.OM

B.t.T.t, Kiu. ud PJ.1o

B.o'll'1f dol.

11mnd

""'"

071

HUJI1Y WQttr. SABOD'T tl Ill., LUTZ and M.I.B, and MmrrSB have porlormed Iligma.pile m6al~mente of the diffusion length of heavy water at room tempenture. Fig. 17.1.9 1hoWll the varioull'Multa at 20 OC plott«lu a funotl on of the H,O content of the D.O. We e&n O&1oulate th e diHusion length of D,D with the formula

(17.1.22)

where PH,o and Pn,o are th e l'06pective molar ooncentratioM of the light and heavy water : p.,o +PD,o=ll. With lJ.(2200 m/aec) equal to 327 mbarn for H and 0.5 mbam for D. one can WIll Eq. (17.1.22) to extra. polate the meaaurod WHusion length to 100% pure z, I_ J

.,

D,O. In thiI way, ~ and LUTZ found 136± 7 om,

Be 2O.8±O.li 1." Nonuand MzISTDfound 168±18om, and S.uwBNT obtained Be GlClUS.,....d cal. 1.78 21.1±1 (2I.3±1 . t • denaity 171±20emfor L(Pu,o =O). of 1.86 g/cm!) The ceuee of the strong deBoO ..e Ko&CllLIll' d aI. 32.7±O.li viation ofLUTZandMsnl:R's """'WK U ." ±0.2 value from the others ia C unknown. We should note, however, that beaid&l the emml in the meuurement of t he difftaion length (lrron .tao ooour in the determination of the D,D concentration. Furthermore , the (}i1 content of different heavy water 6t.mplee ean fluctuate considerably and thereby appreciably affoot the abeorp. tiOD and thu the diffusion length. L ~ T "· 0tAu Modt,rawr8. Table 17.1.4 000- - L ~T '" 1~ n I ,*l1wn1 ~1 taioa the reaultll of diffu8ionlength meelA" 8uremeotll 00 beryllium,berylliumoaide, n N and graphite . There are many more f. meaeuremeotll on graphite than are re- .... ported in the table (of.,e.g.• HJ:~W.l.BD n d al., WINZBL1B. B.EOKURTS, 8cHLtl'TER, ERT.l.ND d al., REIOJURDT, KLoSE • HENDRIEd al.•and RICHEY and Broox). Such meaaurementll have very frequently boon undertaken in ord er to raooount for the effect OD th e absorptioo ... 17.1.10. Tbe dlfflllloa Iolll&h bo _Ill" _ _ _ _tan (Ll.oTD. CLAn'OII N4 Illou'rl CI'088 86Ct-ioo of impuritiee that are diffi cult to determine chemically. Mn.L8. as well as wYO, et...YTON, and RICHBY, hall etudied the temperature depe e, deDce of the diffuaion length in graphite. Fig.17 .UO aho," L(T) in the raogtI I ~ depeodenoe of \b. d1ffuaiou. -tfioilmt O!l the h)'lhopn _ _ tn.tion hu t>M(leoted in Eq. (17.1.22), but in UlIIl resion of hiP deutmUJD OOIlOllIltntb'! thM ". are oonaidering heft tb M permiuible.

W"""'''''

1."

•"•

t•

"

• •.

.,

. .. . . . .,.. ".

b~

:rJ2

01.... ~ 01 'I1IenDal Neueroo. by ~ Kethoda

20 to 600 "0. Themeuured poiou follow a TU'·law quite ~tely, from which . . mar ooocludCI that the diUuaion ooofficient in graphite dON not depend appnci&b1y on the tem.per6twe.

17.2. Measurement 01 the Transport Mean Free Path in Poisoning EJperimenu 17.2.1. Prinelple of the Method

If th e diffusion length and the absorption cross section of a medium are known, ODO can wculate the diffuaion coefficient from th e rel.tion V _DIX. and thua immediately obtain th e tranaport mean free pa th J." c:3D. Now it is __y to m_we the diHuaion lengt h by the methoda of Sec. 17.1, but it i.I not poMible to determine abeolutel y th e .bttorption CnMM MICtion by stationary meth oda l • The poiaoning method oUe~ a way out of this difficulty. In the pwe modentot' we h. n I/V - r.JD. If we now in~ t he .beorption C " * eecuce by homogeneousl y mi.J.ing an .beorber of known .b8orption CfOlllI .ection with th e _ t.terer, . . have (17.2.1) It i8 _ umed here that the IC&Uering propert.iM of th e moderator .nd thua D do not chance upon addition of the e becebee. Th is condition iii .uroly fulfilled if . . tile a IItrobg . . . .bet like boroq eince thon vert aman amounu, wbich hardly affoct tho AVer&gCltoat.te ring ere- -=t1o n. druticaUy in~ the .beorption ertle& aocUon. If we now meuW"O L' at va m ue abeorbe r ooncontratioRl a nd plot IlL'· .e~u. 1:;. wo obtain a etraight line from whe- slopo we can determine l ID and from wboee int.o!'Copt. wo ean determine!'. (d . Fig . 17.2.1). Applioation of thia method preeupJlO&N th at we ean determine th o poillOning nry preciM!y. The lint prerequiaitoo ia th at t he added materiU be a atro ng 1/"·absorber with a very preeiaely known efOllll eectjcn. Wh en liquid modoraoon are u.od, it ia prefon.hle to add natural boron . for example in th o form of borie acid i the boron oontoont ia then determined pycnometricaUy or by titration . The eeeceeaended value of the absorption CfOllll eeeuen of natural boron (19.81% BIe) is 760.8± 1.9 bam at 2200 ml_ aceording to PR08DOOIKI and DlUttlTTr_a. Homogeneoua POl.onlna: ia not poeaible in .,lld modera tora like grapbJtoo. Th ere we mlllt poUoa heterogeneoualy with ....u- or fGila th at are uauaUy made of oopper [11. (2200m/Mo) -3.81±O.03bam]. The oopper thiolm_ abould be 80 I mall that no eelf-ahielding OOCW'S. In order to appt'O&(lb homogeneoua poiaoning .. cJe-ly .. po88ihle. the mutual diatanoe of th e wireI or foila muat be &mall; it .b.oold DOt. uoeod a tnnIport mean free path. Eq. (11.2.1) bolda under the . .um ption that th e . pect.rwn in the modorator ia not affected by the addition of th e abeorber aDd iii al....a,. a Muwell diatributioo witb the temperature of th e moderator. I n th e a beeDco of aouroea - which . . alwa,.. _ e bere - and in pw'CI 01' onl y Ilightly poi8oned modrn.ton. this Ia

.,

bow'.YW, Lbe ~ maUloda d..cribtd in Oapt.er 18. IDdnd ahaolut4 _ -'ioa "Mm...-mY _ M c&o- by utioDary (lOIIlparilioa IDIdIoda.

a~

d.

s.o.. lU .

eseumptlon is alwa yB justified. In the caae of st rong abeorption, however, t he diflu&ion heating effect discussed in Sec. 10.3 should occur. Th en instead. of Eq. (17.2.1), th e more general rela.tion

-1.. = (1;-+ ~)(I- ~ . I'.;~ ,0)

(17.2.2)

bolde (eI. Eq. (10.3.1ge )]. Here 0 is the diffu&ion cooling conllt&ntl. When the ab&orption is st rong, a downward curvature appears in the plot of IlL' · againBt Z: . In principle, it ebould therefore be pceslble to detennine D, 1:. , aM 0 from a poisoning experiment . With the exoepti on of th e experimentB of BURR and K OPPEL (Sec. 17.2.2), the determination of 0 in tbiB way hu hitherto been im potlIIible ; all othcr authors have striven to keep t he absorber concentration &0 8mall that t horo waa no dovill.tion from tho Maxwell spectrum. 17.2.2. Some Exp erlment.B on D.O, B.O, and Graphite The first poisoning oxperimentB were ca rried out in 1953 by K£.SH and WOODS o n ht-.avy flIl'Ikr. Uliog 0. cylind rical sigma pile, tbeee anthoN determined t he diff usion length in pure D.O a nd in boric acid l olutiane with canoent rat ionl up to 146.8mgB.0.lliter. Fi~ .17.2 .hhows lS , I lL' · M a fun ctionafI'; . Thovalue .• 1,, =3D =2.49 ±O.04 em follows "t. from th e elope of th e line I . Th e wa ter temperature wu 23 -c, tho DIQ concentration 99.4%. We 800 from ., /' the Btraight.lin e behavior of tbe 'Slmeasured points in Fig . 17.2.1 that f epecteel effect. can play no role; we ca n alao conclude the sam., ., t from Eq . (17.2.2) (with 0=6.25x l; l()I cm'sec -l ;cf. Boo. 18.1.3). E J:tra- ~ . 17.:t.1 I lL· · .... r:. ,.., __ ooIGUoDI lao 1\0 po!ation to 100 % D.O gives (_ ~ ...4WOON)

,

-:

t• •

,/ ' ,

•

-: ,

'"

1,. =2.62±O.04 em. BROWN and HCtUU:LLY hav., .tuWed th e temperature dependence 01 th e difflllion coofficient of D.O by the poiBoning method. In t hia 06Ifl, t he poisoning

wu with copJl6r wirN. Fig. 17.2.2 sho_ D _ ~'- in the temperature ra ni:e

""88

from 20 to 260 CC. Th e smooth 8U"e calculated a.eoording to the Bad· kowak y preacription (ef. Sec.I7.!.4) and reproduces t he meuured ...eluee lurprisingly wen. HnDRII: tJ aI. ha ve etudied the transport mean free path in gmphiU by heterogeneoua poisoning with copper l oila. The dillUBion length wu meuured in sigma pilea, which were eoeeteucted by alternating 26.4-mm·thick layen of I Aooording to Sao. 10 .3.3, C oont&ina .. contribution due fA) tl'a.nf,pori-thllOretio effeot.&; .. .. rule, howe.er. tb i8 ~ i8 ",,&Iigible compared to tbe contribution due to the .peotnl.mft. I The "'\ue E. tD.O).- (I .&6 ± O.I) X l o-' em- I 10UO.... hom the iDteroe~ Howe.... t./Ie m_l1nlment of tbe H.O oonoootntion In the D.O _ not .uHlciMtJy aooutaM to pennit any concludon .bont the .!:MIorption _ -mcm. of deuMrinm to be cb-wn.

ppbite with oopper loiJI. The oopper thiobell inoreued from 0 to 0.008". The den.ity of the psphite.M 1.876I/em·. The re.u1t of the me&llurementa WM

.t.-2.62::l:O.03 em at room tempen.~ure. ThiI.alue illlluoh hiaher than..u the nJQfII obta1Ded from. DOD-atat.lon&ry meuurementa (Chaptw 181. Bennl autbon (BmKtraTt &Dd K.!.tllln, Rmu. B~1n, Mu..u:R) hu e Itudied o.diDary -'er, BT.ua and Koppa, haYing ouried ou~ a particularly oareful uperimeut. Boron pWooiq .... ued throughout, 011 ooca.uon. in nOO high OOIUleIltnUon. that ,peotnJ effeot.e _re DOUOMb1e. Fig. 17.2.3 p o... ST.ua eed KoJ'!'Uo" . alUM of I lL" M • a fuDetioo of t.he Cl'OM eectioD of the

/

,

/

u

l;f

.:.

/

-

/ V

' " b "" • DJn._lInfI.

I

·· '"'"-' . ...,. --. .....,

• Nu' " "

. f-- -

• /tnf•• IT

&.

-

0"

• e-" -' ........, 0

,

11

""-j"'1 1

.,

-•I

.... " ..... 1 • . 1'. . . . . ....,.....,. n. ..._ .. ....

~

n

c

o

l

'

n

"

O~n.I_W_._"'a.nt'Iu _ __ _ .. ; &M. _II_

... . , . . . , . . . . . . 11

V-

/

, 'u

IJII"( 111

'lSI

Ii fOs a - ~

(I IrrJIMd _

,

- ~ ~_' IIIIIW

u

flcm .., lwol

Ion, ,,.,,., iii

/

..

V ,

V

.

.

11 __ "

E~)-

..... n ...... IIJ,·· • • .1;1.,) ... " 0 • • _ _ II, .......... Zorna., ...-... ..... ..al _ _~

_

..u

_

..., . . ... Ia B,O let. _

~ I• • "'"" ... ~'-u..

-""" ..1 _ thooIl.., _ _

11.1)

....

added boron. The diffuaion lenath meMurementl were ouried out in a oylindrical water taok (1M om. in diameter, 1150 em high) into whicb thermal neutron. from a reactor were introduced from below. The dOWDward curnture cauaed by diffu:ioD b_Una: • oINrly reoognizab1e. Leut-equ.arN e..luation by meane of Eq. (17.2.%) sine .t. -O.fU± O.OOl cm , C _2900 ±3rJOem' .eo-l (.t 21 -C>, UMi cr. (ItOOm/-) _328.9±1.8mbara. per proton.

17.3. DeiennlnatioD of the Absorptlon Croll Seetlon b1IDtocni Comparison Melho4t We _ c:aIoulate the abaotute nJoe cf the .haorpUon cr-a. aeotion of & medium from. the c1U:fuIIon 1eIlath and the tnnaport meu. free pe.t.b. We Dan determiDe it iDdepeodent.l1 by the method of pu1Md _troD MJUI"Cee (Bee. 18.1). In th.iII .ect1oo we .hall become familiar with lIOIIIe prooeduree which make it poMible to relate to one another lobe ablorption en:. aeotiozw of . .riOUInbltanllell

(not neoeuariI.yonly moderator IUbRuoel). By tiling It&Ddarda we C&D then aJ-o obtain the ab.olute valuea of the abeorptloD oroM aeotiona . Buoh method. ha ve many adnntagea oompared to meuW'tl menta with a algma pile linoe we eeed far _ materiAl and can frequently oarry the meuurementa ont muo h more .unply and quiokly. FurthermOl'l. oomp-.ri.lon metboda. partioularly the pile ~tor method. offer nearly the only pc-ibilityl of meuuring the abeorptioD. CI'l:* 8ectionJ of I Ubeta.noel for which the coDditiooa for an abaorption mea· luremeat fla the dilluaion length (0'. < 0'• • good moderation prope:rt1el) or ria • tranamiaeion experiment (0'' > l7. ) are not fulfilled.

11.3.1.The Method or Intesnted NentroD. nux Let a 101U'Oe thlt emiu Q neutroDS per eeoond be Iooated in I medium th.t ia 10 large that for practiceJ purpollCl no neutrone eeoape. Then.moe all the Dlutrona are abeorbed in the medium. (17.3 .1)

when for Ilimplioity ...e IhI1l at lint ignore . peetral effectl. A CIOI'I'eIponding relation holda ...hen the aa.me IOUrOI iI Iooated in aoother medium with the abeorption croea ItlCtio n L:. Then

.z; l~ d V 1:-; """ i7p-jv"

(17.3.2)

Thua we e&n relate the ahaorption croee eections ,o f varioUi sublltanOll to one another by oompa.ring the Dux integrala NOWld th e ..me 1OW'Ce. The flux een be m....ceed with foile. and relat.il'e meuuremenu obvioualy .uffioe. Under 10100 cirownJtanoea. it it nlOee8&lY to take in to acoount the fact that tho foil oorreetion ill differont in different media. The claaiul a pplication of tm. method ill th e m....urelDent of tho ratio of the abeorption of boron to that of hydrogen (d . e .• .• WKrnJlova. and OllAJUM. H.....l..llMUR. Rmoo and W.XUIt. B.I.1I:.a and WILKUlION) : Tho flux inklgnl t.. mea-urod lround th e IOUJ'OO onoe in pm-e waUlr and a aooond time in a borio acid IOlution. Then' E., -N."•. L:,-N l7. +NII O"II ....

th~

a

Nil

-Ni

Nil till + NBai -

J~dV

T(If'lu i .

(17.3.3)

Sinoe N• • Nj,. and N. are known a.ocu.nt.ely. O'.I"B folIo.... immediately from oompui80D of the flnx integnla. 'The aecuracy of the method can be inereued hy making th e meuuremenu at ,.ariolll al»orber oonoent.ratioDa. III principle '"' 0f0D determine the abeorption croee eoctiona of many labetanoea tbia .... y. but in oomparillon with the pw-l neatron method (d. Boo. 18.1.6) thia method t.. ratbll' complicated and ie therefore hardly used any mortl. Ono ean alao UIO the method of integrated neutron flux to determine the abeorption croee eection of an ememoly wealdy abeorbing lubBanoe lilr.o gr&pllite or beryllium. To do 10 one mUit modify it Ilfa;htly linoe tho requirement of MgUgibl1 am.allleakago from thft teat body woWd IMod to abnrdl11arKe loIDount. 1 ~ IIII\hoda,. .hiah - . bo........ limi 8eo. U aDd 1"1) aDd p1IIed _a.- mp p p T1le .t.orptioa. of o s:yptl __ be ~.

ill \heir IP~". 1ft ..,ul'Woa (el. IIItx- (fl. 8eo.18.1.!).

of material. Fig . 17.3.1 .hoWl a pceelble arrangement, The tetJt body (L; ) ill IUrruundod by '" I't!floutor of tho oom pll.n.on Oluu..ll/olil!l.l ( l.~ ) : n " ..trongly " IlolNforb· ins eubetence like paraffin lIel'VeIJ as the comparison substance and can easily be made thick enough to preclude any appreciable neutron leakage . The com. parison meaturement is earried out in a suHil'liently large eemple of t he comparieon lubstanoo; Eq. (17.3.1) again holds fodt . On the other hand, for th e me&llurement on the tMt body and reflector we have

Q-E; and we 6&llily obtain

J

dY +E.

J

dY

(17.3.4)

""'"" lot

\M Iloo17

J

I

~ dY ~dY """,puiooa ... nee "", P ...= L'.

""- ,

_ ...:: ....

(17.3.5)

h&ll done meeeuremcnte on graphite in thill way , and B ROSE has done meuurementll on aluminum . It turns out that cue can nchieve edeq uute prec ision when th o Wilt bod y haalinonr dt , moneicna of about two diffusion lengths. Paraffin was used as the compari80n eubetence. The flux integ ration in th e teat -bodyTtl ! /TIIltI'!" reflector esee mbly ill tedious if the teat eubetence ill not in th e form of a sphere. One mUlltdetermine the flux at many points and sue. eCll8ively integrate ove r e, y, and ::1. On th8 other hand, in a single. auffieient ly large medium, flux Fla. U .s.!. A.. _ hl, "" u.............alol' u...bootpUo.. meeeurement along one radiull _ _Iloa br ......""""- laie'pated "llIlroa .. vector suffices; thereafter 4" x J l1J (r )'" dr ill calculated. Speetral effect. play only a &mall role in the method of integrated neutron OWl:. If the Cl'Ol8 sectiOIl8 of the oompa rilon and te.t l ubata nc(lll both have II,,· behanof and if the activation erol8 section of the detector . ub.ta nce folio... th e 1/11.1& w, the neutron spectrum d088 not enter at all : thus we obtain L; (2200 m/aee ) if we .tart with E. (2200 m/sec). If there are resonances in t he epithermal or fa8t neutron range. the integrals of just th e th ermal Iluxes ere determined by cedmium difference meaauremenu and abeorption above the cad mium cut.off oncrgy iI taken into account by introdul'ling II reson ance escape probability p Id. Sec. 7.2.3). 17.3.2. Tbe Mirollle PUe Metbod B OOKHOFJ'

n a~

Tb eMireillemetbod W&lfiretused by RAlEVSKI and later refined by RE IClURDT. It alIOWlJ rapid m6&lUf'flmcnt of tb e abllorption erose aection of small quantities of weakly absorbing lJUb8tanCCl like graphite and beryllium and is partiouiarly well suited to routine eceeeuremeete (indust rial purity testing). YJ8 . 17.3.2 shows I Some metllod. for .uch int.llpatioM __ diacUMtd in the fint edition or til . book. p. %lJ1H.

a typical &MOmbly for arllophite meuurement.. Near the two ~df&ON of .. 100 X 100 x 2&l·cm prillm of l'OIDpl'riaIon Kra phJte (dUIudon lonKth L ) aro looat«l 110 (Ra - Be) aouroe and a BF. eounte e. The oomparison gr..phit.e can be replaced by the teet graphite being8tudied (difluaion length L ') in a volume V. II Z is the counting rate when compari&on graphite ;. in Y, then aooording to perturbaUop th8Ol}', Z ', tho counting rate alter iNertian 01 the te.t graphite , is gi"en by V .z'-z - , -- __ .4. (7Ji1) .

(17.3.6&)

The con,ta nt .4. can bo doWrminOO by calculation or by nonnaliu.t.ion meuure· menta on different kinda of graphite. Eq. (17.3.6 .., then make. the detennin..Uon of P IL" and t hu EJE. poMible. We ahall conte nt ounelnoe hero with lion elementary des-ivation 01 F.q. (17 .3.6a) u edcr elm pllliod conditaoflll; 110 more eoeurete calc ulation ca n be found. in R IU· CI LU WT. Let U8 _ woe that a point source of neutroM 01 , 'trength Q is l oca ted IIot r . IIond / emit. purely thermal neutrons. ;,1 Th en with compuiBon graphite in Y, the th ennlol flux obey. the --- . ~ J1I:. n.1.t. • •u.m. pDo blJ&Plllto .-tI, _ equation

71

V

V'11l (r)-

yI

l1l (r )=

Q - 1f 6 (r - r . l .

(17.3.6 h)

II we deno te the diflUllion kernel for a point eouree in a linite pile by G(r., r ), then (17.3.6 c) l1l (r ) = Q.G(r . , r l is th e IOIuti on of Eq. (17.3.6 b). In particular, t he flux l1l(rl ) at th e point r l at which the counte r ia loceted ia given by QG (r o' r, ). Z ;' proportional to tm quanUty. II the tNt .ubatanee la in the volume V, the n we have V"11l'(r )- ~
r- I1l'(r)-L~1

- -% 6(r - r . )

4)' (r) ",, 0

ou'tatde Y ,

(17.3.7a)

iMido V .

(17.3.7bl

Setting
V1 dl1l(r l- V 611l (r ) =0 V'd \tl(r) -

~ 6 11l(r) = -( ~ -

-i'I) (\tl (r )+ 6 11l (r»)

outaide Y,

(17.3.7c)

inside V.

(17.3.7d)

On the righ t-hand aide of Eq . (l7.3.7d) Itand the cbaraet.erUti c IIODl'CllI of tb e perturbation 6 11l (r ), which eDIt beeeuee of th e diHerence in L aDd L'. When L a nd L' only diller little, 611l(r )< \tl(r) a nd we can neglect 6t/J eompeeed to 111

011 t.be rtsht-hNId 8de of Eq . (11..3.7d) . Fmthermon,,,e 0lI.D oombiDe Eqa. (17.3.7 c aod d ) iDto 1 V'l6~(r) - £I6"'(r)- - 1J - 7Ji lJ)(r)6 ( Y} (l7.3.le)

(I I )

...hen 6(y} -1 m.ide Yaod 0 othenriM. The M)}u.U on of Eq. (17.3.7.) U. 6 tP(r) =D

(-b -j..) J/fI(r')G(r', r ) flr'.

(17.3.Sa)

•

G(r'. r) • agam the diffuaion kernel for the pile i the lOuroe point r' now lies insid e the ,.olamo V. It foUoWl from Eq . (l'U.80) th.t

z'-z .(r" - _(V -,-II " {T,l

7Ji

_I}. J>./O(T••f")O(r'. ..,) ~J 1)

Ot.... rJ

UT.

• Tbta Eq . (1'703.8) hall been deri..ed aDd the (lI)lWt&nt A determined.

•

(17 ' 8 . ) ,

•

In order to oarry out the int.egr&tioD the diHlWon keme1 of t lul pile lIllu t be known i it can be obtained from Eqa . (6.2.%1) v ADd (11.2.25) . In aD . not o.Jcul• • tioa we muat take into aooount the faot. that the IOUlOe emit. nOD' u thermal DllUUoruJ. However, if tb e d..Ytanoe betwee n the eource and the telt volum e ia lulficientJy larg e eo m pared to the d o'lring-do1m Jeogtb, we obtain the Mme l"lllIult .. for . tben:n&J aowoo. Fig . 17.3.3 oM Utr • lobo... A lor . olamM Yof nriou. me. in the pi1eabo wnin Fig. 17.3.2. na....... n.LL n. ~ • • ,..,.... 1IlInla. ..... ' • • 1_ . . . _ ...... _ w. _ that . "en 101' emeU t-t. 1'Oh UDeI good IIlInaiti'tit1 ean be achiev ed (e.g., A - 0.1 for Y... I30 lit.ar). Sinoe lVe OR detennineZ t.ndZ' with a proeoWon of.bout 0.1%, ..4. ""' 0.1 meanatbatdillertlnoee bet ween L and. L' of abo ut 0 .6 % eee be detected. However, additional efTOri are int roduced.by material inhomogeneitieB (de nait y fiuctuationl, .ni80tropy eHect.),.nd in praotioe LIL' ia rarely determined to better than 2 %. Fluctuations in th e graphite temperature are an importantlO1lr'Oeof error, but they CAn be eliminated by putting the pileina te mper• • ture-eontroUed room . It i.I . 180 wortb while to cover th e piIe.urfaco with e&dmi.um in OC'der that chana- ia the b.o~ttering eo ncUtioDll not a ffeet the OOUDting rate. The pr'O(*lve in the fonD de..toptd bere Ie 0011 . ultabla for' the oompr.ri.ton of .ut.taDoee with IimilK -ueriDg propertiN (i.e ., for the OODlpuiMtn of dif· f _ t MlDpte. of ppbit.e in apphite pile or different aampl eB of beryllium in • beryllium pile, et.o.l. When there .. a large dilleren ee between L a nd L', limple fint..orderpertlll'be.tioa tbeory. which lNd. to Eq . (17.3.e . ), iaDO lonrr appli uble. And we mUit then introduce tenn.I of higher order. Tbe evaluat ioo it then more difficult and If* aceun.te .

-:

"

,/

•

.....--

• ,- •

17,3.3. Th.

ru. Olelllator

Tbere are two kiDlh of pile oeoillator uperimenta, th e local kind and the i~ kind. The local kind .. bued on obeerring the flux de~n neal' An . heorbina: aampM in .. oon.maItiplJing medium. A nuclflU' reactor generally

aN'TM to provide the DOUtton field. although t.hiII ia DOt ~y a p!'W'eqv.Wt.e of the method . OD the other baud, the integn1 method it bued on the effect of &Q at.orber on tbe reactivity of a 1'eMtor; for ita preciae uodentaDdiDg a detailed kDowledg. of ~ theory ill DOOeM&I'J. &ad we _hall limit ouneJ.VM b_ to a diacuaaion of the fund&mentala. FiB. 17.3.4 , bowe achematiOlo1.ly a locnl pile OlIci1lator. In the graphite reflector of a r&&ctot ill 1000ted an annolarionhation chamber which hu been madelOn.llitin to neutrone by boron cceeing. Uaing a euitable me chaDical device, we can make a emaIl aample of tb e tNt ,ubetance move baek and forth through the inte rior cavity of the cha mber. Th e frequ ency of th1J motion ia about 1 cycleJeeo. Owing to the _ ttering an d abeorption of neutrona by the tNt IUl»tanoe. a periodic Iip&l b produced which is IlUper. Wtiz. . ~ fr i.m. ~ on the ateady-lIt&te OW'TeDt of the eham bee . Thb IignaJ oan be . pa.rateel &ad amplified by a 1811· litin amplifier and b ulti· mately recorded. Fig. 17.3.6lho... the typioal time behavior of . ueh a ala:ualover a foll period of OICillation. Th e aignal in _ G ia produced by a eadmium Mmple (pW"ll ablorption) while that in 0&10 b ill produced by a graphite ...mple (pure autte ring). Th e zero time-point correaponde to the inner turning point of th e oecillatory motion. There are two IignalI each time moe the cha mber ill travened twice during a period. The abeorption lignal ill negative. On the othe r hand. the ICatt.erinl signal ill positive ; it aleo h.. a somewhat different fonn than the IbIorpUon Iignal and ia ahifted somewhat in time. The IlC&tt.ering aignal comea mai.nly from thON neutroM which ItrUm through the canal and. in the abeenoe of the tNt _pie ..ouJd PM' ria;ht throuah the hole in the ioniution chamber. Th e Iignal forma in Fig. 17..3.lS are idealiMd, &ad Fig. 11..3.8 abo... eignal fonu .. they are actu ally cbeeesed, The diltortioDl are du e to the freqUeDoy eharacteriat.lOl of the amplifier. which ia giYflrll. the amalleat poIIi ble bandwidth in order not to ampWl the background noiM. ptorticnl&rly t hat d ue to IIt&tiatieal Ouctnatiou in the chamber ourTeDt. AI Fig. 17.3.6 ~ diatortiolUl h.... e the 000' venient effect that a time internl .11 oan be found during ..hieh the "Mage IC*ttering eignalvaniahea. We therefore coDDOCt the output of th e amplifier to a C\lIT'8Dt Integrator that 11 only 1eDlitl.T8 darln8 the time Interval .11. In thiI _ y. the _ttering IIignal can be largelyelin:rlnated. In prac:tioe. a IOIlaitivity ratio of 600 ba. been a.chiend by IUitable choice of the integratioo range .11. Tbt 11 to ..y. the oeaillatioll of a IOItterv with • ginn "aoatterina: .urface "

= -----"" -..,

Iho_.

(defined .. t he product of th e total number of aro ms and the eeomte CI'06lJ section NYa. =YE.) give. about a 600timea smaller effect than the oscillation of an absorber with an equalab&orption eurface Y E•. The ..bBorption signal is proportional to the absorption surface YE. (unless the 88mple is 80 large that seU'lIhielding occurs). We meeeure the absorption eeoeeeection by comparing ite absorption surface with that of a sta ndard substance (gold, boron). Th o procedure is very sensitive ; th e detootion limit in a good pile oscillator is about 0.1 mm'in Y E• . (Smaller Bignalscannot be distinguished from the fluctuations in the chamber curre nt caused by fluctuations in the reactor I I I power.) Thus, for example. we only need about 20 g of aluminum (0'. = 0.24 barn) 1 I I

,""<

1\

-J---tF=- \ I . •

I . I

• Aklrlfi(m l(tI/miIJmI

I I I I

~4/

,I

-l4t I--

II S:I" lf'iIIt (' ''plli(,1

II Stl HlfinJ (: rr; M/ 1 Fli:. 17.'-&.1_1.... oIpa1,."... hI ..._

pIloOllOlI"""

fla, 17..... AoWoJ....... 1orn.. ,... 1oo&l pl io """'I"",.

in order to determine th e abaorption CI'06lJ eecuon to an accuracy of 1%. With mort! IItrongly absorbing lublltanoea (<1. > 1 bam) th o effect of scatte ring can be oompletely neglected beceaee of the lUDall _ttering sensitivity of the method. When the ab80rption is weaker . on th o other hand. we must apply a correctio n that can be obtained with th e help of the _ttering IICnaitivity determined with a lltandard IlCatterer (gra phite , D,O ) and th e known IlCatte ring Cr068 eecnon. We obvioualy cannot determine the absorption of extremely weakly absorbing subetanoee like graphite, D,O. and beryllium very precisely with a loeal pile oecillator etnce even with a aenaitivity ratio of 500 the contribution of the scat . tering to the totaillit!:nal domina tes that of th e absorption. P OMEUN CE, HOOVER tJ aI ., SIIl.u.L and SPURWAY. and FuKET.... among othen, have described experimente with local pile oscillaton. FuKETA towered th e detection limit of th e method considerably (to E. Y = 10 -· mm') by Iargtlly eliminating the effoct of fluctuations in the eeectce power. TIlle Wal achieved by using two ionization chambers and reoordin" the difference of their outputa. In an i nkgrtd pile OOICiIIMtor, th e telIt 1Illllll'Io i~ poriodieally moved in ant! out of th e coee of a reactor. The reactor is run at a constant average power ; this average power is kept very low in ceder to av oid thermal effecta. The reactor power then nhibit. chan.cteristio oaciIlationa eeound ita average valu e with the frequency of the test aample·. motion . The amplitude of th ese OlICillatiOIUl is proportional to the change in the multiplication factor eaUled by the aampl e. The IAmple llffeota the multiplication factor in VariOUI ways : Themal and

~pler

11: Ref_

38'

epit herma l DeUt.rone are abeoc-bed. NeuUoDI are _ ttered and th e leakage i. th ereby aHeet.ed. TNt I Ubet&nOM like p phite aDd beryllium contribute to the moderation . A complete anaJyNi aDd..puatioD of the indiridnaJ eHecta NquUdet&iled theoret ical and uperimentaJ work . For eumple, we can determin e th ermal and epit hermal abeorption ..puat.ely by making mlUW'llmenti wit h va rious reactor l pectB or by using the cadm.i um diffe re nce method. Neut ron etreaming effcetl can be kept Imall by oacillating t he teet body between poin tl with vanishing flux gradie nt. (e.g., from outsi de the reactor , where the Dux va nishes, to t he middle of the core, when it h.u a. muimum). We ahrJ1 not go into the det.a.ile of this meth od here ; more information on th e method of the integral pile oecillator can be found in WJW(BJ:RO and 8cRwJW(LI:R, in L.UIIO&00 .... in BaJm>N, and in Ro n , Coon a, ud TUT&RU.u.. The limit of detection in the m...urement of t h«mal at-orption CIl"OM _tionl t. aimilar to t hat of the local pile o.cillator, lIowever, the effect of _tterina Oan be more eUecti't'ely ..pa.rated, 110 that me..lmlmenu 0 0 p phite and berylliulD are poeai ble . ReIonance abeorption integra_ and 'l'"valUeil (01 filw,jonable lubetaocea) can alao be determined (d . Ro n, Cool'lla, and T....tTJ:R8ALL). Th ere are a number of reacton which were exprellll1y built for pile oecillato r meuuremenu and have l poeial facilitiea for tb em [e.g., OLICI:P, MnfICRv.). Wit h th e integral method, oaci!lat ion of t ho aa.mple i.e in principle not neoeuary ; the cha ngo in t he multiplicati on constant du o to the introduction of th e aa.mple can be meeaured otberwi8e, e.g., by compensation with a calibrated ehim rod. ThiI ltatiC motbod, tho so-een ed dangor coefficient meth od, ..... formerly used very ofte n. Ho. ever, it ia much t - lOIllIit.iyo than th e oecillation met hod on aooount of Iong.te rm drifts in the l'NOtor poWOI'. 111_ drift.l are ca U«d by te mpera t.ure and air,p~ure ef feota that largely ca.ncel out in t he oacillator mea.suremenu but. that. limit the a.ccun.c:y 01 ltatie meuuremenu of th e multiplication c:onat&D.t.

nil""".'

Chapter 17: Referentet General T "KrLIlC. L . J . (ed.) : Reactor PhY" l", Conttaot.. ANL-MOO. Sooond Edition ( 1963); NpeoiaUy

&cLiun 3.3: 1'hef1ll&1·Group Difl ll.iun Pwa1Pll~. eoall(J()U), N. (ed .) : ~np cd \he Brookha ftll Confere.- on Neutron, TbenDali&aUon, BNL-111 (1882) : NpocirJIy V« ume Ill : E z peri_taIoUpeet.t of'r'ruw>eotand At)'lll ptotie

............

Speelal1

c.:

B... uo.So W . BSL-1 11, m ( 1M2). Bd&Ov,L., V.K. L1aD:, aDd K. N.IIVJ:lmf: J . NMil.ED.w:1 to IN. (186'7). BIt(X~ It. H., aDd O. xce...: Z. NMat'foncbllll£ l b . 8%2 (1868). H ..nn d , L. R . : N\lCIrIoniao u., No. a, lOll (1866).

Ju

J . A. 11......1M. U,.101 "". .. : J .lb . N.L B Ul . SlAnd. 51, 203 (1(1111 ). M.• • ,,0.1 J . A . Ju : J . Nuol. !tr""lO" A If. III (loo l). Roc1r.u , K . S •• and W. S IlOLJl lCIl : Nue!. Sol. Eng . 8, 60 (1860).

H

u.

Rollll, 0. : Unpubli. hed K..rl.t.ruhe report (1962). S18It, F. J .: ORNL-Il33 ( l~ ll. ST...... E., and J . U. Ko,nt.: BNL.l1l, IOlt (1162). W WlOII. V. c., Eo W . BIU.OOOII, and H . x.........: cp.t3OIS (1\loU). WIUOIn'. W. B.. and R. T. FMn: KAPL-lI.WBW t (11l5ll). I Cf. footnot.e oD p. 63 .

M....llIe_ll t of ' " DiffiWoo. Loog
~

R. W.: NIJCl. 8ai. Ezlt;. I.W (1llr6e).

~.B.I,,""' I1"tvp'.N"'EawvS..I.

Vol I . P. 2:07 (1_)Eaca, 1..1. : N-.aL 8ai. Eq. 1&. 1" (lM3). hnD. Co D L.1bouI. aod P.". Zwanr.: NDCll. Sci. Eac 718 IIM7). RAlta OWQY, A.I Al'iL-U7'" (IteiO). ~.. L.: ArW... P;rMk " S3G (111M). EaT06'CJJ). A.." • . 1 R.JIPOIi CEA.J (IHB). H~uE, J . II., d 111.: PJogr. Nllol. EJleI'lD' Ser. r. Vol. S. p. 270 (l~). u.o \be d1p101p thMUI of H. WDlULD, OOtUnaeo (1862), K. H. B.:snn., GoWngen (I BN ). W. RmmuADT. Kariaruhe (11160), H . XLo... KarImIhI( It62}. a...-dDw H . 0 ., 401., c.n.d.. J . Be.. A 15 11lH7), w auftDll. R. A.: 0-.. 1868. P/SH. Volt!. p. e8lil• Lu, IL T. : BW-611'l'5 (J867). JtK:.::rr. c.1L, aDCI E. Z. BlDcI; : HW-U0305 (1llr6e).

a.

u..

LLoYD. a,c, Eo D.Cu.no., &ad. C. B.

8It::qy:

I

DiU....iou.l.en&th X-uremeDti OD. Graphite with~

........

Sigm&-PIle

T_perMaIe Defll'Gd- of the

EA,." elIO ( 1868). . . . Knu. J. E. c.: AEAE-RP/R-I Otl (1t65). Diffua>oa. I..eoctJa ia Onophita. B.6'U, L.. I. ~ Ul4 1..1. PLt.: ae-. lNa. PfI7JO' \ Vol. II, p. 6Oll. N IJCl. Sci.

BaoWll. W. W.o If al I Genen laM, P/~. Vol. 12, Diffu8ion Length Heuuremetl.w p. 614. la Varioul HydrostInoua r..o.. 1.. J .: Muol. 8ci. Ena;. l ' , IN {1ll63}. liIoden.ton. HEll'TU, L. R.l NlJOitIoni....14, No. 5, P. 108 (196e). 1'n'ru. W. : Pb,-. Ret' . SO,7M (Iaro). B ..... d ! ) . H. 0 ., d-l.: Can. J . Ree. A H ( I N 7}. Lvn. H . R... and R. W. )fsum: NU1eoNk f., 108 (1M2). Si. -Pi» Ueu\1ftlment of the )(DITD, H. : UIlp.blDed ICarianIbe nJ'llri (I MO). Difbioa 1.eo(t.hill D.O. SaoD"f, B. W.. .... : Can. J. Rea..& tl,IM (1&6'1).

c.

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u..

GuulT....L.A .. da:l. : ae....... laM P/M%,Vola, Poll.! Ko.sLul', J. e.. J.I4uTJu.T, and V. P. Dooou.: 8Ipl&-PIle Keuwe_\ ~ the 0-..... 18M, Pj36a, Vola, p. JO. Diffaaioa IAD(t.b. in Be aDd BeO. Noaua, R., aDd J. WAl,UCIB ANlAO'lll, 10 ( IM 1).

J' l

AtroD, P ., .&.II.II\1IOI',....s B. 1'o"lOOa"o : Cao. ft.. A &. lU (I t "'). H ..-DAla, J . )f.• d 1IIl.: PJosr. Nllol. Entqy Ser I, Vol. 3, p.!TO ( 1060).

)Iou

of btl

Er. \lftI~~; Eo:. I trapo..... po n\ of TbermIo1 N.utro....

HOIla, D. W. : J . Naa!. Energy A II, S.fo (lUi). Sua, F. J .: ORNL-hS (1061). Born, W.. aDd Eo h :a: Z. PIl,.• • !!, 'Jet (IOU).! Bonm, w.. and P. 1 1 Z. PIl,.. Itt, 74a (IOU). DifflIIioo. I-rth W-~U by Frri. J . G.l n-i&, 06ttiD&- laM. KeaDa 01. SlIIUoI ~ ~ H.: DI,..... n-ie. ~ 1866. LuDwa, and w .R..Ko.o : TnnI. .&m.NI>Ol. 800. 1, 281 11"1). B_ftT80 K. R.. IMld O.lWl : Z. M&twfonob-. II .. m (1068). BiIOW1J, H . D.. aM. Eo I. HDJlIU.Y: BNL-7I a, an 11M2). DsnD, A. a., ANW168, 14 (1061). 1blrDaa, J .II., d"' ; Prosr. NlaOI. ED.a:r 8er. I, Vol. S. p. Z'7O (1m ). x.u.. 8. W.. aDd D. C. WOODe: Pb,.. a..... to. lI04 ( INS). JriI..tz.La., J . ; 1ftM. ,Ap). Nad. Soc. 4, 282 ( lNl). R au, X. : J . Naa!. Enqy A 14, 188 (INI). E., Md J. U. K orru.: B~71a. 1011(1M2). PaoIl'!!OCQC\ A., MId A. I. navnonal J . Nuol. EDqy A. AB 17, 83 (1M3) ('I'beAt.orpdoc ~ s.ao.. 01 Botort.).

w.e..

er......

B...u.:a. A. R ., aDd D. H. WILKDIO. , Pb.iL x.,. s.. VU .. N T (18G8). B&:luo n. K. S . : Diplom6 '1'bNia. GOuiDpI. 1M&. B.oea, H.: Ull.publlahed Kart.uhe Reporl, IlMlO. Baosa, M., and K . H. Bsonau: Nllkleonik I, IH (IMO). ~ .....s. B., O. R.. Itnfoo. and 8. Wu ua: Pb)'ll. Rev. 10, eoa (1963). Wmn, K .. and K. H . BM:Il1rJlTll :

m.m.o....... NOlitroDeDph)'llik.

06ttinrn-Helde1ber; : 8~ tw. WannoVla, W.J.. aDd O. 4 . GUIUIC: &1aYut. V., dal..:

~

Badill'

J. Ree. A U. %81 (IH 7)} ...........

o-•• I~ P/J ~ VoI.IJ, p.06 .

R ~ . W.: DipbDa TbMw. KaNn.he lIMO. I"nft.. T.: NoeL b&w. Xotboda . 1, S6 ( I ~ I). HOOTU. J . I.. d el. : Pb,... RoY. ,(, 8M (t wa).

Powuua. H. : Ph,... ReT. sa. &U (l iSl). 8luu.. V. O•• 1Nld A. H . 8PvBwu : AERE RPfR 1438 (1966).

I

l-a po. C/lM:iOatar •

Buro_, D. : Genna 19M Pj366. Vol. 4, P. 12'J. } L:1l'OIDOQ'. A. I AECD 3194. (19151). In tegra! Pile Rosa, B. t W.A. Coor... and R. B. T...ftKllU1L : QeDeva 1968 P/ t4. o.ci1Iat« Vol. JO,p. U . . Wano.uo, A. K., and H. C. BoIrwaIJfLU: Ph,.. R..... 74, 86l li Me}.

18. Inv es tigation or the Diffusion, Absorption, and Thermalization oCNeutrons by Non-Sta tiona ry Methods During the lut ten yean non.at&tion&ry methods have gained lncte&ling llignifieanoe in neutron pbylic.. ThB i8 because they freq uently permit cleaner and more aoeurate meaaurem ent. than at&tionary method. and bec&UMI tb ey aJ.o gi...e much dee per inaight into the physieIJ nature 01 the probleml. 'I'hW eonelw on iI tllIpecially tnJ e Irx the method of puI-! 1OUf'CIN, whOfle great potenti&li. tiN "'eI'e first recognized by VOK DanKL .Dd ...hieh hal aince become. Itandr.rd method. We IhaI.I become more lamili&r with it in 8ecI. 18.1 .nd 18.3. ao.eJy related to the puleed-eouree method but leu W1ual iI tha eeeebcd of modulated lOuroeI introduced by R unalU and HOROW1TZ ; it is diaeuaeed in Sec. 18.2. W. once again limit ounelvtlll to the dellCription of nporimenta on nonmultiplying media. Bot h methodl men tioned above, putioularly the method of paleed lOuroeI, have a 1ride rang e of appUeation in the Rudy of aubcritieal

....-.

18.1. Measurem entl in Thermal Neutro n Fields by the Pulted·So1ll'ee Method Fig . IS.1.1 aho...a .chema t1eaUy the Ipparatul for a pu1Ml.eouree eJ: periment . To begin with, a pu!Bed lOuroe emit. a ahort (compared to the time duration of th e phenomenon being atuwed) bllfSt of neutrons into the.ubat&nOlbeing.tudied . The time beh• ..-ior of the neutron flw: iI follo...ed with deteetol'll in the medium ee on Ita .udaoe and reoorded by a multichannel analyzer. Wh en the InteMity hal d_pd lUffieiently.another neutron. pWae. in jected and th e enUre prooedure repeated.. After ea.eh the neutronl are tint. 110nd down (lOme pulled I1owinl-down uperiment. ha.. &1ready been diIcueIed in 810.18.3) and ~ therma1izedi ulUmlt.I, we are left, with a tM maliled neutron field t hat deoiya

pw.e.

by 1eakage and abeorp Uon aooording to th e .... 01 dillueion t hoory. Th o th flJOry 01 thil decay h.. alrMdy been ~ in detail in Seea. 9.2. and 10.3; here we only need tho aingle reewt that after t ho decay of all higher t pati al and energetic mOOea, tho field decaya aooording to t he law I1'(,)_, - CI. wbere (18.1.1)

Th e I oU of a pulled neutron experiment in a therma liud field. ill the determination of the 1l (B')-curv e aDd ita anaIyaia to determine the dilfueion pllnmeten of t he medium.

--.. .. 1I·. . . 111f1t

" '.1 &.1.1. TM

If

II e,Clti _ liHtIc . . . , VI Il.ntVU &D4 I U.. .. lor peW .... l.n>lI upert_to

~

18.1.1. Inltnlmontatlon or a Pubed Noutron Ezperlmont We reetriet the Weeullllion here to the mOAt ne0eM4ry inltrumentatioD ; more detailed information on tbe apparatul UIfJd in pulsed neutron uperimentl can be found in vo~ DnDaL and &&TlUl4D and in BII:OKIJIl.TS . Today. the moet frequently used Mwnm _rcu ani .w.aU, flexible deuteron ~Iefl.torl with 't'oltagea of from 160 to .wo keY that are operated with tritium targeta. The pulae length mun be nr:lable in the rangtl lrom 10lL8fI(I to I I1ll1f1C for U8e in thennallyatemt ; the duty ratio i8 in th e rarlle from 1:20 to 1:100. With a pulle CWTfInt of about 1 mamp. Le., an anrar current of from JO to 60 Ilamp-. quite adequate inteoaity can be obtained from a lreah tritium target . Ar1 ummely hiah eignaI-to-baoksTound ratio i8 important. TbUl the CUI'nlnt during th e pw.. .hould. be very large compa red to th o current In tho interval. betwoon pulaea
in TbenuJ. Neul.rQu. FieJda by tbe PvJ-l·Souroe Xethod

ll~w

S86

300 OC &re oommerci&Uy AYwble; at higher temperatUI'M, ooo1ing appllt'atu mlWt be pI'O'rided.. It ill importaDt that the detector aod iw .-ociated circnitry hu e a small dead time that ill aIao accurately mown. The amplifiec mtwt be able to handle reliably th e high coonUng rate that oocun immediately after the neutron puIae. The ",1dtitiall ul ti_ aw pr ehoald ha n about 60 to 100 ch.nnela, wh-e widtha are varia ble in t he range from I foL lOO to I meec. The aIlalperi developed for tim e-of.flight meaaurementa are usu&1ly not v~ euitable bec.UlIe they have too large a dead time. There &re .. number of efrouita that hav e been built e:I. preul,y for pulaed neutro n experim enta (VOH D,UDIU" OU88, O....'JTI). Very good inetrumenta ani aIao oommercially ava.ilablo. In puleed neutron e:lperimenta, neutron ~ttering un produce • disturbing backgro und. I t ill therefore ~ to take eve that .. little Mck. llC&tte ring .. po.ible occun. FOI' thia reaaon , a pn1aed neutron facility ebould not be housed in too amaIl a room. The room temperature mlWt be cartlfuUy monitored and perhape enn held CODlt&nt.

'1''''

18.U!. MeulU'tlment of the II (lJI ).Cane

In order to determine the « (BI)-curve. we must meaal1rtl th e decay constant « for auembliee of various aiua and tben caloulate the geometrical buokling IJI for theee asaembli .,.. Some formulaa for the caloulat.lon of B' weregiven in Sec.9.2.1. We uaually UlIe t he value 4 = 0.71 0\, fOl' th e extrapolated endpoint . Ho...evu, in .....ter and othe r hydrogenous moderaton, 4 depend. on t he me of the ueembly bec&u8e of the atrong energy dependeoee of th e eeattering et't* .eetion. In thiII cue, it is better to ue t he reaulte of Ou.Jl nD (Sec. 10.3.4). II ill tb ee determined from the decay etlfTe. Thia determination ill oomplioated hy the fa.ot that the decay curve ill purely expo nential only onr a limited time interval (Fig . 18.1.2). FOI' abort timee the effect of the higher l patial harm o.ue. and oDergy' tra.Wenta ie appreciable , while fOl' long tim .,. the b.ckground contributee aignificantly to the flux. The infiuenoe of the 11&DV1' tranaienta can be eliminated by waiting long enough after the boginning of the experiment for the neutron 601d to beoome oortain delay time '- to elaPM completely thermalhed. We must thu aIIo bew eee the -a uroe put. and the tJme from b1oh the decay ClW"Ve iI meulUed. We -n detenDine '- by ob.errinl the ~y of the field with two detectof'l with extremely different energy depeoden,*> e.g., & thin (" 1/.") aDd. a thi ck (" bla.ek") boron oounter. lalong aa the .pectru.m ItilI ehangeI with time, the ratio of the DOWIUng ratN of tbe two doteeton contin uee to change; it tint beoomee ooutant fOI'timee 10bgW thall '- (S,........ a.Dd D. Vn.un.I). We cu aIao determine '- by detflrmining « from th e decay ourYfl, beginning the evaluation ,u~Y(lly with late r and later timflll It . II tleerea- at lint and only beoomee conata nt for It > ,-. In thi8 w"y, we find, for exa mple, that for graphite ,-"",2 m.ec. In larger ... RlmbliOlJ (linfl&l' dimoneionl greater than 2 to 3 diffusion lengtu), .pat.ial harmonic. nthor than t hormali%ation tlffoota are mainly l'OIIpomlible for the delay time , In ntremoly emall ueernbliea, there are ciroumatanDflll under which there iI DO IUtionary energy l pectru m even after an uhittaril, long delay time. Tbia" clearly erident in th e exam ple of a '(err eoId beryllium bloolr. (Fig , 18.1.3; SILYn). 1leoll. ......1n&, Jl'_

~

III

«..

.hen the Talue of • de«1 _ -..d11y with iIIoreuina: 'I'biI pheDOlDeQOQ i.II 0ClGD"0ted with the faoi. diIoullOd in s.c. 10.3.3, that under oerWn extreme oooditioDl DO ..,mptodo ~ of tbmnalhed OIIll.t.IotuI eDt&. The effect 01. ~ harmoab tau be .ery s:r-tly eedueed by a IIkillful e..rn.naoment of the oeutroD Mnlf'Oe and detector. In. parallelepipedal _mbly the .aurae Y plaoed lot th e midpoint of one of the aide .urf~ .a that in th e ~ - aDd ~ only odd (I, ", -I, I 3, 6, ••.) h&rm0Di0l are e~cited. U the T I detector 11 now p1aoed at the midpoint ,< I of &0 adj-.t Iide alU'face, theD.ofthe humonlOi ..aited in the ~on '\ .< oo1y the odd (. - 1, 3, 6 . . .) ODM will be deteoted. TheD it followa that the nen higher harmoniol &her the fundament&!

"

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-..-.I

_ _ b ._

~-

mode II I to be det.eeted are the 311-, the 131_, and the lI3-mod.., which uau&lly d_y OOOIiderably 'uter. U .... UN two deteotorI oonneoted in panlleland plaoe them in th e moderator u ahown in Fig . 18.1.4, we O&D. alia eliminate all hannonic. with r,_, or . ... 3. Thu the nut mod. . .fter the fundAmental mode lore tb e 611-, 161-, and lI5-mod... BimiIar ooDtideratiora are poeai.ble for oylindri cal geometI'J. A.. genenJ rule, the elimination of tho highor h&r:mOniCi 11 more diffioult, tbe Wpr tho IIy8tem in oompubon to ite diffu.li.on lengtb. Under .alOe ClironJ:rat&noe.. .., muat oornince o_lv.. by evalu. ting the decay curve with a variable delay time abov e) that th e barmoni01l have decayed . ufficientl y. In u. e:deDded geometry, one DaD aleo peeeeed by th e method of Fourier aoaIyaia. A _all counter 11 lOOTed through the ey.tem and th e d_y curve 11 meuured at alarr oumberof pointe r •• Now

<_

4>(r, ' ) -1:A.R. (r) 1tI.(,).

•

(18.1.210)

Here the R.(r ) iI the epatiaJ. eigenfuDetion of the IH1 humonie lmd 1tI.(' 1 11 the ueoeiated et-y eurve. Obrioully,

4'.(I>-/4'(•• ' IR.(p)" ,

(TU .2bl

Th_ we Clan ieol&te lhe ~1 ClIU'VIII of the lndiridual humonice by integrating the -..eel epace-time UtribuUoo ... iDdioated in Eq. (18.1.3) . Lora and

Bn nu ha...e done thi. f~ B.O ';p\oeml, and M.zurna hal done it for D.O . y. t.flml. The ~114 it due \0 neutloD em.iMion from the I()UlO(l during the lnt.er'rall betWeeD put- and \0 b.eboettering 01 neukoJlll . We can 1IUt.t&nt.ielly r-iuoe t he effeet of baobutt.flring by oo"",ring t he . urlr.oe of the moderefA)r block ..nth cadmium or even botter with thick boron , heet.. Nevort.he..l leM, tbeee . lmOit a1war- reo maiJlll a . Iowly varying beck , ground caused by f&$t neutron. I I that we muet careIully deeeeI mine and ' ubt ract before the evaluatio n of IX from th e mea· eured decay euree. 1__ ....._ _ After the background hal - ::;7 been IUbtncted and the delay ti me determined , we C&D deter. r-Jrr 1 t.iIIr , mine IX from the deea.y eurv e. II II , ·fl In order to obtain good accura- na. 1L LA. .. _ _ _ _ cy, the _ble pvt ofthe decay . . - ~ " ,'ji4IIW _ Wr .. _ _ _ ..... .w. __ wIUu..r I. . eurve ahou1d Itret.eh over leT· era! (2 ee 3) dee»deI with good ltatlatioal aocuracy. '!'be calculation of. iI uaua1Iydo ne..nth eithw PaUL8'. or Oouau,.', meth od . In fnorable ClUM, one can achine an accuracy 01 0.2"4 ln L

,

-'

,

....

--t

--

,..,-i.. l·t

•

,

z-r-f,.

at._.

18.1.3, BenIta or Pn1lwMI Neutroll 8tuilM 011 Variou. M04eraton The pulaed neutron method primarily yie1dI th e « (8')-eurve for a moderator .utNltanoe . However, we are u.ually not oontent with thi . cur""" bu t Uling Eq. (18.1.11 t ry to determine from it the diHueion parameten «e-iI';Tij, D" and 0 , For thil pUI'p<*l. the meuured curve iI approximated by a parabola .hoee coeIfielcnte are determined by the method of Ieut aquane 1• 'I'hbI method giVIlI lle and D, 'll'ith ..tisfactory aceuraoy. In oontrMt, the achievable aoouracy in 0 ie only moden:te Moe the devia tion 01 the «(8'j.curt'e from altra.ight line iI,ma.n.nd linee in lIl1allUllCmblillllti1l higher te rma (FIJI+0.81 + , . .)contribute to IX. Either we must limit OW"IeIvea to a range of B' in .hieh th_ oontri butiora are not importent, or _ moat apply e&1eulated oorftlCtione for th e higher tenruI. However, theoret.ie&1ll1timation of theM termI, . hieb depend IIenaitbely on t he t.henn a.liz.ation aDd tranlport propertjel 01 the moderator, ill di..ffieult. L'gAl Willer. Table 18.1.1 cont&inl th e difhuion parameterl determined by ....n oue autbon from tbree-pu-ameter fit.l to th e « (BI)-wr't'e. In the regitm 8' < 0.9 em .... the role of the B"-t.er:m ill amall , aDd there 51 no atrong dependenoe 01 the panmeten on the e...aluatio n .cheme. Th e ...a1UIlI ha "o been corrected to a temperatar8 of 22 -C. Tho rea wte of AJITolfOV d cW., of BuOCl aDd Cocr"" of DIo, aDd of K OCBU ~ withill nperimental elTOl' and all agree welllrith the 10M UII6Ily performa ~ fl' In ROIl, _ y dIM die _ 01 IM .q_ 01 Lbo (tllCK tile .-MUM) dPiMiou 01. die - . - . I pob:lM from tile o:v9. 5o., (IIln.1m1Ull..

26'

..-u.. .

poisoning ezperiment. of S'I'.liLa and K OPPI:L (Boo. 17.2.2). Th e result. of Lo PEZ e ed B I:Y8'I'U are ~ewhat higher . In Fig . 18.1.3, we oompare the « (B').curve meuured by th _ latter authon with th a t obeerved by Kocau (which alm ost ooin~ with thoee of the other a uthon abo1'e). Th e 1'alUN obtained by Lo:Pltz aDd B....8'I'U &ftI luger througbout than tb<.e of K OCKLK, elIpecial1y at large buck.liDp. A poeeible ezplanatiou toe thia d.iaCl'epancy ia that Lopa and. BKYsn- u-i cubio goomotry Iee their . -mbliol while K ccau: (and tho ot hor

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.,

u ~

u

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II:IJC.SU'I~

n "D.. -"-_~

authon .. bon) ueed Oat cylioden. We migbt oonclude from theee facti th..tthere ia altrong depeed eeee 01 the extrapolated eDdpoint on th e geometrio ahape 01 the ...-.mb1r (of. H.u.L. Soort, lAd WALDa in t hbl ClOnt.e.tl.

..

T ..bl. 18.11

.....

Dil" " ~.

Y.DaDlIL aDd s,,6naAJm. AJrlooJlOY ~ .I . . Ba.(Q Md CooP... Doo .

,.. w....

,,... ... y-

("~)

,or'

....

,.., ,... .... ""

... . ..

· · . . . . ·· ·· ·· , . K ltOKl.l • . . . . . '000 LoPD aDd Bun.. ·· · · '08'

' 831

A-.li",lo. PtJ".d

NrvItrm JltlW

D,(_ ' --")

C(om' __' lO

36,.340±7M 34,OOO± 1,000 "'.8<50 ± 1,100 34,450± 8Oll 35,400 ± 700 38,!m± 6OD

7.300± 1,lJOO ol,ooo±I.000 .1,000± 1.000 3,700± 'l'OO ol,200±800 6, U8± 7'78

broKo1' d rd. ha ye in9Mtig at.ed th e RClOtl'oD diHuaion in R.O up to temperaturN of 280 by the puleed IIOUl"08 method. Their experim eutal neulta lor the diffuaiou. CItlOUDt c..n be a ppNzimated by the lollo-wing lon:aula :

-c

r/J.;~T:c.

_ (O.934± O.028)+ (0.289± OJI09) x IO" T. + + (O.I06± O.03)( 10 '" ~

I

(

18.. 1 . a)

I In the ~. of Fig. 18.1.6, 1J' ill e&1ca1&ted with the .:nnpo1&loed ehdpoint d of Gm.• • aD (d. 8eo. l0 .3.th bad we aimpllM\ 4 _ 0.71 .l,.., we 1f'Ouid h. .... obtained _n\iall, \he

NIIle ...u1L

Here p. is the water temperature in "C. The meaaured values of 0 can be repeeeented by

,*(io'b, = (O.987± O.098)+ (O.619 ± O.031) X 1O-

1T.+

)

+ (O.348 ± O. 104) X IO """ ~.

(18.1.3b)

HONBCK has calculated the diHUBion parameten in HIO at room temperature by numerically w iving the tr&nBport equation for the Nelkin model (ef. Table 10.3.4). H e obtained Do=37,460 em ' 800 - 1 and 0 = 2,878 em' eec-t, We can also estimate o by the elementary method of Sec. 10.3.1. If we UIIume that in . HIO molecular rotationisstrongly t hindered but molecular traml..tion is not hindered at all, we can .. treat wate r in firatappro:rimation I -:; aaagaawith A _18andu. =80 ~ hamIH.atom. Uejng themeeeured t' -" value of D. and 8.8IIuming I {v- ~ behavior for (11" we obtain 0 = 4250 em' 800-1 (d. Table 10.3.1). ' 1/ Ht4lJ1/ Water. Fig . 18.UhbowB the « (B'I)-curve in heavy water GnmtfritfI WIinI 8' (99.82 % D.O. 22 "e). KUSSMAUL Fl,. liU.a. taat .... ~ bI>ol
•••,, * ,

-:;;

V

1

,• •

•

7'

"

..

.. .

'1'1>1_, ......

end MEISTER were able to now that in the region of 1JI covered by their meaauremonta the B'_term waa negligible. H OliSO K obta.ined theoretically (t.e., by numerieal eclutdon of tho trallllport equation for the Nelkln model adapted to D,O) D. = 2.069 X IO' om' &ell- I and 0 = 4.8lI2 X10' em' 600 "1, d _ Table 10 .3.4. GANGULY and WALTNER have invl'Jfltigated the temperature dependence of the diHUBion coefficient in D.O in the temperature range from 20 to 60 OC by the pulsed neutron method ; their tellulta hav e already been shown in Fig. 17.2.2. GrapAtt~ . AliTOliOV d aI., BItCKURTS, L.u..u;:DK, ST~ and PRICB, and KLoa., Ktl'CBLB, and REICHARDT have dono pulsed neutron upcrimenta on graphite. Fig . 18.1.7 shows the ar. (B-) -curve according to two recent expcrimente l . The t wo

KU88!UUL

1 Th e experiment. 1VerII carried out. with graphiteaampJe. of differeDt.puritiellolld denai~ . In order to elimina te the differenllllllt.hereby produced, the nIIIult. were oorrect.ed to" (hypo-

t.het.ioal) uou..blQtbing graphite of dentit.y 1.6 gJcm' , Le., ( '; ' )'lI'.

~ (OI- ~l

e

" .. plot.ted ag&inat.

IeriN of meuUftlmea.ta &re mutua1J.yoonUtent i lOme of the _rlier meuurementa abowtld .. weaker doWDwaM eur....tUftl at Iarp Bt which ia prNwoa b1y tn.oeable to tIlrOD8OUII «-m-.uremtlllta rewJ.ting from. a n Wufficient delay time. Th e eT&luatioa of the diffuaion paramtlterl from thu (BI). CUlTtI in graphite hu lJtill not been fuUyclarified . We • A.- n a.-AM """" obtain diHeRlnt yal ut18 of D. a nd C according to t he length of the Bt-inte rval we o' ~ 11M in the analyaia. Thia is ~ demonmated. in Fig. 18.1.8 >4... ... for a three-and a fol1l'parameeee eyaluation of the da ta of KLOU ,K tlOBu a nd RUOH4RDT. In the thf'tltl. parameterevaluatlon,obvi. ~ I ouely it ia only Mnaiblo ee Ulloll t be eune out to ValUM ""1" 1.7. n. -.,. _ . .. - - . . ....... .......... .... .. I ..... of Bt of 8 x 10-' em" i we tb en obtain I ~ _ 88.3 ± 1.2 D. _ (2.13± O.02) x 101 om.'j_, and 0""" (28±61 x 10' cm'faec (d tlllaity = 1.8 IJom.·. T. =20 "C). A four -parameter fit ia obrioualy poMible o ver the entire Bt-interval eed

,..

... .,'..,.._,J _

.

0

0

I""

•

,

0

0

_ -,/ .. //

0

0

I

,

~

.

r._

.'".

(f 1" -

r I ~ U. ·

_-I.

gi,..

....

~ =88 .6 ±1 .6 l1t1C 'l

D. _ (2.I I ±O.02) x 10' em' /eee F _ _ (20± 10) X l Ot

0 -(18±6) x 10' em' jllbO

..

·~ ,I ~""" . ""...,.,

.. ,.

t ..

• '.

~

,

,0

,~

0 0

0

I.' ' ••

,"

n,

0

0

0

r~

,

,

,

..

em'f _ .

"' ~ "...... .' " .. , . ... ':*' , .... . ' " or , . " . ,,

I'

• '""""'" fi1

'J

,f ,

0

• • " n.•1ta-'"-_

""1..1.1.. _ ..... 110 _ ar 1III

"' _

. ..... III

tJ.po.llllot-..l 17 1.• • · _ ... .. . ,... II ~ l1li

.u. '" r . .. r .....

"

"

"-"

"II- I I.1 .all. 'I'Iot 4lIfw6ooI Ito '" .s-1, 1.1 'Ookle cI

r_ .

"

•

•

....

_ _I II ~ ' haoIloI ar lM .......l • ••

~1IlJ1III

r ...,.,.

Table 18.1.2 OODWn. .eme additional da ta which _re obtained by malting threeparameter fita t
.-.-cera.

III ~ . nhaa&iDIl 01 \hi diff\I&ioQ the pubed _ troa da&a .... HPMUted of &he diffUlio1l.1eacth obWrled in a . . pile.

by • - - d ...

'"~ mentioned . th o", were probably extOl'll in th e m8UW"fIment. on am&ll .-mbliee ; in contrut, the low O·value reported by BJ:O&UBTS can oonceivably be flJ:pl&ined by hbi use of too 'mAll .. BI·lnte"al.

..-

1_

A~_.~'~ 11161 B~ .

8T~'"

Pale. • •

"..., 82±'

It7 ±1

Do I_· ...... J

" 100_ '--"1

0 1_ ......1

(U J ± o.ot)x l O" (IU ± U) x IOO

(1--40) )( to""' (0.7- U ) xl()-,

{2.07±o.(3) x 100 (I U ± !.OI)( lot

1G60

71.2 ±0.1I

1%.06± 0-0%)( }O"

IIU ± U) x )OO

1l.6--r U i) x J()-'

1M2

'I u ±o.e

(2.14 ± 0.0I) x I0"

(3t ± 3) x lO"

(1.7&-11.8»( to""'

8T~'"

Pluca (GBf

gra phite ) .

I

4

'1- 1- -1--

...

'

.-

n. &n-' _ fNo po.llI 01 ~ III bwJlU... .. • IuDcUoG 01 tbo 1IIoOdoNf,~ .. " ' ~ .. _ _ br 1l. ...vavu ....I1.~

'II. 1t.1."

_wo.

I_

h-

... .. .......

AIrToIlOY .. IlL

""""~ erm..o•

••

. .

lnufto. DJ: & n.n. 1IDd

Bu.f'U •••• • A>n>_. ....

-. 1Mr'11

110 1_'_"'1

.~

........

±O~) x . O'

c 1_ ' __' )

...

, ,

"'l.~

IU 3± l .G) x 10"

O~.CllI2

1:10 1.15 x 10' 0 2'JO± 1 ( l oU ±0.(4) )( 10" (U IO± O-',x l ()f

O.005-0.03e O.Cll»--Il.Of. l

188±

0.008-0.07:

{

J8G± 8

"

(U .s ±0J:l8) x lOi (1.40± 1.0) ( 101 (1.t35± O-OI3»( )(lI 1S-80± o..s) x 10l'

o.003-0 fl71

Utina;themethodofTA.UB..UHJ (8eo.l0.3.2Iwefind O_3.36 x 9 X lo-cm·MO· 1 ....

3 xlO'cm-_-1, while 1M" IU of m- 9 (Table 10.3.11 _ fiDel 0 _ 5.8 x I0' em' .eel -I. D. B.uTuUllIi and Sn... ... and ..180 AlrDazwa haft! Itudied the tem . perature dependence of diffueion in beryllium by tbe put.ed neutron method.

Fig. 18.1." Iho.. the obeerTed temperature depeodenco of l" comp&red to a ealen1&tioD of SIllQW1 aod KO'f'BllJ. Olitr J[odnolar,. A variety of ftllIU1u of paI-l neutron e:zperi.menta on additionaJ moderaton are oollect«l in Table 18.1.4. The temperature dependent(! ... aJ.o meuured for Dowtberm A and BeO up to 200 '"C.

.-

.........

Do........ BoO

Do"'' '

ll le..')

I.1.18' "

......

-, .., zu.-ivm

B"""" ....... ....

.. 1_-' )

....,.

1,,870±40

Do 1....' __'1

0'...·__'1

fl8,200±800 11,(100::1: 1100

".000 (1.18::1: 0.0!)

-

..-

B'-bienll

lem-t J

0.~0.26

Kttcm..

3.783±_

0.Of. -0.1 .~ (3.u::l: O.08) 0.02 -0,0f, I .....ou. x 10' x 10' m.UOO::!:3" (U 8±0.271 0.G3 -0.11 XIIAJ)()W'Md

1I,e20±,lliO

nPlO±800

131.11

0.118 11.100::1:10

»ro

' OOO±l.""

lO,eoo±1IOO l,eoo± 800

W~ .

0.0'7 1-0.872 K ...... 0.(l9g- 0.IU B.roen...lfD

....

18.U. DetermlnaUon of t.be Oeometrkal Blltkllnr ., t.be ruw Neatren !IMbed

The normal peceedcre in paI-l neutron e:zperimenta is to meulU'e tbe decay coNlt.anl ll. of a g«tmet rieally ei.mple aMembly for 'lll'hieh B' ¢an be eu.ily calculated. If we h....e once determined the o: (B')-cuTt'e for a particular moderator in t hit way. 'lll'e can then inv ert th e procedure. We oan build &Memblielwith an arbit rarily oompticated geometry, meuW"fl their decay constant., and by means of tb e lI.(B')-curve determine tbeir gtlOmetric buekling. Such experimental detennina_ tiODI are of intereet for aMemblie. in 'lll'hich a 1I01ution of the equation r-4'J + .8-41=0 is eitber impoeeible or poeaible only at the e:zpense of a grNt deal of computational labor. Among .uch aMembliM are bomogenooua bodlee with complicated .urfaoee (e.g., cylinden with rounded emil) or aMembliel in whicb loca1Ued .trong ab-orben are found (e.g., roda of cadmium that are supposed to limuJate the effect of control rod. in reaoton). Such experiment. have been carried out by BSOKlJRTS.nd by s", 68TRAIfD rJ cd., among ot hen. We . hall not go into the rNultt, 'lll'hioh .pply on.ly to .pecW oonfigurations, beee. 18.1.1. A.."UoD. Meanremtrntl with the PulMd Neutron Met.bod tI) LArve Autmbliu. We can doloenn lnCl th l' a1:Morption CroN .actio n of " _uhftanoe hy m_o- of tbe dCleay of t hCl noutron field in lion .. infinitCI medium" of the nt-tanoe (practically .peaking •• uob " Infinite media " can only be realiz ed

in hydrogenou 'UbetaDoeI). If 'lll'e iDtegfate th e diffusion equation Dr-~ -r.~

.pace.

~ ~~ _

OTer all the diffutlon tenD ...aniahetl and 'lll'e obt&io the foUowing equ'tioa for th e integrated nw: : 1 ,.

Le.,

- -

J -~T - - E.4'J

(18.U a)

j (') =e-~ .

(18.1.4 b)

ThUll we muat determine t he f1ul: 4J(r , ' ) in the vicinity of a pulsed 8O W'Oe, integrate it over all epece, and plot the logarithm of the integnted flux againat ' ; the result ill a straight line with the Ilope vI • . V ON D ARD&L and w...LTlfKR, MuDS tl Ill ., and RiM.I.Nll"i.. tl Ill. ha ve carried out l uch e:zperiment8 on weeee. n boron counte l"l are used for th e flux meeeueemeete, under lOme circumltanoee a correction iI neeeeeary to aeoount for t he effect of the additional It:.orptJon in th o counter. We can cireumven t t hill correction (MU DS) by uaing .. a detector I liquid acintillat.ol' that it mounted on a plenglaa light pipe. ThiI deteetor doN not detect. neuUoJ18 but only the t .2-Mev f -ra,- produced by neutron capture in hydrogen . Since tbeee f -ra,.. han a me&D. free path in wat.er of w . 20 OlD , an int.esntion over a oert.aln limited rejion 11 alrMdy 11Itoml t.io&l.ly performed. Uilinc thil method . M.... DI d Ill. found th at in wlter l /VE. 203.3± 2.6 .... eee . b) M eMUffl1Ifl7IU OIl DUMIlvtd Ab4orbe,. S~n
..,_",::=,--_

." "".

o.l - ~ +N ..O". +LfN.;a;+A D. B'.

(18.1.5)

Here 0". ia th e ablOrption croeeeeeuo n of the dissolved eubsta.nce, N it the number of absor ber atoml per cm l , LfNJI it the change in the number of H .ltoml per cml du e to t he a(Mitio n of t he ahttorber eubetence, and A D. it the change in t1;le diffu Bion conltant . N Ind A N s can be determined by chemical lnal)'llia and density meaaure ment , and Lf D. can be calculated from th e .cattering crou toetion, [For strong abeorben, even a email ablOrber concentration h.. a large effect on the decay coDltan t, and th e Iut two term l in Eq. (18 .1.5) can be negleeted compued to Nlla•.] Thlll, ba ring meuu.red ~ and ai , we can determine '10". from Eq.1 18.1.5) and thul obtain ' O'. (2200m/_ I_ _~O"~I"'" VO" D i..IIDa L and

SrOBTlUJI"D

(el.

iloIlO

determined the ab.orption croea IOCtion of boron in thia way THOMSOIf, and Wal ORT). Recently, K UDOW I and Wa.ll.P curied out 'Ierr oarefuI meaaurement8 on a number of nb.tanoee. Some of th eir ftIIIUlt4 are l umm arized in Ta ble 18.1.6. Additional COrrecti.ODl mut be .pplied to the data for eubsta.noee wh<»e ahlorption croea IOCtion abO" 1 deviati0n8 from th e l /lI-law. To begiD wit b, at

S
1 In principle, Mparat.e de t.ermination of . 1 and. .. ia Dot ~. h IJIIffk:el to J!Io' lobe rat io 01 the t ...o d_,. curve- point by point ; the ourre 10 ob!&lDed ClIJr!'Npend.t to lob. oonatant .'-.-. TfLi,. prooedwe P'6"- that higher .petiaJ harmoniee do Dot. ooatn'bute _tronltl)' to the I'lIeUUr'8ment.

ot-,.

hia:b OODCeDtraUoo. of • .uvoe _-IJ-beorber (like 01. Gel, or 8m) tben •• c1eu' eHect on the De1ItroD l peoVum. We muA therefore ourt out l:D.lUW'etDenq of the a~ cr..- IeOtIoD at vlol'ioul ,bIorber oonoIDtntionl aDd then

eZU'apolate to . - 0 abeorber OODoeDtratioD (d . Fie. 18.1.10)_ HoweTW. the ab.orptioa em- eectioD obtained in thie ...y iI DOt aD a~e over a Ma xwell lpeotrum. but rather over the ..ym.ptotio neutron IJ>l'CtrwD in the experim ental v-I. ..hich aooording to the IIize of the vNIIII • IIh.ifted by th e diffuaion oooling eHect more or 1_ atrongly in t he direction of lo..er energiN. F ORUDately, it • quite limple to apply a correction for th e diffuaion oooling eHeet (of. MU DOWS aDd W~KIf) . We then obtain ~ -lI. ·,,(T) a. (2200 m/teC). HeM "IT) is WU1"OQft'S factor. and ". =2200 m/tee. Table 18.1.6 eontaiDa eome reeulq of m...ure menq by MUDOWS aDd WIULn'.

....

~

..

,

P\I. 1&.1.10.

....

- I.'"

~

r

~ .......

•...

"

' Jr,.,.,.' "...I

~

I-... I .

I.

..

.

,

no ....... &Iooir-..I _ . - . . . - _ _ fill

...".....

~

t _m n.,'"

.... 'trhU:01

_(II~

8m E•

04 H1

A.

II<

1.338 I .... 1.038

..... ..... ..... ..... I ....

..IUOO-'-o1

'-I

&UI± O.'

"" ±9

194± 2 6828±30

....± 30

4M17 ±IOO IOl.4 ± o.a VlLHCU 37f±a

l 8.!. Meuorement of th e DlffuaioD Collltant by th e Method of Modalated So...... The .tOOy of diffUl10nby the propagation of neutron ...... through 8Cltt.ering media .....uggNted by W ,Gle " (ei. &leo WEDf 8l:SO and N ODIUI. .) and firat oanied out by R~.V91U and H ollOWITZ. The prineiple of the method follo_ immediately from the treatment of neutron ..aYelI in Sec . 9... At. diatanoe r from a anuoklaUy modulated IIOW"OI (Q(,) _ Q,+i1Q" -'} in an infinite medium th" ""ponM ia rPno by ' .(-.1'

IQ <,"

- .1') - Q;

[ esp -

(IYtD.,}i. +¥:) +.( I (I- ~) )] r • cu - V2DJ- ' .

(18..2.11

Eq. (18 .2.1) holda WMieI' the ooodition that tbe modnlatioQ freqll llney cu ia larse eompued to the ablorpt£ on rate ;I;. but .mall oompued witb the oollWon rate .E•. It .... derived in Seo. 9.• for monoenergetio neutf'OM; In the following ..e . ban auume t hat it aleo hold. for modulated therma l neutro n fieldl . 11'0' th e cJ»olllk val", of tbe reepon-e we olearly ha....

'. 1

In -

1 -.

_ 10 __ IQ

0.

+

• L

cu rro;- r-;( or.z. ) I+ ~

(18..2.2)

while for the ph_

anele _ han

~ _ua~ _ _ ~

r;(l-!i:) .

(18.2.3)

U we meaeW'O 6 f11/fII. at .. fDed point ,. for VIlrioua frequencies w and .,jthor plot

.

In I ~~ I " function of ;:11: = or f .... function of y _ YW(l -~). th o l'Nult will be a straight line witb .Jope - - '- . U we ropeat the meuure·

VW(1+¥:)

vw,

monu at vari ou8 poin t.ll,. and. plot tho I lopee.o obt.&ill.ed ag..uu.t r. we ag..m obtain

we mUit know t he ab.orption Cl'06I an ap proJ.imate value ill enough.

yI

• I n ord er to ulcuJate:r IUId ,. ' D . aectio n of the moderator, but aineo ttL:< 2Cl1.

.. Itr&ight line, but thia time with the a!ope -

UU.l. Produ.ction and lleuunment 01 ModulaW Neu.tron Fleldl Fig . 18.2.1 mo... t he appuatu used by HoROWITZ and RnnllU for fill· perlmenta on graphite. By periodiotJly nanowilJ@: .. thermal DeutroD *m, ..

,..

r"JM'

I'fdtmrlli,Jif1'

"I

ewfII.

J'It.

~L

n. . _ _ ... .

r._..

ru_

""",.

~ "" ....,.

_

""'" ,.,.,

. . __

2J .... .... ~ ....,w..

bum .. produoed whc:.e integrr.ted. Intenaity ..ariN liDuoidaUy to .. 1000. degftoe of approdmation. The frequ ency can be yarieo.l in the range from 60 to 300 oyelflll/'*" Tho neut ron b6am W Inoldent on .. .cattere, In t he mldd lt> of t ho moderator boins Itudied . I n fint ap proloima tJon, tha O&n be ooMidered AI .. IinUlOidally modulawd point. 1OUl'Ce. Niekel (aDd later poIyet.hyteDe) wu ueed .. t.h. 1l»tteNr.

_ Ue,.,

We canalao modulate aradioactive (y , ra j 8011l'CfI. To do 10, we can, for eumple, mak e the source out of an antimony core and an outer mantle oonaieting of 86veral tegmenta of beryllium. Between th e core and mantle we place a rotating absorber (Fig. 18.2.2) which periodically attenuates the y.intensity falling on the beryllium . Such a device ball been used for experimentll in heavy water. Also thownin Fig . 18.2.1 it th o elootronio appar.tUll for recording and analyzing the time-d ependent neutron Dux. The neutron nUl:; iB measured with a BF,counter Wh086 pcteee are led into a bank of four acaJet'll after amplification. These lIC&l.erl&re turned on and. off by an electronic lysWm photoelectrically synohronized with the modulator in euch a way that they reepect.ively record ~ +Z. , Z,+Z., Z. + and Z. + ZI ' Here Z, u; the counting rate in the 1.11 quarter of th e moduletion cycle. Since

z..

,,..

Z,_

f

(18.2.4)

4) (1 ) dt

,.

( ' - I) ..

we hav e and

lp =aret&n ~ZI j-...zI~=.lZ. j- Z,t (2:,+ Z.l - (Z. + Z.l

1

_6 ~ 11 == [(~Cf-~~)- (Za t~.IJ" t [(ZI +Z.l ~ ~Zl t !"Il" 'PI ((ZI+Z.) + (z.+z.U"

(18.2.6a)

.s;

(18.2.6 b)

"

In this method of integration, all even harmonica (which ca n 00 ex cited by deviations of the ecuece modul.tion from . t riet dnulKlidal form ) are automatically eliminated . • , - -,----,,---,- --.=<7""1

•

.•

"

~

PIe. 1&1.1.. ... ~ _ "bow.! (Ib -IM, _ (IW'STRI I

18.2.2. ModalatJon Experiments on Graphite and D.O Graphite Wall first etudied by R ...IaY8 KI and H OROWITZ and later by D ROU. LEU, LJ.OOUll., and RAIJ:VIlIU. Tho experimental apparatu8 hall already been mown in Fig.18.2.l. We MSume th at the graphite pUe u; Ilufficiently large that it may juatifiably be considered an infini te medium. Fig . 18.2.3 ebcwe th e eaeesueed "alUN of

lnl ~.;1 plotted. .. a function of z=J'W (I + .!~) for VarioUll diatancee r.

The plota are quite Itraif!:bt 1ineI. whoee elopee are plotted. u • function of r in

•'"

YII . 18.2..... From tbnlope of th e line in Fig . 18.2.• • W CI ' obt&i.n D. = (2.09±O.03) x , 10' Cf1J.1/~ lot graphite of « denaity 1.6 gfemt at room temperature . We caD.JBo ' III determineD,from th e phue ~ .. I- III angle 'P' To do 10, we plot ~ a. th e meaaured nlue8 of f AI againllt r for variolUl valuee

./

'" I '"

/' ./

.>

...

/'

.

..

•• • ••

~f V""l'W (I- ~}

(Fig. 18.2.6).determine the dope of th e ~ultirll: linN. and plot tbeee a!oJ- .. a lone tionof r (Fig. 18.2.81. From the elope of the rMwtins line foUo," th e va lue D. = (2.09±O.03) x 10' cm· &OO~I. in good agreement wit h the value d erived from th e eb, 1I01utciv.IUIl of t he f8lIPOIlJlll. The ~n that we proceed differently in enluating tb e ph.., m eaaUftlment. than we do in evaluating the r&' POl1ll(l meuuremontA i. the following.Tbemeaaurement of the phue angle i8 ob, Tiou.ly done relative to th e phuo of th e mechani cal modul ator. However, Eq. (18.2.3) doel not a pply tc

1M

/'

/'

. I

• "

•

u

,

............

$

•,

<,

"' .. ~ I

~

............

H

" " 11.1.1.TlIII,.... ...... ... ""'i;".. - - '-'>d"'_~IIo ..."""

'"

t he neutron flight. tlm o bet1n!llm the modulator aDd the ~tterer. anadditional. frequflocy-dependent ph.&ee ~}

1 • 1i ..

ahiftont.el'l .Thialattert.erm ia eliminated., howe ver, ...

when we plot against r - fint . In a IlimillLr way, RA.

••

I&VlllU andHORQWlTZfound

."u

... -

/

tlu. ph...e &nile ; owing to

D, ... (2.OO± O.05) X10' em' eoc· 1 for .w-,uuter (100%) at 13 00.

~

I'-- ~ • ."

• ,-

•

..

<,

•

,/' /'

-:

V

. .,hi. l&J.a.

•

"

'- ~ I I~

...

"

The mea.flU'ed ...Ioe of the di.ffu.sion conat&Dt in gnphite is 80mewhat lower than the nJoe obt&ined in moo puleed neutron meutll'emen~. This is oonce.ivably oonneoted with the fact that apectraJ. effeota alao ooour in modulation ezperimentl IOthat the limple theory developed here no longer hold. and additional oorrection8 are~. A thorough theoretical nudy of . uch effect. bN only rooently been done (01. PKUZ aDd UDJo).

18.3. The Study of Neutron ThennaUntion by the Pulaed geueee Method Fig. 18.S.1 abo_ achematically the "life history " of a neutron in a moderating medium. We differentiate the epithermal or Mowing-down region, the much loager.1Mt.ing tberma1iution region. and. the uym.ptotio region, in which the nera.ge neutron energy no longer t cha~.,. with time. Thepul-'- neutron r meU.oo. III "".. IIIOlIl.ly woll .ultcoJ lor the Itody of thi.I liIe hiltmy .moe it &110_ UII \0 determine th •• peotnlm. or at leut oer1&in ave~. oyer th e . pectrum, aa a f!mctioq of the chrono losica.J .. lie I t 01 the Deutl'ona. We hue already dealt with ecme eueh eJ:perimenta in the epithermal region 1ltnt t (time depeedeaoe of the &lowing down Y!a-1U.L n. .. .....,... _ ............. to indium and c.dmi.um. l'NOJl&nee) in Soc. 14.3.1. Th _ tttwliM lMlI'Ve principally to COI'TObonte the elementuy theory of tIo wing down by free, ltation. &rJ atomio nuolei. In oontrut, the eJ:perimenta in t he tberma1iution and uymptotio ranpa to be dieo~ here haft a deeper physical aignificance In that they demonatrate clearly the role played by chemical binding and. with in certain limita, allow UII to check the ....riOI1l tbeories of thennalization .

.,

.'

--

IIU.l. lateenJ IUeIUpUoD of the Tlme-Depeudent 8pedram The claeaieaI method of atodying the tim. depeedeeee of the thermaliution proI:*I ill to o_ne the time-dependent tnnamj- ion of the Deutrool. To this ead, the time ....riatioa of the intenaity of th e oeutrolUlle&'l'ing a medium u. cbeereed OIloe with (Z·) and once without (-Z-) aD abeorber foil between the medium and the deteotm aDd tbe ratio Z'(')/-Z- (') formed . The abeorber of choloe is all" abeor-ber Rob u boron-oont&1n1ng gla.. AI ..n example of the relU!ta of euch an uperimeDt. Fig. 18.3.2 abo..... tn.namiMion curve. of ...t.er and toe at Tarioul temperatwel (vo. D.unlllL) . We can _ elearly the eet.t.bli.lbment of tbe IItationary lpectrum. at the end. of the tbermaliution ~• • hioh witb d ee:reuina: tempen.ture require. more eed more time. Similar atudiN have been carried oat OlIo beryllium by Anol101' d on graphite by Analfo", d al. ..ed by BIIOKt'J;ft, OlIo beryllium ozide bylYoo.... d al.. and on &irconium bydride by

.z.,

MUDOWI aDd WIWJtlC.

While tn,nlllJliMion eIpCrimenti give 111 u.tefu) q uaJiu u vo infon:Patioo, it ill difficult to obtain reliab le quantitative data from th em. Ma.t .utho~ ha ve

taken .. Maxwell energy diIltrihution for gran ted and calculaW neutron tempera. turel from the meMured trani_

u

(d . Sec. 16 .2.1). TbiI pl'OOfldure yioldt curve. of the neutron leltlperature T as atune- ~ I. tWn of the time. A. all. uampl., ., T (' ) in graphite ill . hown La /U Fig . 18.3.3 . Uting the elementary theory of Sec.lO.4.1. we e&n approximate t hil curve by mitIldOD

of\: •

-I

!

T(t)_T._e - 1i1r (10.2.161) and tbu obt&in th e kmptmhlf't reltJZlllu". Umt. ' T ' T"ble 11'.:1.1 aho... valuM of ". obtained in

~

nther tbu at the . m ace of the medium, .here mOO experiment. beee curled out. Alao the a.ngulu dimibution of th e MU· tIon8 lea ving th e . l1lfaoe, .. bout which we know little, affect. many trarlBIlliMiOD ex perimontl. Spati&l. barmowCI and timo-of· £light .Uecta infiuenoe the mN.urement in .....y that ill hanJ to a.oooun t for. FinaJly. the ClOD-

h..,..,

oe~

of tho temperature re1u&tioo time 1a iteelf very crude.

-""'1'""""1 • 'J-'-+-< '+tr

.......

•

1u.L n.u-.topo

1M '" .. ... .,••.: _ .-----.

uu. .ay.

Th _ 1a .. wbole IIerieI of nther 8eriow objectlou to tha procedure. The ....umption of .. MA. well .peotrum 1a oert&lnly only permis&ible in th e immediate neighborhood of the eq oilibrium at&te, and then only in tbeinteriol'

"-

,..",

., ..

~

~

~

..

1-'

_ 011 _ _ .... _ ., 1M ' . - - " . . _ _" lu- c_ ~ ..,.

'::'\ r ....

'" '"

·tl" l'} """*''-'-. TftJ-1,~,._wiItt Jr ·IH ~ "

''N

1'1'1- i-I.r

..

, "., I ......

,. 1'" J'1_.lbftIoqtllllh. f)_...,.,.... _ ~

1Xl. . """ lI-.1.,.dl-l

11,0• •• • • • Bi (1.86aJem'1 • • Graphi\e (I.l!l~1

YO.

D~I:L

AnoIlO't''' . . B _ _d Ill . AICTOIIO't'

InllO... .,..,. One dlould not t here fore att&ch BoO (2.08 110m' ) • too much Iig nilicanOll to the rNulta in Table 18.3.1, with whioh other m euuremen u (_ bellow) &r'e in contradiction and whi cb furth6rmore do not agree at all well with the theoretioall)' ealoult.ted value. (el. T able 10.• •11. In oontnat to tho tem~ture re1u:atioo time 'r. the ~ioIt Ig ill • ..u-defined quantit y . According to Sec. 10.• .2, it UI l/~. 1t'bere ~ UI the

'i""

fun eigeo't'alue of the thermaliution operator for the tim..dependent, at-xption. free problem. 'r.a can be cleanly meuured. To . . bow thiI . done, let ua llODIider

au infiDite mediwn with 1/_beorpUon. The tim e depeedeace of the .pectnun II thea liTera byl

I

tz.(E. e)- .M{6)e-~+ ""I (B).- (~+ ':)1 + + hish.. te~ whioh decay rapidly.

(18.3.1)

U we obeerre the OWl: with .. detector wh~ energy-dependerat IelWtirity 11. E (K ). then the tUne-dependent lXIunting rate 11. liven by

z(e) =z.t-~I +~t- (iY,;+ -~l l + + hlcher te~ whioh dec.y rapidly wi"

Z. - !E(E)M(E)dB:

I

Zl _!1:(8)4>I (.&')dE .

(18.3.2)

(18.3.3)

We DO'" obt&in tu by determining th e decay co nat&nt of th e finlt energy mode in.dd.itioD to that of the fundamental mode. Tbe accuracy of thil determinatio n caa be sreatJy ~ by m&Idng meuuromenta with . eeeced detector wit h another -wtirity function r iEl. Then

Z·(e)... Z;'-~' +Z:'e

-("',-'"-I'' .

Sinee Z~Z; +ZJZ. we at.D find .. DOnR.&n\ G . uoh. Lbat

- (",, -!.).

Z'(r)-aZ (')_e too + + higher te~ ...hich decay r.pidly

I

(18..3.')

Le., we can elinlin.te the fuodamentU mode. H6~a and &6a'rB..ufD hue det.ennined t he therma1iu.tion tim e in • lara e

....te r _ mbly by thil method. The neuuoDl ...ere detected by mNl\I of the captllnl y.ndlationl of cadmJum and ladoliniQID; thflllll .ubltanOllil bav e .bIorp. tion CI"OM MCltJoM with enremely different energy dependeneee, They ...ere pr-nt ... IOlutJone in .. 260-cm' ve-el which oould be moved freely irWde t he ....ter ta nk. Fig. 18.3.' . hoWl th e oblerved ca pture r. teI . They be ve .Iready been normalli.ed to yield t he MIlle funda mental mode decay for timN > 20 peo. Th. if th e diHerenoe of both Cl1II'l'N 11. fanned, the fund. mental mode
tu-'

•n. _

of • _-abeorbq; &Dedi _ ~ la s.ee.. 10.'..t. W. oar-!... tIW tao. iDWdVClt.ioa 01 a I/~ cb. POt ClbIDp ~ aad -.Jy ~ .......~

_nae.

"1;zo;.

e&I:II

~

Neily .....

and once with BFI oountera covered. by .uver &baorbef'l in orde r to change th ftir .enllitivity funotion . Line&r utrapolation of the data lIho'Wll in Fig . 18.3.6 to lJI =O give. ~1 ""'6liO I l - (for d eMity 1.8 gfcm.I ) . The calculation by the method ofT AX Aa ' 8 HJ Jie ldl lu - 3 x HO JlMlO -420..,.ee (cf.8eo. IO.O&.2andTable IO.U).

u

I~ ••

-.

~

I

u

"

,

•

,•

-... •

111 ,.• .,

"

• - ... u-_..,.. .'-

I'll. 1....1. n.. tlm Il/tbor ~~ .. !II snPbIIoo ... tIaMUa 01 ~ bIor~ lfo.- Cbol'

1'1I.1 8.1.'. Tbo um.4opeodoU _ _. . .100'" ~ .-...u-tn4 hi

_IDly,. .... .,ter ()II OLLD. &D4 ...

""

n.l_ 01 ., ... . boft 1M " .-1 UGo.1 _

~)

M

(01..

or..... lO.a.l ,

In view of t he rough e:ltra pol6tio n procedure and of the great inacouraclee of the meuured data, the meuured and oaJculated ValUM of Ir.lI are compatible. MbDow. and WlUUrr have done aimilar .todiel on &iroonium hydride ; they found ~ -IH ±32 JIMlO at .. d enaity of 3.0&8 8lcml . 18.3.1. Dlreet Oblen&Uonl

or the 'I1me-Depedeot 8pedrum

01. have curled out d etailed inveetigatiolw of the time-depeodeclt noutfOn .poetnuD in pphit.e. Th oii' a ppant w la abo,,", in F1g. 18.3.6. The JlUl-l lIOuroCI in thia _ .. !ineu .-lent« - iI .ynohronized with • cbop per in .uch •••y that between eacb pu1le injection and chopper openiq • (.....r iable) B.ABJl'dD d

...

I_

ill 61f'f1t " ' . J'II,.lU.a.

n.~1Il_'

", 11 "rtI

- " b,

~.UII"sl.1or

lIo_t!IUbII

t.pqbtw

u............

~.

..

,

___

delay time 1 elapeee. The neutrollll are tak en out of the middle of .. graphite block (60 X 62 X 11
.' -••

~:~ -

r-

f- -

-

...-,.., "',

, -

"' * HI

.

'" -,0< • NtlIIrlfl 1r¥"If

holU,f . 'nil _ _ .............

""..,pIlI..

.-

U4lflwgt _

""-

,v , u.....""" """' IaloooIi
diRorted by the Ileutro n Oigbt time between th e middle of th e block and the ehopper, Le., If) (B , , - tiM . Ho_ver, it '- poeeible to get 41(8, ' ) from the . poetra ob.en'ed at yariolUl delay Umfll I. Fic. 18.3.7 mo ... mflUUl'fld n.lUtll of ¢l'(E , I) in the r&DIe from ' -300..-c to I -=I000..-c. W. ean _ elMriy the in ereaaing , hilt of the.peetn. to lmaDot energiM ; eeen 101' ' -I000 ..-c th e th.,rmaJization ~ • DOl- ca.a.pIetely finilhed. FOI' reuonA 01. inte nlity. m-.urementa at Iooger tim• ...", DOC. p<*oible. In FigI.18.3.8a-o the meuW"ed Ipectra ani compand with .JoWatioM that ...", done by .. analtigroup Method oaU!.fI; ex_ perimeDtaI .....1 _ of the iDelutio _tt.ering eroee ~n of gra pbite ; eJ:08pt in the cue I =IOOO..-c the agre8Mellot. good. U _ wiah to know th e IlW.-P'o''c .pectnuIlo at the eDd of the thermalizalion ~' _ caD peeeeed U _ did .bo..... laYe that we muat W&it a nfficiently long time . In media with .hort neutron lifetime. like wate r, bcwevee, another procedure ia poMible. lnItead of a ehcppee, alim ple rotating .hutter it used that

preventl DClU tron 1eabge out of the medium durin g the .Jowing-down and th ermalization ~. When th e aaymptotic . pectrwn ia eetabliahed, t he thutter opena. Since th e field deceya in a tim e that ia . hort oompared to the Oight time of t he neutrons, it ia not I1eOell8&fY to clole th e . hutter again . B W K'D'IlTB haa Irtudied the aaymptotic .pcct rwn in tllOlu in this way . The eJ: perimentaJ arrangement waa otherwiae IIimi1a.r to t hat of Fig. 18.3.6. The length of th e Oight path wu 335 em i a high.mment deute ron acoelerator with a tritium • target eeeved ... the pulled neutron source. Flight-time dela)'1l al80 occur in th ia experiment, hut th ey oan euily be eliminated by calcul.tion. Fig . 18.3.9 tho.....th e . pect.rum of the neutron flu. in a " large" veeeel of water 170 ~ ;;; after injecUon of a Dl'lUUoD pulee. AJ.o plotted .. a Mn....ell diatri button with t be te mperature of the moderator. The meuUftd pointot fit to COrTe very well ; .moe no diffUlion cooling oocun becaUle of tbe large Iize of th e ...~, thia

,

,

-

I

--

•

0

0'

•

0 {-

• 0'

f

•

=~

r.;

-

I

0'

Ii;

--

..

0'

~ .' ;;

.

• •

tt.

•

£-

.... lu... _

.. _ _

[-

•

ee....- "'_ _ ... - . . . __ ..... ..

"""'~

~

_

. ...._

.:...,_ 1000,,_

. = = _", :

. :

~,

_

•

~M

.~ . _ _ ....

_ . '": . : ....J

_

!e'

_

. _:

agreement ill to be expected. ' In oontl'aat, t he l pectrum in a aman uaembly (Fig. 18.3.I0) abOWI very olearly the effect of diHullion oooling. Fig. 18.3.11 abo. . the letJ.iugra ",..m- from the Inrfaoe of a Iarr water tank. Beeauae of

I

•

"""

L.

I ., • -

•

."

-

-

,

•

,,.

_E_

"" tu... n. _ _

........ ...

~

'" •

U•• l • • 1.-"l _ _ ll'O.. _atw

~

,,.

• E-

M

M

_

_

-

"*'

" " lLU(l, " . ..,.""'" _ M _

M

~ ( .". "

•

; - ~ -- -

..:: •• •

...,

I~:

_

'"'*'- ...

Ilwo•_ _ • M......,;

_

- .

•,, I• ,

.. ,

. ~""""hf_. ":: = --"-"- I I I II " I I II

, . , . . . . . . . . " aM

" " ILLIL

-I--

I

En.~'"'*'-'"- .........

<Uf ..,

u . ...

"'.M ....M_ .....

U. " II.lI...,

the ItroDi: -rx::r depeDdenoe of the traD8port meaa free path, thiI lpeotrum II much hotter thaa a Muwe11 lpeot.ru.m. The leakase lpectrum ill reproduoed. well. by a oaIoulation of KBnuB.. in which the Nelldn model for lMlattering on BIO wu 1lIOd.

Chapter 18: References

Gon.'"

bAull, E. : 100. cit .. NpeciaDy I I~132. B:IQ;tTftI., It. H. : Ree.otor Ph,.u. ae.e.rob with ~ Neo.trozl

eo..-. NIIlaL lMnza. II, 1" IIMI). DAallIll., a. F . l'OJI' : 'I1le ID~ 01 Neo.tn1D8 ""th K&uer 8tadW ""th • PII1Ied NeutloD. 6ouft:e, Tnu. Roy . b L TeabDaL fltookbnlDl " II~). D.ulll., a. 1'. " O'Jl', .00 N. G. s,,~D : Diffaa:ioa x-...._~ with Pm.d N..WoIi Sou-., Procr. NItCl Enerv 8er. I, Vol t, P. 183 (Jt68). ~DP 01 tb. Brookha.- 0clIl'-- OQ. N _ ~ BNL-lIt (INt) N~ Vol III, Ex~!.ll! Alpe0t8 01 'J.'raM_t .00 Aa)'lllp&otiD ~ }UaftQ, V.I Ph,..a'P' _ PiIet Ato~ PviI: P . - u ~ .. J'Iaaoe Ieeo. )flIthod.

B"

8peelal 1

I

DAlUl:EI-. G. F. YO'Jl' : Appl. Sol. ReoI. 3li (11163). ) IIWUumeot&tion of Pu1eed a...1Tl, E. : &.,.. Sol. bU'. U, U6 (1963). N E ' Gr.ua, F. M. : ORNL 2480, 22 (I N7).

eutroD .penmeaw.

B_nft, K. H.: BNLlIll, RE 1 (1M2). OouJ:U., R. c., ORNL.2120 (INa). PDu1a, R. : Proo. Roy . Boo. ~) A I U. 48'1 (IW). MM._'ofthe Sn. E. G. : BNlr7111, esl (1M2). c( 8")-Cgne. 8u lt.. UMl1. W.L. VILl.DU I BNJ.,.7Ill, 997 (1M2 ). AnoJl'O't', A. V.. . <11.1 Ome-n 11M PjOSl. Vola, P. 3. AJrToII'o", A. V.. " aL: AtoIrmI.:r- ~,.11, ft (IlleS). BUOCl, A., -S c. 000P...: ND01'O Clmmto 4, 6Q (111M). The BWd, DAaIlIII., o.r. " o. , .m. N. G. &loer--lI : Pb,... Re.,.. ", 11'5 (1M6). oINeo.troII 1>1:0, W. H. ; NukIrroD.ik I, 13 (Ill68). HAu.. R. 8. A. 800Tr, -.ad 1. WAl.&D; Proo. PlI,.. 800. (LoDdoa ) 71,

».

s..

147 (1M2). K ttCJlU" M. : NukleoDik I, 131 (1geO). Lop llI., W. M., uwi 1. R. Banl'u: NlIOI. Sai. Eq. l l, 1110 (1M2).

"""' """ lDB.O.

G.uOV't.T, N. K., &no! A.W. W~TlI''': Tnu. Am . NlIoL Soo. 4 No.t, ) The Stud, of

282 (lMl) . K VIIoIIlUVL,G., MdH.lUvn'n: J . Nuolet.t EDera:y A &: B17, 'II (1ll63). A1tI'oIlO,., A. V., d al I Gene.... 19M P/M I, Vol.. 5, p . 3.

Ene."

B_lTJI"ftI, K . H.I NlIoI. Bal. 510 (1967). KLoe.. H .: Dt~ 'fbe,U, KaNruhe (1M2). ~ .. H., M.Jtttea.a,.m. W. ~ I B1fL.7Ill. ll36 (INS). ~D" I . : lDd ..tri. Atomiqu." No . 5/0. 71 (IM I). 8'ru,a,E., aDd. O.PaIcJl' I TnoM . Am.NIMll. Soc. I: No . 2, 12.5(19054iI). lh'u,a, E .. 1oZId G. hum: BNL-7Ill, 1034 (1902). AJmu w.. W. M. : tJClU,8083 (1900). AJrTo. 01'. A. V.. d ": 0-.,..1ll65PJOOI, Vol 5, p. 3. ~1llL, Eo Uld P . H. enuoII' ; OR.NL-207O, 3S ( l aM). )(01101'0, T.T.. &Ddr.~. :Tt-. Am.NIMll. Boo.l, Ko.I,18(1968). S...l l. .au, G.IHl,.oo E. O. Sn.. .. : ORNI,S6IoI (l llGih

c..

ORNl,.2842, 116 (lll6I1).

....

8rJl'Q'IIt'I, K.8.. aDd L8.Koonuar : ae....,..l968 P/UI38,VoI. IO, ~

1 ct. foot.Dot.e 011 p. 53.

Neutroll. DiffuloD. ill D.O.

h'D'Q'" S. B. D.• d oZ. : Proa. IndWi Aoad . Sci. U. 2111, 2U (19611. Ktlcm.a, II.: NukJeonik t, 131 (1960). Kawo...... J . W.O N1d J . F. WIUUIf: Nual. Sci. Eng. II, 230 (1962). BI&JU1flf. Jt. W. : 'I'razuI. Am . NaoI. Soc. t No.1, 69 (1Q69). SJOITa£1lI». N . G., J. )ISDIf18, aDd T. NIL88OII' : Arid ... Fy.ik U . '11 (1969).

B. .trJrra. Jt. H. : Z. N.turfonch. Ita. 881 (1Q66). &.!6ln'a&JIn, N. 0 ., J.lbDtmI,1.I1d T . NILIIlIOIf: Arlr.il'

I

Fyaik U. '11 (1968).

B'Meu

I

The Study of Neutron Diffullion in Other . Modentonl. t b th

Ie::: I I""" .

Pu;--" Nure:n~~ U"!'L_~ ........ till.............,w........

CoLIn. C. H. , R . E . )fUllS, Nld E . E . L()(lu'I"I' : Prce. Ph)"_ 800. AbaorptiOn (LoadOll.) A ", f6f, (liM). )f fIleRt'II Du.J)~ O. F. V01l'. and A. W. W,wrua; Ph,... '1, 1284- (1943). . )l&UI., R . E ., dal. : Proo. Pb,... Boa. (Lo Ddon ) A (19M ). ~mb~"' , R.uuxll.f.o A., dill.: J . Nuel. Energy e, I" (1966). D&U1:Lo G. F . VOlll. aDd N. O. &!08TIWfD: Ph,... Re" . H, 1666 (19M). ) Ab.orptioD Kuoows, J . W.ot.lId J . F. WIULlI:K: Nuol. Sci. Eng . to 132 (1961). M:_~mllllta Soorr. F. R. o D. B. TaOllllllOI'I, and W. W-.rOJIT: Ph)'ll. Rey . . . . 682 011 Disaolved (19M). S" blItAn.-. Daot71.D8. Y., J . 1.&000''' and. V. RupU!: J . N ucl. Energy 7. 210 (19118). Tho PDsz. R . B.• and R. E . UIfllIO: Num . Sci. Eng . 17. 90 (1963). Mc>tbod of R.uItVUI. V., and J . HollOwrrz : Compt. ReDd. !18. 11193 (19M) ; Geneva 1966 Modul.ted

Roe".

It.""

Pf360, Vot lS. p.42. Neutroo WEnlBEIIO. A. M•• toDd L. C. NODDKB : AECD.M7I (196 1). p. 1- 88. SoIU'Cell. AJIT'Oll'ov d Ill. : Gene•• 1966 P/661. Vol. IS. p.3. BM:lJI:nTlI. K . H .: Nue!. Soi. Eng . I, ISle (19117). E ~lOn D.L&D1lL, O. 11'. YOJll', toDd N. O. 8.JISrra..tIl'D : Phy•. a.e... N, 1246 (19M). Tixpe~nt.d01\ IUlIod, S. B. D., dill.: Proc. Indian Acad. Sci. U, 211S (1961). N 1II~ ,;n KuDO... J. W., and J .P. WaRD' : Nuol . Bei. Eng. tI. 230 (1962). ell n pee • XOu.n, E., and N. G . 81Own..1.Il'D : BNL.711l. 966 (11162). } Meullnlmellt of th e ScawI1CI.Eft, E.: Diplom. n.e.~ Karitoruhe 1964. TbermalbaUon Time. BUIf.L&D, E ., d GI.: BNL-71Il , 806 (1M2). 0 ' -• I Ti BM:lJI:V1lTS, K. H . : Z. N.turl~. II., 611 (1961). .....rv.tion 0 meKUU'lu,na. E.: Nuol. Sol. Eng . 18, «I' (19M ). D6pendent Neutron Spe ctn..

%'..

1

Appendix I

Table of Thermal Neutron Cross Sections of the Isotopes Th e following table containl mouurod n.luet of tho thonnal eeutree .. b.orp. tWn, ruetion. act1v.. tion, and _ ttoriq ~ ~iOM of the J.otopNl. The data ..ore taken hom BNL-325. 100000 edition 1968 and . upplement 1960 ; only in .. fe.. p1acoI _ro more recent ValUM used. The unit of CI'OM aection. unl_ other_ ..-i.e DOted, i. 1 bam _ IO-w em'. Tho fint column of tho table Wt. th e elementA to ..hich the iaoto poe liated in the lIOCOOO column belong. After oaeb. iaotopo, itA .blllld&nOll in tbe naturally occurring elemen\. is listed . For radioactive iIIoto poe, th e half·life iagiven . The third oolumn oontaiM VaiUN of th e abeorption C1"OlIll eeeucn 0'. at the neutron velocity 11, =2200 m/&ee.. 0'. Incledea all roactioll8 in ..hich a neutron diMppoan (a, r ia, Pi a , d; a , «i fiMion). The listed n.luel!l ani the l'e'ulte of direct ebecrption meuurement., i.e., pile ceci llstc r meeeuremente, pulsed ..nd other integral oxporiment., or trantmiaaion e:lporimente (t be latter witb .. l uitable OOITllCtion for neu tron IC&ttering) . Th e no:lt h ro oollllDDl contain data on reaction crou 8OCtioM. U neutron ..b8orption IeadI to ndioaetivity. column fonr lista it. half.life. UlI11Ill" th it ..bere activity is ca...-i by radiati ..e ca pture of neutrona j in tboeo few _ oth er roactiona lead to ra.dio.ctirity, th e Iym bol of tho roeu.lting nucloua is given. after the hall-life. Sometimoe, radiative neut ron c.pture JMdI to IIOVeraJ. activitiM owing to the formation of iaommc l tatea. In this table th e upper (metMtab!e) ute 11 listed .. bove the ground nate (whero th e order iI known ) aDd th e era. IOCtioWl rofer to t be direct formation of each ltate. Where the d_ , of tho metaatable ata te doee not go completely to the ground , tate, the branchina: ratio ill given. In &Orne ~, roa.ction C1"OlIll ll)CtiOM bave been determined directly. Le., not by mea.uroment of th e neutron-induced activit y. Thia i8 the euo for fiaaion, for eome (7l, eillond(,..,
(""1

[d. Eq. (1.<.7)].

r

I Bo.uerinc aad "beorptioa ~ 1IlI(ll!onl MotopkI mlat.un WIll l iYl!lll.la Tabltll " .l . p. 20.

'Of \be 1IIl.ment. ill their

~1

oooani"l

... ...

-

.K

~('

..

.......... ~"

H" (.. IOO)

H" (0.016)

(U7±21 '1~ (O." ± o.lOI · l~

".. H" r..ooo1S}

At7 ±10 0

,Ll

""H

,"

," -

•S

B" .. 1(0) Li' fU12) La-'(DU8) Bo'(M d) Bet(lOO)

10 Illoba.nl

Bil (11.8)

J.UO± II

:au (1O..t) CU....., Q'I (l. 1I~ ...,

CM (U1O I

(U± o.!) ·10--

-eecc

N"' t"U3) ,.. (0.>7)

,0

L

()Mt".70j

.,.. lO.lI37) 0" (UlOl

r(loot

< lOmbua

w~· N'-(OlUS) N" (UI)

~

. . (a..a) - -N .-

.'"..... JI&"'

(100)

..... (13.." (10.11)

1:::

~

0 1.21'

(U ll mill)

o.6J8 ± O.ooe

Al"'" (100) 0.28 ± O.Cllil

L

p i (100)

o.~±o.o2

0.4*0.•

..B S- ~.o(8) SO" 0.'760)

8'" (4.111) 8" (0.01'7) a- ('75.011

'., 0:&_ ..,..,., ,

.....7."_, ..1.3_

±o.ol) . I()-I

1.0 ± 0.7

0.028 ±0.008 0.033±0.006

U ± O.2··

M,OOO ± 8.000 < I

0.009 ± 0.003 0.6±0.2 < O~

< 0.000 O.Q038 ± 0.0002

(0.1 ± G.S) • l()-l

H I 4.0 ±0.6"

.... ± 0,3..

6.6 ±1.00'-


6,.&70,«(.'1011 0-4±0.1 (0-21 ± o.Nl · 10"4 1l.J_ o.ooe ± 0.002

J-!'16.1711. --

HI

0

O.OU±o.OlO 0.631 ± 0Jl08

U'miIl

0.02'7 ± G.OO6

LIO m1llo

o.ll±O.02

lUll

1I ± 0.2

•.O±o.o

< O.~

U2 h 1' .3d

o.tl0±O.OIO 0.19±0.01 0.0018±0.0010 25.1 4 (P") 0.015 ±O.OI0

......

< 8 ' 1~

o.2lS±O.06 ± 0.04. .6.1 ~.~:7"" 3.08 010" 1 'H2O 87 4 (8"') O.UI±o.06 104 (1"') < 0.05 ·10-'

.

U _

0" (' U )

,,~-

"

• - (O.U.I . - (O.OU1

AG (100 ...) 1.N±o.I.

"'±'" 1.... ±0,10

_o.l"

OO ±'"

0.006*0.003· OM±o.U

H'

'"" >,..

O.8±U O.A±G.OS

1.1 01011

'±" < .

lOO~

. - (ell.8OO)

XOi (" '1)

2.'l' • 10' v

'7d

a- (S-08 0101, )

x,- (O.OlJ)

0.890_

(0.5'l'

....±e0<>

U ± O.I

0.080± 0.030

81- (t.l18) Bi" (3.06)

.,s. r .......)

11.'1

o..t8O ± 0.1»0

o.oeo ± 0.0lIO

8P' (12.11)

..

.... , ...

Fa 1""1

0Jl3f, ± 0.010

u Bi

,'a

_ c-.)I-"I

"

,,(-')("'1

...

lUI ..

> 0.000

1.JO± o.lI

H I

_....,

... ~ (~ ·:rv

.0. C."(9U71 Call (O.M) eaR 10.1") 0."(2.06)

...(_Jlt.RJ

.v . o.

184. 8.3 min

24.0± 1.0

__

Vt- (O.U) VJl (99.76) CrH (4.31) Cr'I (83.7t1) CrM (9.M) (JrH (2.38)

~ lin" (l00

."

Fe"" (U 41 Fe'- (91.68) Fe" (2.1'l') F e'" (0.31)

-"co- eo- (I OO)

U±0.3··

"'.8d

.._- - ---- - ..Ti" (7.9(1) 0.8 ±0.2 Tin (7.75) Ti" (7U 6) Tin (6061) Ti" (6.34)

,. [batIJ

42±'

0&" (0:186)

n Tt

_lllolrt-nl

O.22 ±O.Oi

Ca." (O.OO33)

.So SoU (100)

"

20_ 86' 20_+

86'

0.70±0.08 0.26±0.10 1.1 ±o.I

10± 4 12±8

22±2

1.7±G.3

8.3 ±0.8

l.9 ±0.6 < 0.2

.,

1i.8miD 3.78 J:DiD

I.'

17.0 ± 0.76±0.06 18.2 ± US

< 0'

13.2 ±0.1 2.3 ±0.2

0.14±0.03

37.1±1.0

2±2 .±l <± 2 I ±l HI

260 ±200

27,'1 d

4.6 ±0.9 l U ± l.li

....7.

lU ±o.t

U ±0.3

2.7±0.4-

2.6 ±0,3·"

Umin

2.911y

O.38 ±O.Of.

2.7 ± 0.2 2.8±0.2

U ± 2.0

"±2

,

12.8±O.J"

4Ud a.tlmiD

I.
(c.;") 10.4 miA

IIU±l./J

. ... r

20.2±l.9 10.4 nain+ 8O.3±U

2.0 ±0.6··

H I

6.287

(99.'7'.lO 0110.4 miD ..... 6.28,) l.76h l00±6O 1.7tl h 8± 2

O:l- (10.4 DllD.)

-,.N- Co""

(U 8 l ) Nil' (67 .711)

Ni" (26.16)

N i l1 (UG) Ni A (3.36) Ni" (UII)

-

Ni"

~.G6 h)

. 0. Cu· (119.1) Cu" (30.9) Cu" (U6 JILin)

.,zn

-U ± 0.3 2.8 ±0.2 2.0 ± I.O 16 ±2 U ±O.I U±O.2

24.4±000" 1.0 ± 0.1"

'M'

2O±.

U.87 h

UI ±O.23 U ±O.4 ISO± 4O"'

6.111 =in

'"

UUd

Zn" (48.89)

I.6S ±O.14

88'

12.8 b

O.4'±O.0li
("""I

16 ±IO...bam 86y(Ni" ) < 20 ...b&m 1I± 4 ...b&m I3.Sh O.09O±OoOIO 112 mID 1.IO±O.11l C,' <110 llbam 3.2 min O.IOO±OoOIIO C,C

Zo," (27'.81)

"c

Zn" (4.11) Zo'" (18.66) Zn"(O.1l21

a..G. o.n

(110.2) (39.81

U±O.2 6.1 ± O.4


0.00. 1.4±O.3 U±O.S

9± 1"

41.

--

AppeGdia: I ~ ('... r~

o.82±o.oe

ee" (1.0')

O.3fI±O.G'l'

.,,--

A.n (100)

4.3 ±O.!

27 b..

6.'± I ,O

OO± ' M± '

10_

I'"

26±6

0.8 1±O.o&

"min 18 miD

ee"It'l.3'T)

--

_(~ I""J

"

Ge'I ('UT) Ge'" (3lL74 )

. 0. Go'"' (!0.66)

....

..f-.>tMn l

. So Se"

(0 .81)

U±G..3

O.88±o.ot 14 ± I

800" (8.02) Se" (7.68) 81" CD.6%} 8e'" 148M)

'2±' 0.4±0.'

Be"" lU I )

u±u

." ... --

"'

3.42±0.36 0.0.0 ± 0.008

"min

0.21±0.08

0.080 ± 0.020 li b 0 .080 ± 0.020 (ow 60% of 67.eo _1 2 h)

_-

-_

,,......

0.030 ± 0.010 O.6 ±O.1 O.()60 ± O.O2l5 O.()l)f, ±o.OO% of e'J _ -+ U miD)

' .lI h 18 min

" .0 h +

J8 min M.t h

-. x,- &-' (0.3.5)

(48048)

}(r" J{,-

34.5 h

13 _

(2.27)

+

! x)ot y

X"' CI U I )

"

X .... (1 1.66)

.. y

KfM (54.110) Kr-

--

4.
.y

18.'Id 1'7.8 miJl min '70 min

... iii.'

Rt.- (17.8 min)

<'

4lI ± III 210 ±30 0.10 ±0.03

0.85 ±0.08 0.12 ±0.03 l .o ± 0.2


1.1 ±0.3 (80 % 0170 lIlia -+ U d)

St- jU I I

.... b

U ± O.2

Br-" 17.02} SrM (82.64)

".

O.()()5 ± 0.001 0.1±0.1 l.O ± O.S

Br-153d) &-UU,.) Y" I IOO)

HIJ)'

1.3I ± O.08

Y" (SU l bl

.:u

2.0±0.5 t.S ±15

ID"- of .... Ito _ IU 7J < 16 Tl ..in O.oeo ± 0.020 Ub < ...

Rbi' (%1.86)

- --

3.2±0.•

-.y

1tr'" In I JU-' 17T miDI

Sz60I (O.M)

2.I ±U 8.6 ± 1.4 10.4 ±1.0

0 .0&0 ± 0 .020

t .• r

(UjJ

.,ab Rb""nUll)

.8.

Zr" (5 1.48, OJO±O.07 Zr-I II U 3) U 8 ± O.12 Zr" (11.11) O.25 ± O.12 Zr" (1.1 x )007) ZrH (17.40 ) O.08±O.oe ZrH (2.80) l.I ±o.l

«

-.N> Nb" n CO)

NbN(U x l 0071

. ±I

H '

IN_

,...."

.....,

I.IS ± O.02

11.7 --SU

)a

b

US ±O.08

Ol'

<,

...

O.09± O.03 O.10±O.06

1M b

e.S miD

".

I.O±O.6 16 ±'

' ±'

. ±l

....-.

.11

1.-,. ( 'Iio .

'.

)

_

(-.) [ ' - -1


-co.coe

67' 14.3 miD

0.20± 0.06

2.8 d

-0.21 ± 0.02

41Uh'

0.7 ±OJ:

7, ' ..... 1

I U ±U U ±O.II

U±0.7 0.4±0...

Mo"" lUll) "10"" (12.711) MOl. (9.62)

-- ---Tc" (2.1 x l()Jyl

'-"-'. R, -RII"- (6.71

... ,.....


Mo"" I I ~M)

Mo"" (U %j )10M ( UI.'7O) lIo"" ( 111.60)

.-T,

.... (

r,>

-

0.6±0.15

"±3

0.1iO± 0.08

6± 1"

au" IU I

R ,," (12.11)

Ru" (12.7) RII'" (17.0) R u- (3 1.3) R u Mi (lU)

. Rb Rho- ( 100)

--. .P

1" ±4

N '" (0.8) Pd' " (9.3)

.....±ue

12 ± 2 14O± 30 (9ll.ni of 4." =1" _ 42 _ )

4 .4 IPUl

42 -

17.0 d

U ± 1.5

Pd'· (27.11 Pd '" (28.7 )

4.8 miD

Pd'" U 3.1I)

%3.6 miD

0.28 ±0.06 10 ± 1 0.26± 0.06

H I

Pd'· in.., )

13.8 11

...

-- -.-\i.... llU.36) Ai- (• .06)

31 ±1

"±'

U h

< ....

u

"3.1±0.4 ±'

.u_ "'. IIIln

01- (0.87)

1.3)'

CdUl (12.39)

-

llO± 10 W"-ofWd _ U .2_1

~7h

. Cd Cd" (1.22)

.

l.o ±o.a

1.'1 ±0.3.5

409 miD

0.16 ± 0 .06

6.1 Y

0.030± 0.016

Cd w ( 12.711)

Cdu. (20'.07) Cd11l (12.26)

•

ce»

(28 .88)

ca»

(7.68)

In"' lU 3) 1D"' (8U71

• S•

Snu. (O.N ) SnU 6 (0.66) Bnlll (O.M) SIl.1lt (lfo.24o) Bnu , (7.117) Bll.W, (240.01) SnU* (8.118 )

Bll.'*' (3!.P7j

20,000 ± 300

... J'

<3 . 0.14 ±0.03 ± 0.3 M' (None of1.1Ud_ U hi

60 min

H

O.3 < 0.008

49 d 72 _

M± U 3.o ±1.O (M..a 'lro fl8 d_ 12 _ 1 6Ut IDln UIO± t lU _ flt ±l (None of M .lt miD _ IU _I

I

H 2d

1.3 ±0.3

IU l d

O.OOlI ±O.OO2

200.
. ..

0.010±0.00lI

l ±lmbt.m U fo ± 0,(13

IO± Z··

.± 1"

...

_.-

ApplIlld.iz I

.... l ... ,.~

...(.w [boon]

Sn" (Ull

10 ....

O.lll±O.04. O.2±O,l

IOd

4 ± 2mbam

II.9±O.5

SbJII(D.7/J)

4.I ±O.S

,. d 3.Smin

21 miD.

TeU"(18.11)

'JO ±70 U ±0.9 4.IO ± 30 U ± 1.3 UO ± O.lO O.8±O.2

TeUI (31.79)

O.3 ±o.a

Tell' (U .49)

O.5 ± O.3

[11' ( 100)

Ull: ±O.~

T~ (IU9)

.;J.

.x.

lUI (l.7 )( 10"y) p.q (8.1 d)

XeD' (G.OGe)

x..- (0.080 ) X'-lf.(8 )

Xe- (2U 8) Xe>JI (215.89)

8.8±U O.19± O.03

O.03O ± o.o13 O.03O±O.Ol li U ± O.6

nO d

1.1± O.li

.. d

6±3

H Od 0.090 ± 0.020 O.90 ±O.llJ U h (98 % oI llOd _ U h) "72 dmiD o.ol6± 0.006 0.16 ±O.
I

25.0 min 12.8 h

t.n

701 ±1

Jte.UI (UII) x..- (28." 1

XeD' (1i.3 d ) XeI" (10.") Xe1" (U 3 h)

1.O± O.6mb&n1

60d

1.3m.iD

Tell' (U l )

.... ,, -". , .. ,

U ± O.3

IIO ±.. <• <•

4olI ±14

Ud

11O ±11J O.2±O.1

U Sh

O.2 ± O.l

Um.iD

O.lli ±O.8

190±90

. e.

(»l" (100)

...

or- (1.8 xI0"1 ) 0.- (215.81)

2.9 h O.OI7 ±O.OlH 2.3 1 3O± 1 (.. 911 % of 2.9 h -+ 2.3 yl 2.8 )( 101., 134 ± 124' 13.7 13 ±"<2

• - (0.101) . - (0.087)

u,

:&U' (7UII)

...

st.O± 1.0

~ (2.31)

. - ( 7.81) • '" (11.32) • -(86mbl) &- (12.84)

Lalli (0.08t) I6- (tUII) 16- (..0 b)

3.8 ± O.6

20H

(2.72 ±O.Il) '10'

Xe1" (8.87)

. . . . (U2) • 1II(UD)

To [bAnIJ

(None 0110 miD -+ 10 d )

-oSb SbDI (67.24) .T> TeD' (O~) TeW (UO) TeW (0.87)

(",,",1

...... ' >ld

Bo'" (lUll)

_ (-.l

"

" .... 12.0d

. ±•

U ±O.t O." ± O.• 5.1 ± O." U ±o.I

...... 12.8 d 18mm

...

3.n

10 ±1

H '

0.6 ± 0.1

4±1

12 ±"-

U ±O.8 U ±I.O

7 ± 1"

_ -.

z....". ct. ,

:r.,

•e. ee- (0.18)

26±"

Ce- (O.26) ()elOI

..1... 1....)

'H

(8U 8)

eew (l U71)

....-_. ""'''''J

0.156 ± 0 .08

1.0 ±0.2

Ce"" (32 h)

_PrW ,._0. _(Ii _._ _hi-

"N'

Nd u ll (2'1.13) Nd'" (12.32) Nol'" (23.87) Nd hl (U i ) NdNd- CU t) .Nd- (5.00)

ur.n

..... Pm'"

18±1

32' ±10

1I.0 ±0.e

OO± '

10 ±1

U± I.o J.O±U

(1.5 y )

• 8m SIZl" " (3.16) Bmw, (16.07)

SmUll (I U 7) Sm- (13M) 8 m-

(7.• 7)

BmW. (73,1

.E.

lU ± o.J

87±8O

Jo.OOO± J.OOO .

I J8 ± .

Eu W (47.17) Eu w (13 y)

7,1OO ±80

Eu Ul (52.23)

"'" ±"'

EllUl (1.7 , 1 • Od Od"' IO.2O) Od- IU lI) Od'"" (14.'J3) Od'" (2IU 7) Od- (l U 8) ad.... (2-f,.81)

-..r>t Dy....

....

IUb. IUd

.. ,

U±U

0.007 ± 0.006

0.8±0.3 0.31 ±0.10 o.N±O.cl6 t.O ±G.7.

1.8±0,6" 1.8±0.6"

1O.t± I.o

6.0*0.4··

240 ±60

'76±' " 1' ± I "

18*'·

I.8 ±G.I 17 ±1.J l.5 ± o.J

"..., '"

U ±U

""

.

IUh

'., 1'"·'1

. ". 16. d

M,!OO ± 1,lkXl

"-Y

242,000 ±4,OOO

'.,

"±3

(G.062)

OO ± ", <,

140±40

l,tOO ±3OO'" 6.GOO ± 1,600"'

ol2O ±l00" 1,&IO±40014.000± 4.000"'

< 126

7O.ooo± 20.000"

18.0 b 3.8 miD.

l eo.OOO ± OO,OOO" 3.8 ±0.4 0.8 *0.3

1 .0d

626 ± 100'"

>"

:0,-- (0.090 ) Dy* (U98) Dyl&l ( 18.88) DyIOl (26.63)

Dyllll (24..97) Dy* (28.18)

Dr'" (I .

.,J!!. .E<

mill)

&"' Er*' (o.I3e )

Er''' (1-") Er- (3U) ~.

" ±I

..".-,

U ....

2.000 ;:1:200

13.....

800 ;:1:100

,...., 10'

OO±1S

8.U

2.0 ;:1:0.(

0.000 ± 2,000" 2.03 ;:1:0.20 1.&5;:1:0.17

(22.9 )

Er- (2"1.1)

:£rl'" { I U j

U_ + .±' ,~,

......,'"

O.8±o.J

.. .., IUd

1$ miD

,.,

OdIM (2U10)

• Tb .",.. (1".l ~ (73 1

,,1... ". :n, ."

~800 ±goo

Sm lJ,ll 12U31 SalIM (22.63)

Eu Ull (161)

,...,"

_ 1-.) C'-nl

•••

_ -. __l" f'''

.....

'l'ra IW (129 d )

.Vb

n- (0.1.0)

TmIll (l OO)

AppttDdU. I -" ' J I"''''I

!2'7 ±4

.29.'.

_ <.J (banll 130 ± 30

1.9 1

l tiO ± 20"'

32'

11,000 ± 3,000 '

YbI" (12.73)

101 b I.8 d

"'HI

Lui.. (91.~) Lui" (l.tIO)

3.7 h e.8 d

4,ooo± 800

Vb'" (3.03) Yb'n (JUl ) Yb"I (21.821

if lbanl.l

, ±3

YbI" (16.13) YbIM (31.8f,)

n""

.Bf HP" (0.18)

MPH ((l.I lI) HfI" (18.39)

HI'" 121.011) Hf'" (l a.' 8)

{36.44' -_. - - -T..Hf'M u, (99.1188) .T.

""HI 'l'lI ± 10

116 ± 15 14 ± 6 21.0± 0.1

W1U (28.4)

W'" ( I U) WPli (30.8)

11±1 2.0 ±0.3

73 .

WI" (Uh) JleIII (3T.0'71 .Re Roe'"' (eU3) JleIII(17 h i

"'

" ±3 IM ± 8

" ±.

0.... 10.0181 C.-IUIlI)

0.... (1.6') 0J1M (13.1 ) Ot'" (18.1 ) 0.... (. .. ,

,"

-

- _ ..

160 ±160

'±6

1.2 ±0.G

Au'- (Ud) Au'" (U lI dJ

I"" -.,.

37±3

0.1 ±0.1 4.0 ±0.6

GU ± 0.3

Ud

UmiD 1'.1 _ 31 min lI .6h

-

...... I .U 3.16 d

..-

2.20 ±O.~

70 ± 7

.. ..

H I

0.022± 0.001

"llO ±' ±oo

.. y 18.0 h

1'\'" (31 miD)

,Au Aullof (100)

...".

10 ±10

M' '" --

Umia

x.- (74 dl

Pl.- (1.2)

185 mID.

". 700 . ".

tr- (38.6 ) U- (IU ) J>t1III (0.0 12) P\.UI (0.18) Pt- (SU) Pt- (U.1) 1'\- (205.4)

DIm

10 .... 18.0 d

0.- (4 1.0) or- (31 h)

n"

10 ± 3 0.030 ±0.01 lll d 19 ±'I' (.. 96% 0 UI.4mln _111 d) lI.lI d 11,OOO ±2.00018.4

I'"

WUO (JU)

. 0.

.

...

"' ±'"

W- (0.14)

36 ± U

J.MO ± 1,000 16±16

T.... uu d)

.w

lUS± 1.0

lJ O±lO

<3

< 300 0.008 ± 0.002

H 3 U1 ± O.• 600 ±300 "" ±'" 860±100

700 ± 200 130 ± 30 0.78 ±0.10

llO ±oo

0.81 ± 0.0ll O.069 ± o.ol' 0.028 ± 0.006 U ±0.8 16 ±10 GO ±IO 28,OOO ± 1,200

30± se-

U ±l.O

-

_....

loocopo (~ .

7',1

T,

.... (• •) [bu1:l ]

-.r "-.rr -.r

.Hg Hglll (0.1'6)

H,:lII (10.02) Hg1" (I 6.M) Kg" (23.13) Kg"1 (13.22) iii""' (29.80) lIg* (6.~)

_

(••) lbanll

2,OOO ± SOO-

< 60' < 60'

IUIlllill.

3.8 ± 0.8 U 3 ± 0.IO

lU ±0.9 0.80 ±0.08

2.7y Umin

8±3

.,Pb Pb-(U8) Pb'" (23.t1) Ph'" (22.6) Ph'" (62.3)

0.8 ±0.6 0.026 ± 0.006 0.70 ±0.03 < 0.030

6 xlO' y

0.7 ±0.2'

3.2h

0.0006± 0.0002

• '"

O.o:u. ± 0.002

6.0 d

0.OI9±0.002

"min 1l.7 d

< 0.2' 0.72 ± O.O'l'

.Ro

n "(29.60} '11- (70.60)

Bi" (IOO} & - (610_)

R.n- (3.83 d)

IOlbanl]

3.IOO±I.000'

".

..TI

...

T.ble of Thermal Neutron Oro. SectioM of the I8oloopee

(Ra...)

O.IO± O.03

' ±1

T, "Ra Ra... (11.2d) Ra" (3.M d) Balli (1,620y)

__._

~~~.7 y)

ttAo

Ao""(22y)

10

130±20< 100 12.o ± 0.6' 2O ±3' < 0.0001 5 I~_ 36 ±6' U3h 620.±60 .__ <2 3.M d IU d n.2 min

" __ 6IO±~

<'"'-, IC

"Th Th'" (18.1 d) thill (UlOy)

l.600±I.OOO :; 0.3

Th-

(8.0 x 10' y) ThIllUOO) [U6 x I0It y) Th'" (23.3 min) Th-(2Ud)

P.-

26.eh

36 ± 10"

:iii O.OOI

23.3 min

7.33 ±0.12

< 0.0002

24.1 d S IOmin

1,400 ± 200' 1.8 ± 0,6'

16 ±2' < 0.010

P.-(17.3 d ) (3.' x 10'Yl

27.' d

P.... (2Ud )

6.7h

(e..~'~h)!... I

.

_

_

"U

U1" (21l.8d) tJ- (4.3d) tJIII (73y ) 73 ±lS U'" (1.61 x 10" y ) UIM (0.0067) 106±. (2.42 x 10" y) U. (0.114) 0R0 ±8 (7.1 X IQly) (UO x 10' yl

,±,

. _ _._ _

1.62 X10' Y 300 ± 200" 2.62 xlO'y 62 ±2 7.l xlO'y 9O±30 UO xlO'y I07 ±6

e.t a

12.e ±0.2

700 ±IOO < 0.1

1.18 miD

P.-(UX2) (1.18 miD)

U-

.. 1

I,OOO ±260 O.QIO ±O.OO6

1.31 d

P.'"" (1.31 d)

(1!:~J

_

" ± ll

(7.3 X 100y)

Th-

uP..

I

[bull]

e± l

••

:s~ ~.o~_+ oo

_ _

26 ±IO 400 ±300 8O± 20

m ±. :;;0."

10 ±2


...-

Appmdis l ..... 1....

l'joI

.... llfo) (Mnl

U- (IIll..3) !.7 1±0.02 (4.60 X 10" ' 1

U- (23.6I1l1Q)

,.Np

N~(U cll N~ ('7,.60011

'P"'

10"11'

11O ±1

(J .2 x NpO- (2.10 )

...

%3.61ll.iD.

".

Pu-

(4.8)( } OI y )

hili (13.2)'1

Pu-

(1.7)( 10" 1 )

2% ±6-

!.IO d

10 ± 0

... ±,

13.1 ,

l,tOO ± 80

U X)0" 1 4.• b

30±'

7Jl dO",. lOb

IU d

(10 III

Amllo:l lU I y )

030 ±"

1&.0 h

100 r

« . ..

......... · (181101 AID.... (100,1

8,OOO± I,lXlOl'

X J()I'1 )

.. """

"" ±.. 390 ±'" 19 ±. I

~*-!~ -

76O ±80"'

eo ±...

01. 1.0 110_ 100,)

'14 *4

. ...

( U X t o" y)

Bk- {2UOd)

900±_

3.1 b

"'± 200"

lOr. _ OOr

270 ± 100"' 1.&00± 1,0»" 3.000 ± %,000-

::1

a- (66 "~!JII CP'

E- (20 d) E- (t80 d)

2,800 ± 800 0.0 19±0.003

J.700±eoo-

... u.

UQO±llOO l U ± Q.J

' 'll. J ± U

U ±O.lI

O.O3O ±O.~

l,010 ±13

< O.t

1'70 ±8O"' 1.8 *0.3-

Cm" (112.1 ell 2O± 10'" em- (35,> ...± .... "lO rr "" ±eo re ± 10"' em'" (18 , 1 I>< }l)IJ em'" (I )( Ul", ) 1I.lhcl 1 llOO± 1000' c.ICI ± 10"' < 10' , (8.8)( }OI I I 8± 4c.-

a'''''rl a(lOr.!. a-(... 11

14± 3·

1,800 ±- l oo

%."x )OO 1 0603 ± 10 1.1 )()ot, 3 111 ±18

(1.11 )( 10' 1)

- ..A A .-!!:! --

< 0.0006

900 ±300

1,030. 1±8

Pu-

.!!-

~"'"

170±34

Pv- (,U 8b)

..

2.7. ±0.06

.... , (Mn)

3CI ± lo<. %6 ± 16eo""" « 6 % of '7.3 11U.ll _ eo min)

IKIUI,)

It... )( 10' 1)

." .a

(110)(.....)

PI/.- (1.71J1 pq- (.e I ~

. e-

_

'U mIn

Npa- (U ell

. Pu

"

"±7 <2'

300 ± I ISO"'

< ...

--

3.l ± 0.2

UCO± I.ooo

8,600±600< 0.015

<6'

7oo ±60 1,800:100

ooo ±_

Appendix II

Table of Resonance Integra ls for Infinitely Dilute Absorbers (from McABTHY, PEllllWO et al .') The following table, which 11 reprin ted with kind permiaaion of the Argonne National Laboratory, contain. meeeured and calculated valuea of reaonanoe a bsorption, acti vation, and fiMion integn.ls for infinitoly dilute! ebeorbeee. The tabulated quantity 11 .-.

.. f.. O'.(E ) -z-

1. -

a Dd t hus include. the I JI1- p8rl" Tho value- of E.u a nd E. depend on th e lIJ:peri. mental conditione. No att&mpt b.. been made to reduce the u perimental l . to t he .,.a1uo 8 . - = 2 Mev, which . .. propoeed in Soc. 12.3.1; however, thit neoglect 11 permiaaible eteee only • .err emaIl eontributWn to th e reeonanoe integn.l Item. from the high IInergy region. For th e calcul6tiOM. E~ _ 00 . M . .WIled in most CAlI08. A nlull of N. w.. eetima t«l for each IIxperiment .nd is listed in the table. The va lu611 of E. employed in tho calculationllare alao listed i 8 . =0.44l1v wu used in moet cuee. Measured. valUeI are listed a.ooording to the mothod by which the meaeurement w... made, i.e ., aooording to whether th e actintion or alwiorption method wu u.ed. Brief oommIlOt.. pertinllot to the reeultl. are indicated by litcl'alluperac ripta.

and they are peeeeted at the end of the table . lWIfertlnoel are liated at the end of the tab le.

._. ..,

,

. ... 114~

~~-

LI B N

...

N.N."

~ ....

.

po

8

100 ' 00 ' 00 ' 00 ' 00

. 0...

--.

......1....

280HO U± 2.4 U± O.5

...,

0..30 ±0.01

0.' 0._ .. 0.18

' 00

a

...

< 0.18" U,

<S

0.'

,"

12.8 ± 1.1 :U ± I .I

Cl

K Co

-~

_...... _ .

'"'

...

0.'

0.49

0.49

0.'

O~

.. O.

0.'

O~

0.'

0.49 0.49

0.'

,,, , •, 1

I I

I I I I I

I I 1

1 l4cABTIIT. A. E. . P.J. Pa.alUll, B. I. 8P1X&AD ud L.J. T&JlI'LDI : Araonno N.tloaaJ Laborat«y. RMetor Pbyac. e.:-t&nlol Ceo1' - NenleUer No. I (IH I): Puauln. P. J ., J . J . K.t.OIoJlOv" aa4 A. E. lllcAuBy : Ibid. No. 10 (1M3) . _ _ _WIN, M.........., , -

n

_. ...

."

AppeadiI: n

'-. •

So" Ti Ti

...-

100

... 10.7

..... '"..... ..... ..... ..... ..... ••

.."""••

to.'I.

. U

100 100 100 100 100 100

16.7± 0.&I

"Ni Ca Ca Ca Ca Ca

""" """ . """ """ """ Zo

ze

0. 0.0." 0. Iu" So"

x.x.x.-

U'

0...

.

"

81::1::41

........ ......"... .... "'3 .... 100

,... .....

1.3::1::0.•

75::1::5

72.1 ::1::6

113

l1J::I::O.07 1.11::1::0.1'

• •

.....

100 100

U ::I::U

0.411

u±u 33 8.8::1::1.2

n 8±H

' ,0::1::1.1

" •O_ ::I::o.oe

... .... ....,'1.."

..... ",.

l O.O± U

U ::I::o.& D.N ±o.I6

0.73" O.U ~f

I 3

I 10

•

0.'

,

0.' 0.'

0...

I

31 I

U8 M'

I,..

.

0." 0.'

O~

U ±0.8 l1.i'::I:: 2.7

•• •• • •• •• •r •• •• ,,

31 31 I

O~

•

1 I

0." 0.'

0."

1"

. 01

M

Me

5.1 ::1::0.1&

U

........ 0.' 0.' 0."

u

U2::1::O.tl U U±0!3

0... M

0.' 0.' 0.'

11' U±O.6 3.1::1::0.3

U... U...

... .

'"'" '"'"

U

•.r

"" " """ """ n ...

Y" Y"

10000::I::o..s

.

M.

11.4 11 ••

U± ....

100 100 100

..... ........ ..... Se Se"

0.'" M

U± I .l

IU ±G.I

In)

0•• 1l

'HM

I U±O.II'

.... ... .-. 0.' 0.'

4.1"

Y.

""" N. """ N.

...

"""'

U ± O.8

V V

V"

.•

........

0.41l

0.'

0.49

0.'

M

o.•e

M

0.• 0.' 0.' 0.'

0.41

... ... 0.' 0.' 0.'

0.' 0.• 0.'

• •, I I I

,

, • I

13 13 I 13 13 13 13

• •

13 13

I I

13 1

••

32

_...... . ..- ...-

~

•

z.o z.o

Nb"

Nb" Nb"

.... Nb"

Nb"

II, II, 110

11.23 11.23

'00 ' 00 ' 00 ' 00 ' 00

.... .... ..... .. .... .... .... ....

.

6..f. ±U'

", }4..6 ± 2.3

..T'"

.'"

RbRb-

33." 23.711

fUI2

'00 ' 00

...±....

...

3U ± U 107· ·..

10.8 ± U J.'1ll ±O.JO

...

2,816±126

.... ">,.. ".

....

SbW

T. T.

To'" To'"

"" To'" T.To'"

I'"

p-

I'" I'" I'" I'"

0."

..

t.&30±I33

1,1d-

1.4fOb·1

3,'Ioot'

3.1(ll)tl••

3,300 ± 860

'-I ll

3.l1lO4

871...•

''-

..",

1.lb

-

5.3.... .

6.'7±D.7

0....

... . '38

10&±13 1I11 ±12

6l:t.O ±e.o

e." 18.71

..

,

38±.

'2'

.... "'" ", ' 4b, J

~

' 00 '00 100 100

,10.1)4

1,91Ob,1t.

0.... U J 31:111

1M".• IM.td

71UII••

32.97

..... """ "0."...

.

2U&' ".

""

IDW IDW IDW

Bb Bb

"" I~U'"

1,160

BoBo-

Ulb"

II.UG

~- ..... " "',.,."'- ..... ,.,. •."", ......"" ...... ..."

..w S. S.

....

n ..f,G

IIUli

... .

""'"'""'

8.82± 0.$6 U ±I.34

15.'10 1&" 0 18.6

11-

...

130±18

'03'

> .f,p. ,

....... "",

....,....

1Mb-,

...

I1S-

.... ...... ....••'"'• .. ... .... "

~

13

....... ....

..

0." 0." 0... 0.' 0." 0.• 0.' 0.' 0.' 0.' 0.' 0.' 0."

a

33

" " ,•

.

..,• ..,,, ,, ,

a a 38

..... .", ,• .u 0."

.. ••• ....

0.'"

••• O.olD ••• •• 0.'• 0...9

0.'

•••• •• •• ••••

....

.~

0.'

, ,,, "",, , ...""" ", a a

7

1

""

W

-

x..-

"'"' ,,,,-

., ...... ,.... ~

.... ....

A~

••_

21.18

... -- ,., "

00"'

100 '00

--.-

AppeadaD

400 ±26 ' 8! ± !

I

....

I U ± I .1 II

M .II11

<>-

""' "". Nd

Nd lt•

"" "".. ....

..

Nd-

Om Smw Sm w SmSmSmSm-

Sm-

..Sm-

SmSmEo_

Od 11>'"

~

' 00 '00

1'l ± 1.1

z.oO ±G.2&

U.4 ±S·

....

. ..

< 21Sl>

12.32

1661>, 11

lU lU lU lU lU lll"

l1l"'

T. T.

'3' '3'

... > 1811l, t

> %'731>, 1

...

U 20± 13O (to PIa"'- ) 1,6:4.7'1 1,790 ± %70

l.5.0'7

<'' -

16.01

IS."

,.......,, ,..., ..." ""

2M ±2lS

T.n

41:77

.... 100

1- ,M'"

.<,"""

3.2'"

" ±8.0

.'"

1,)80 ± 12O

, .....

.. ' 50

",

..,.. 2.373'

U40 ± 1110'

W±50 "'±" W ±15

"'±"

1,7P '

I ....

1....•4 94.7.8 2,8104 l,u&.8d saUd

2,llOO ± eoo

2,11lO ±'*'

2,OOO ±SM

0.18 18.•

'....

2 ,2lI84 IllG.!)!

7,n04

'.-

" ...

l UI

4!\l .'1l\

'U

" O' 1.110-

'''±O

....... .... en ,

0.' 0.' 0." 0.' 0." 0.' 0.' 0.'

Ull O~

0.' 0." 0.' 0.' 0.'

...

7~"

12 12 17

•

....•, .• Ie

12 17

,

..••

U

••

.. .,• ...... .•• " 0." 0." 0." 0." 0." 0." 0... 0.'

17

0.13

0." 0." 0." 0." 0." 0.' O~

11O±70

'.....

_

0.019

1l7s-ell

3,249'1

1.47

-- ........ Hf'" Hf'"

J94b.•

(10 Pm,1OO)

",

. . .. '"......

....

2<10'

~,

1.1OO ± !60

i§ ..... ..... ..... ~

"""'"'"

OJ

... ..••. ... 0." 0." 0." 0." 0." 0."

II

,• ,,•• " Ie

,,,

- ...... ""-- ..... .......

.~

' -'

•

TaW

W W W W W WW WW

,...

W'"

30.'

""' ±t3 310 ±60

37m

7.6 ±2.0 420 ±IOO

3M

"'±" ... ±" 1,1110 306

00

b

......

A,w

A.A.A.J!<

Au'"

es.s 61.11

Tb-

T. T'-

ow

v-

vow vvv-

'-"" ..... o.s

100 100 100 100 100

p.p.p.V

vvvvvvvvv-

l ,lI33 ± 40

'"

p,p,

Tb-

1,370 1

"1-"'"

13 ±lI

~

Tb-

3'-

... 1891>,!

100 100 100 ' 00 100

n-

B;B;-

180 ± 20 2.ooo ±490

320

333." 631.04

..... ... e.a

"'...

...... " 0' 7.In" n i b,)

'.....

0.07'

.....

.oc> ± 100

1,660± ti6

%24 ±"

to to

.,

..•• 1

•• • re, I I

3 3

10

~. O~

7

•• • , •• ....... • ......, "• ... "• I

3 3 3

dO

20

0.' 0.1

128.1(0)'1

""ll'

0.0067

1181.' 188.3(0)'1 2fU.• (I)4

0.714 0.714

... ... . .. " ....... ,. .......... ..." ~.

812il)'

"",n 2'7 1(1)

31

~

0." ~

0."

I

214 ± 11 tfl

.~

""I»

'l6 ±0"

360 ±lKl

0.' 0.' 0."

to

20

8 13 1~

0.714 0.714

0.' 0.' 0.'

37

0.'

IOO ±4 Cll) J.ooo± 200(o+ Q

0.'714

~

0.' 0.' 0.'

0.13"

Tro ±lKl

...

0.'

~. ~

I

••

O•• i

••• 700 ± 2OQlr;

0... 0." 0." 0." 0." 0." 0." 0." 0... 0.'

1,691b••

0.1

"soc ± ItJOk

~.

0.49

7U ± tl

BOb,"

._........, . .... 0.ol9

389.3 4

14.4

W'" W'" W'" W'" Roo

,..,.-

3M'

290 ±"

.... ".....• ""..•• .... .... ..... WW

""'""'"'

"'±'"

"'± -

21

...

-

UUU-

....--......-.... -.......--..... U-

N"..

_

Appemli:J; II

."• ......... ....

.-. ...S ".S " .S " .S

Am" Am" Am" Am"

280 ±I(llI:

281±tl)

.-

""""""'

30" 8Mb, a

271l ±'·

3,2eO±il8O

111.2'1 181.6 (0)4 280.2 (1)4

1,137 (0)4 1,&&3(1)4

3,GOO ± 1,000

,.

,", ....

0.' 0.' 0.' 0.' 0." 0." a..

a lO a lO a lO

30 S

."

S

" 20

(1l+ f) lI

327 ± 22(1)

O~

8..... 8._

0."

1,0344

0.' 0.' 0.' 0.' 0.' 0." 0." 0." 0." 0.'

8,400 ± 1,100' 1l.46CJI>, • 8,700±800 9.000 ± 3,000

.... _

IO,OOO ± 2,800

367 ± S3(f} l,2'l'5 ±30

1,614 (1

··.,W 1.27

2,t90 ± 1JO

0.133 0. 133

" ae

3

.."

27

30 30

30

Comments . ) EAimated from the paramllltera of th e firet large resonance and the t hermal eroeeleCtion ; doee not include .. correction for unreeolved levelll. b) The reeonanoe integral given g not aignifieantly dependent on what out _off m wed since the material b.. .. CI'088 &eOt.i.on dependence that is ol0&8ly I /fJ in t he cut-off region . 01 ValUN were deduced from m e&lUJ'MDenta in. the Dimple Mazweillan epeetrum. and with the Gleep oecll1&tor. d) Valu. oomputed ualng IBM 706 oode ANL-RE.266, in oluding negative enlll'lY lenll.

III) Ettlm.ted from level parameten ; dON not Include to oorre otlon for unI'tlIIOlved 101'&1.. f) Calculated from the parameten of th e finJt rceona noe only ; includes un reeolved ~D&JIoe contribu tions. g) CaJoulated from parameters given in BNL·32lli ; includes unresolved level oontributiona. h) Mouuremon~ on aingle eolUtiOfUI only ; eoeeeeted for II(lreening. i) A J'MOnaDoe near thermalleadJ; to oonaiderable dependence on the detalla of the oadmlum abtKwber that ... u-N. j) Only one Mmple oftbMe materiala wu available and the 118timated IOI'Mning wu large. ValQll& Uated mat be treated with caution. III VaJ'QflI preferred by the anthorl after analyail of the available data.

'"

I) Value for .. I.mil foil; the value for .. 2·ntil foil ill 16.5 ± O.6 barn. m) c.Jculated from level puameten and the th ermal eroes eectiOD i inoludee an unreeolved levellXllTeCUon. 0 ) Caloalat.ed by P . PJ:R8WU of Argonn e from lovel puameten given in Supplement I , Second Edi tion, BNL.W (1960); includ e. unreecleed le't'e1 001" rectiou. The ently for Dy'" includ. the bound Myel. The p&ramet.8n for thi8 level _re obtained from R . BHn (prin te communication). 0) Calculated by P. PJUUIUIU of Argonne from Jeye! puameten given in Rof. 28 j inelud. Ion ~,.ed InN ~. p) c.Jeulated by numerical. integration of the fi.lon era. ~OD. q} Pre liminary eatimate of . · .Iue &lUi NTOr. r) A goki r'IIIORAUOO int.egr&l (inoluding the l /"·pu'i) of 1634 hama .... u-t ... atandard. I ) Caloulated hom level pare.meten in the second edition of BNLm or Ita wpplement (1960). The Dumber of Ievele listed ia the Dumber of f$lOn&ntle lel'~11 for whioh .epa-rate ca.lcul.tioDl were ee.rrled out. U th e number of reeolved leve1l ia three or more, the reeonanoe Integ,.,l listed include. .. contribution from the unreeolved ~nanoee calculAted UAing aver'ago reaon&noe parameten. t ) An avenae nlue involving utn.polt.tiOD .Dd toreening oouidorationl.

Referenees 1. 1UanD, R. 1... aDd H. 8. Po.Ka&Jf(m, Jl-maw- c.p&an IntepU. Proo. 1. IA....

a-...

CoaL Pe&oelal U_ Atomia EDerc. Pj833. '. M fla56). I.lUDrU'TO", V. B., III:Id V. IL 0 1lUU"f': Some NMltNQ AbeorptioD.lDWIpaII, .1. N IWIiMt ~J'7 (1ll6t). 3. T.f.TfDUU.,R. B.. fllal. , PiJe~ M-tAoI ~ AbeorptioD. IDWpU AERE-R- 188'l' (Alii. 1868). &. Fmen, r., KAPL, penonal oommWlbtion (IMI). 5. D...m..uo. a . lIlII.: ~.. 01 Some ~ ActiYMioa 1Dt.egrU. J. NDClI.r EP«o'. 1& (No. 1), 63 (April lMI) e . Ca.t.Jrp...u.,J.L. , S...annroh Ri ..... penonaJ oomm1Uliaation (1860) (Won h, 0 .1(. J.f.CQ). 7. Srn..... P. E.• d lIl., HeMllftImea" of tM a-nanoe A ~ IDtoegnLi ,... Vwiwt lIIAt.oorW. and •.a _ the llultlpU..tiorI 00effi0ieD.' of a-ma-ae NMlVolia Irw FI8IiOD&ble aotopow. Peoe. l .t In t.llrD. Coni. P-tul U_ Atomlo EMra::r. OeM .... P{&MI. '. IU (leM). 8. FlElna. F .. ..nd L..1. EtomI, Coklt R-n&DOlIlDtegrai. KAPI,2OOI).12 (Deo. 1860). g. JO,",II'I'OII. F • .1.. d aI., The n..na.r Neub'on ere- 8eoUoIl. of Tb.- and the ~ In~ of Th- and Cd". J . NlloJ-r En..,. A 11. a&-IOO (lMO). 10. ButJlftT. R. A. : EHeot.I... R.:lnanoto lntegralto of Ca toDd All. HW.lI367' ( IteO) . 11. S..... R. I BNL, ptor-.l oo_1IIlio&t.Ioa (l lM5O). II. W.f.U:n, W. 8. , new. tond Effeot.l.. ere- 8ect.ioato 01 n.uoa. Produm. aDd PModo FiItoioD P!odocrtA. CR RP-gI 3 (llarab 1860j. 13. horn, V. I A : W - ' 01 \bto R-on.._Intepal 01 Zr-. JUPI,2000-8 (Deo. INlI). I ~ hr.... F.: ~_ ~ of M n p - . ~ ... NiobiIuD. XAPL-ZOClO-I I (Deo. IMl ). 14. C.f.lIa.r., IL J .: ne Tberm&I Ne.tNQ (&pkIre en- StoatM- ... u.. 'R-.._ a.pwr. InIepJ 01)I‫ס‬lII. J. N\I~ EQqy A 11. l1S-17e li MO). 14. LnwOOD. T.A. .lflll. l Rad.........."-l1l8tbodto Applito4 to \bot ~ 01 e:-8ea"one 01 R-olcw Intend. Proo. .QfS fa ..... Cant. r-:.tlI1 U_ AtoaUo Eaeru'.

F.Dq;,'.

Jl-oa._

Til-.

o-..... I· j"j03. ... M I l lKlB). 17. ElLup. R.IL. del. : ~ oI"1benDa1er.. Seot.loM toDd

a-.._latepaJI 01 Some I'IIIioo ProdIlCC&. XAPL 1000-11 tSept. lMO). 18. JIIoLOW. K.. and K. JOILUIMOlfl ne ~ Ia.tegral 01 Gold. J. N1IOlear E-v AII . 101-1 07 ( IMO).

Appen.ti,m (I ~). D.lI.t.LrJaIp.J.. """ Tba R-a_ lateeral far Neutroa " ' " 01. U-. 0 R.'iLM3 (Sept. lMO). M. JUJlu,.. J . A.., aD4 R. B. &sw_l Tbermal and er... 8ectJoN 01 H. n ,. NvoW. ill ProoF'- ill N1IClaat X-V. PwpmoD Pr- L&d.. l.-doD.. B.;.II Pb yai.. aD4 ~ po 61 (18M). ~ IUADT Jr.,J., ., 01. : 'I'bo ~ I'\-'on IntegnJe of 0-. Pu-, and Pu..•. NuelMr Sci. En,. t . :U1-346 IMaroh l 1EII ). 26. 8111NO., P. B.• d ol : ToW Neutron era- 8ectJon of P.-I. Nuclear Sci. &lid Eng. U:. U3-UV (February 1962). 27.00.101I11, F. W. , and M. Lo 'lll
era-s-uo... er.-

a-._

n.

ea... J . I'll,.. Ii. 147 (Feb. la67).

31. BPlI"". R. A,.; Nole OIl the HW~

P'IZ-.

ero-

R-._ lakIgrU of NatunJ Copper. ea- &Dd Clt".

(J - . IM l). at. H ~ll D, E.. If 01. ; A StGd, 01 the R......_ In. . . . 01 Ziroooillm. KA....OO (OR) lEi "0- (J_ IMl ). 33. H~. E.. Stodiea 01 the Effective Tot.aJ .00 a-.noe Abeorpt.ioD. er... ~ far Zift,Uoy 2: aDd Zirooa.i1Im. An.i... Fyorik 10. 41 (1M2 ). IntesraJ al Niobillm. A.E. -81 33. H..a..U Ta£llD. It.. aDd O. L tnlDOU . l 'I'bo (A~. 1M2). M. ~. R. P.: Allti....tion en- SeotionI al Nb·U and Nb.M. 100-16760, MTR .ETR TecluUolJ BraDobIa QulterIy Report, t:lol. l-Dec. I, 1961. S6. B vn.... J . P.. ., Ill.: The Neuuc n Capt ure Croll Seotionl of Pu.... . nd Am.. •. CuI. J. Ph ya. Ii, I,n ( Feb. 1e67). 30. T"'1T~ R. D. : n-mal ero..800tion .nd n-:.n.noe Abeorption I ntegral of Pu .... AEEW·R 1111 (Maroh 1M2). n . LoIIO. R. L.l a - - In~ )f~enla. ANlA6lIO, Re&ctor D....elopmCInt Ptoosr- ProDc- Report (JIInO 1M2). 38. K.u.ur. R. A..• .-I T. F . P..l.UtJlIOW: TbeNvcle&r Propertaeeol RbeniWD. Eight Qual1erIy TecbnieaI Ptacn- Report. Ull..i......n.1 01 Florida., Depu1menl of Nuolear Ensin-iDg NP· II D8 1n...-bet 11161). fi. Soo'l'U.U,J. J .. It. Fur. aacl D. W.KJnon: ~ At.orpQoo. Int.egrU of Dyapr1IIIiwa aDd,........ Tr-. No. N..... Soc., No!, ( INS). 60. KLoI'I'. D. A.. &Del W. E. z....aon...: Effecti" Re-oa.a.noa Inteeral 01 IDdiIUD. r oilL TnI'lL Ammoua N.... Soc.'- No.!, m . "1. 1.AJfn. P . M. : Then1Ial Neutroa c.pt.o.are en- 8eIltiorI .00 a - - c.pt1Ire Intesral of~. Nuok.r Sci. and En,. 11, &-fifo (J1I11 1M2). U. 8cIn7JuJl, R. p .. and J . R. D.....na: Neutron Alltiutior\ Croll Seot.ioD8 of Pm,", Pm.~. &Del Pm. ~ • . NlI'*-r Sot aDd Eng. I:. 61a--6%! (Apri l 1M2 ). U. HAUUDI. J •• ., "'. l 'I'bennaI Neutroll en- Seot.ton IUld R.:>nanoe Intesral oJ 8rn· I60. Trana. .Ma. NlIcI. S«!. l, N... 2, J7e-377 (Nov. 1M2 ).

.,.t:

Reeoa._

Pu-.

m

A, pendlI 1II

Table of the Funct ions E. (x) The table oontaina vel uee of

-

E.(~l - f·;'~4. ,

.Tfl, f',--I,-

0

(d . Sec. 11..2) for ,, _ 0, I, 2, and 3. A general dieeu.Mion of the propertJ", of the E. (%) functiona and ValUN for higher" can be found in the Canadian report N RC.I M7 by 0 , PLACU K.

• .00

m

.ll2 .03

... .M .M

.
.08 .08 .10

.11 .12 •12 .14 •15 .15

Ko (~ )

......'"

49.Q0993.6 32.348184 24.018'738 19.0US88 15.89llO'l'8 13.319912 11.63896' 10.164791 9.Cllll374 8. 1~7

7.3ll10lM U ..... Ul'702 U 3l1053

4.037830 3..364708 1.969119 : .G81M4 2.467898

•.""'"

2.11i0838 2.026941 1.91874li 1.822924 1.737107 1.669642

1.5,,"" U24146

.19

4.lW0390 4.:tlI2"UI

.eo

' .0Il36M

I....... U 09I 87 1.3$7781 1.:KJlr11l6 I."""'" 1.222600

3.80171113

1.11:12002 1.1463MO

."

.10

.21

.22 .23

...... .. .....

'.325ll1l9

' .1l82134 :I.IIMlll'.!1I

3.4IS«1U

3.%77618 U 16!03

.27

"''''''3 2..82"7332

.30

' .om2ll 2.580219 2.469394

.21 .22 .32 .24 .M

........

2.269218 !.I78M7 2.0Il3442 t.Ol»H U3791lO

... ........ ... .n

...

1.71Ml83l1 1.738OU

. , (%)

•• ( %)

1.10ll&l.3 1.076%M

r.1.0l3889

r.eoecoo

O.94M70 .9131Cll .88187%

.......

.827834 .0>404& .781836 .760981 .741Ul. .722Mli .7(H,7112

..,.,.,. .....,. .""'" .....,. .

....,.

.871638

.

.641039

.612842

.67'201 .662171S

.....,. ..,.,..

...,... ~""' I

.&17730

.967308 .9JOll18

..,,'"

.88 1606

.460180

.ll6833li .836101 .1:1147' 6 .794216 .77.... .1M+1I .13nll: .71N37

.461482

o.e84933

......,.,

.' 97448

.478297 .469116 ,«3010

.4347M .428713 .4188'10 .411121

..."" .....77

. '(')

..•""" .scocoe

.-

..."

.471W8

.4M919 .+ee781 .438833 .431120 .423610 .418292 .409168 .602194

........ ....... .3lltt7•

.n_ .389741

.seseec

.3li'M"9 .3li194ll .34&254

.. ....... ..""

.U0700 .336262

..",

. .4834

..3000<, ..'"

.soseos

.eesee

...... .m... ....... .2OOll37

.281976

.272442

.....15 .M Ute

... •

..

,~

...A' ..... ........ ..

U,....

U llUI

AS

U I.12

.

UA71.

..". ....,,, ....... ........ ...... ......,.. ... ...... ....... ..•., ....... .. ....... .. ....... ....... ..... .'UI_ .n 1.41_1

U,....

U881S2

1.113081

~,

UT74d l.J~

1.)10&11

l.o7tlQ

$1

UdOll., 0._1S)

_

.I I * ,

.02

.80314'

... .eo

.10

.n

.72 .72

." ." .16 .77

,n, •16

.,.eo

...... ...... ...... .. At

.In

•11

.os .os

...

....,

APJ*MiU.m

.,..... .""'" ........ ....... .n.... .7UT"

.MOI. ,

...... ,644.' 48

.115361

.eom e

.087700

$1_

...""" ......, .MIMI

..531112

......1 .61~1

.4t2Olt .U laN

,"11'" ." 10111

."17" ....... .usns

...... .. I6llU

."""" .....,. ......, .....IS

.em" .02633' .enase

. .... .........

""'I,m ....

~,

.......... "" ....... ...... ....... ....... ..... .""" .,.,., ~14I'"

..."..

..13113

.U7W7

.41Ml12

........ ..... ....... ....... ..,,, ....,... ....." ....., .4U lIi17

...",.,

.un13

.3MllG

,...... .328032

......,.. ,~

..,.... .3ll24!'I

.......

.-",

....'" .""""' """"' " ..""""" m3

.......

.321108

.Sl&e88 .110381

...m. ....,00

.

....... ...... ....,., .J86118

.27111M

.271ll88 .267278

......

...... ...... ......, .268715

. .

........ ..,.."

....,

...-

....... ....... ....... ...1lO31

.....14 .!M13t .231M! .2282G I

..

."" .22,' J,_"

.

J,_ ..... ....... .....,. ....... ..fUllS:

.renee .,..".

.181M ! .188811 .188117

.1.....

.180867 .178291 .1767• .173281 .I"lOllOO

.238721

.1eM29

.!S IUt.

....... .......

.'02"130

.217111 .213739 .210.28

.IM7117 .162612

men

.1&0081 .l' l~

.l P 178

.r..... .16M 92

.. .......

.207178

.sceeee

.1..... .1.....

.~ II

. 111'71'

. IIM760

....'03

.r.....

.1..". . 1U331 .1..... .138<30

.r.....

.174177

.183183 ,l80ft7 .1'T1eti

.1.....

.s""'" .311m

-"....,.. ........

.1lltHo1S

......, .l lt1781

.......

- -

.!:e01M ....,14

""330

.........,

.l'l'602V

.l'l'U
.. ..

.IM7tl8

.,,.... .t:S2Im

.r.....

.127f.fO ,126703

.,.'22300 .....

.r.....

.1l1l()I0

•

..... .n .. ,.00 .OS

rm un

........ ..,.00'..0.8 '.03

'.08

1.10 I .ll 1.12 1.13 U ..

l.lli U ti 1.17

1.18

1.18

UO U l

.... .... •.23

. ,C _'

. , (_1

....... .....,

.238138

""'08 ..,.,00

....... =.00

....,.,0 ...."

"'ll33O ,.,n.

..... " -""" ....... ...,..... ....

.! 1a384 .J l1i7U

.1021534 .10Il N .0CKl81fo

.'40007

.3332"1' .a.....

.,oom

.201813

........ ..... . 130168

.......

.l sm! .1829906 . 11lOO62 .1711f'J . 1701335

. 12828 1

.008&1,

.1'7166li

.1Ia3f8

-""'. ....... ....... ....... .......... ...,

....... ..... . 1636!%

. 116971 . 11013%3 . 11J701

.J3337'

.166821 .16347t .lCi l081

.1..... .1..... .1..... •1N Mof.

.t.....

......

....... ......

.1" 180

.302S 10

.~1 321

......,

..,....... .286870

.

.188828 .1&8150

.2"JOU'l'

.217217

.l m Ul .1" 6&0

1.32

."""" .

1.33 1.34

.r""'" .liM07

I.'" 1.41 l.4 %

.16M13 .161811

.1aMG l .133313 .131331 .129326

.206970

."",

.......

.

......

.1273M

. 126411

.188721

.12M 13

.r""" .1791.1

.111&'2: .11e803 .117M

.17t141 . 17SIU . I'1Ot21 . 1f'f34i . .."

.

.... ....

.

1." I." l.4 7

.r. 15641%

1...

. 151257

....

.r00334

JI6238

.213388

I."

. ..... . .....

.330800

.... •.36

.1",118

.116811 . 11"2407 . 112'708 . 111188

.100706

.J" I"

1.38

.r"'" .100708

.lI1m

.1"180 .J'2070

.223122 .221127

...,

.1.....

....,.

..sun1

......

1.3,

. 1576'l'S

.Ul8I I•

r."

1.30 1.31

......

.

•• l.-j

.r..".

. 1'81%.

'.38 r." '.38

-,1-,

. le 1773

,."""

.IIU II

114413 .11t'l'U

...... ...... .111(lOS

.t.",.,

.JOlIn 4

.1030f,5 .101520

.137ln l .1"...

•12tWo3e .IUll21

.1228311 .121078 .1I1Me

.111104

.091173

....... . .093362 .0921U1

.""'"

.0..... 8"'" ..08,,'"

........ """'" ""'" J186 1811

....... .Q81744

-07M16

.1..... .1""" .103IlOO J)ge18'l' .($'7811

.sn..... ""

.095102 .093'778

.072&10 .0'7 U17" .0707"

...... ....."

....... ....... ........ 0."" ...... , ....... .08.600, ...... ..,,'" ....... ..,,'" .ogU IH

..."..

.O'1el 0 m5131 .0'14108

•

.07 8312

.1m ' "

...... .073670

."""

......" .....

. ....., .. . .007146 ....."

...... ........ ....... .eeseo

.eeeuc

.0lIl3J 1

.000023

.0W745

.

",

.06'747e

,

• ' .00

U I • .62 U 3

,..

.. ....

U ,

Appendix III

.... . 1487," •1.....

.r.....

..lum ..... .1.....

.IU703 .132lS13

" , Ia)

..r..... 00020

......

.09700<

.o•.,,,

.osess

.09162-4 .0901S8

I

". (a)

.073101 .072108 .071130 .0'10166 .069218 .068281 .067369

.066<00

• .<JIl 1.<JIl

.1.....

.128266

.087680

.0ll881'

.""'"

'.00

.126186 .12-4 IM .122160

.086308 •0800"

.063800 .06.... .062102 .0812'70 .06001O

1.81 l.62 I ." I ." I ."

1.8. 1.87

. ..

.. 88

1.'10 1.71 1.'12 1.'13 1.'14 ) .'16 1.'16 1.77

1.'18 1.'19

' .80 U I U' ' .93

.... ,...... ....,,..... ,' ....93 1.8'1

UI

I ." I ."

I ." 1.9, I ."

' .89

, .o

,., ,,.a.• 2.1

.120202

.118280 .116311' .1I4M2 .112'723 .110937 .109183

.0838" .082613

.081' 21

........

.0'79093 .0'7796'1 .0'18838 .076738

,)04108 .102.(78 .JOO8"

.O'J_ .0'1" " .072639 .071506 .0704.90

.007'153

.ceesoa

. 107' 81 .106770

.-

.06....

.0M673

. ""'''

.06DMI

."""" .""""

.067285

..OM771

.05IJ02lJ

.OM"" .0ll3679 .062869 .062169 .OlH479

J)M'1.(() .0932'7'

.067l>:W .066579

..,..,89

.oesesa

.060128 .0494.67

.091833 .0Il00l17 .089026

.06f.713 .083802

.048815

.086233

.081'" .08831' . ..,

.... ....... .0IlU 19 .081166

.O'J8932 .078'120 .0T7629 .O'J"" .0762Ot1 .0760'7' .072981

.071886 .0'7U71lO .06trJ32 ......1

8.7M78 (- 2) 6.83126 6.036(11

.......

3.77991

..062021 ...,.

.061162

.oeosa, .06 ~2

.06ll621 .057803

....... .cse"" ....... .......

.0Mf23

=

100

.062-41' .061600 .UllOM7

.00027' .041ll1ll2

4.80006 (- 2) ' .26If3 3.71911

,a.eeosa ......

~ 173

..... ....""" .......

.Of.7539

.04.6692 .Q4,5093

.

.Q4,3920

".I .f)

.0667.-0

.060014.0Mm

......

..064691 .033206

.0421528 .0l518119 .061199

..•"..,06 """ .049272 .041lM7

.oecao .047' 21

..'" ...esscer ..... ...... . ....aeae .

.046821

.04393'1

.."., "..,

.04176'

.04.1239 .0W121 .040210 .039706 .039207 .038716 .038231 .037762 .037280 .03681'

......

.....,

.....000

. .,'"

.036010 .QU1I76

.042780 ...2m .oue72 .Q4,II29

.QUI.:! .033718

. ....... ...,.

.O:lIlMe .U:lIlO:lll .03S626 .038027 3.763.(3 (- 2) 3." ' " 2.891127

,...... 2.m13

.033"" .03"" .032-478 '<1320'73 .0.11e'1l' .!l.1 12Hll .O:lOH9of,

.03061l 3.013:U (_2) 2.66136 2.3:52m

.,.""""......

•

....... U

,.s

••• ,.e

,., ,a.e.• a.i

3.3

U

U U

,.• e.o

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••• •• .., •••• •• • •• • ao

.. " U

3.3

..., ••• "u ••• e.e

.. •••... .... e.r

U

I

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............,.

3.28:UO ( -I)

2.1'71'1' U lM'M

U GH'

1.46320 1.27382

11.17072 (- 3) 8.81l167

,....., e...,.,

8.112782

1.8

, .r

., , ~

"....... ....

U lI I4e (-I)

1.01 330

8.9390(, (- 3) 7.89097

8.8'7014 1.IIIOU

6.""82

U3601 2.07340 1.84101 U1MU U 6M

1.71453 l.61D11

1.:W759

1.19Ufo 10.e088 (-40'

...... ~"" 181

•."'"

'-"306 6.870 1 5.2199

' .0431

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3.11708 3.2733 2.010&8 :1.Ii1lM

•.eeces

•.

U8716 ".,.

- "'"

2.961178

%.83291

-

7.1980

•...... . ...... ' .8008 ...... e,...,

' .0318

3.1109

2.6647

...".

U'"

U 211

l.2022 1.1821

U~

, .e

7.31"

M t .•

e.""

1.1168

...... 1.4410

1.1MB 1.031'7 U l llll (-6) 1I.t3K7

,....

...,.,.

.""'"

1.1U3'7

0.89306

0.711290 O.70U5

'M'"

,,....... 6.801811

,..... " .... ....... ' .00383

1.62878

....... U..,~

1.7'7889 1.68321

O.lXl&fo7 0.8881 2

1.Q8e2 ( _ 4)

'.'"

UZllM ( _2) l .oWUlI

1.267" (-3)

7.38433 ( - 3)

l.29 148

....,

I

'.83368

1.1'll3O 1.02130

I ....,

l o.:J60'J ( - 6)

1.&41"

I ' -Ill!

r...... r."'" 1.11831

U"O

1.8379

1.74430

1-OM"1 O.9'IM

2.46867 2.18518

U 3517

U 7977 (- 2)

I ." ' "

2.70030

".5781H

,~

1.304840 1.1"lU4.

.

&.ID023

.

.......

U I81t

...001:2 167038 :U6648

6.tIln06

...... ...... ••• ,. ,,..•• ...... ,., ...... ,., ...... •."" u

I

.....

1.t173 ( -4) 1.<)691

U IS8 6.0 118

...n m

....... ....... 6.50190

UI8eO '-48310

...... .. """ "0082

1.'76130

I.1t l 58 1.96316

1.'4110

..,'" •1.10Un ...,., I ......

. .....so

.""" ......

0.9978 (- 4)

................. ""81'7 um

.................... . 3.9922

$.&&U

U 82e

...... ...... .... a.l830

U T63

........1.8211

1.8115

lAin

...... ...... 1.....' 6

U li73 1.0341

1I.2lI31 ( - 6)

7.0111 2 . .Mlll

e.61131 6.8869

6.(1363

•.seaa

U I26 U aM

".7O'l'!

60247&

4.210f,

U IZ2

U ...

-

I.J884

•.

1.0" 1

c.....

1.6100 ( - II)

............ fUMI e.oU7

u ""

"'S2S

:U 71IO

.

AppmdisIV

•

...•.,

"(t)

'0

...

U933 (-6) S.1414 s.34H

...,.. S.DIM

2.8'1'10

......

,...,.

UIM

U916

'2.lfOli .2937

Ue21

UI48

1.1331 I.MI9

1.9689 1.1M2 1.6810

......

..•••, ••• ,.,•., ,., ,.. •••••••.,

U"'

......

.....

1.8191 7.0661 8.3179

......

8.....

1.Il/lO 9.9881 ( -e)

,.2.'19409

202461 2.0138

U067 UI92 U~21

UI69

r.302<

1.2700

1.1682

1.1384 1.O'lOIS 9.1492 ( -II)

0.....

1.<1479

,.....

U331S (_ II)

,.... ...... ......

8,0119

.....

8.18911

8.0921 6.46'1'1 4,{I071

......

5.9180 U061 4.71196

6.'1'109 6.1127

......

S.1I81 (- 6)

8.2033

1.1848

•.s.eee, ...,

_,(S)

UI38 ( - 6) 3.06'13 2.'1384

,.....

1.3'112 1.2211 10.9826 ( - 8)

••• ••• lOh

S.1686 ( - 6)

I._

1.1129 U326

' .0

•...,

",(tl

, .eeas

,....

,-

4.1570

AppeDd.l:l IV

Tables oftbe Functions '1'. (v, ~) and '1'. (v,

~)

The tablee oontain valUfJIl of 91.(11', Ill. 9\ (11', III and 'PJ'P•. These functions were introduoed in Sea. ll.2 [ef. Eq. (1l.2.2la) and (11.2.21b)) . 'Pt('" P= 0) is identical with "'(11')-1-2&.(,.).

'i-..

...

.

...... •• ....., ..... .....

0.0 03

0.' 0.' I» U U

.

'.8

U

.0

J<

••• J!

so

.

IV. 1. Tala" 01•• u

••

0..... 0.81" 0.7114 0..... 0.lIlI91 0.lU47 0.1lO81 0....' 0...., 0..... 0..... 0 _ 0..... 0..... 0.4911 0....'

0 .1873 0.1'. O.ltlOll O.laG o.I378

&

•••

FundleD 01 v aod

••

0.78011 0.8321 MOO' 0.7a.& 0..... 0.688'1 0..... 0.82'16 0..... U'1155 O..lO7. 0.3911 U31e 0.411M 0.6298 0..... 0..... 0.4170 0..... 0...., 0.3338 0.38411 O.U49 0..... 0 _ 0.3101 0..... 0.3961 0.....

....-

...... ....,..

•••

.-

0..... 0..... 0.313'1 0.4'168

0.....

...." ..... U." .....

0.1857 0.2376 O.3CK9 81080 0.1'148 0._ 0..... 0.28'18 0..... 0 .11d 0.2113 0.1724 0..... 0.3101

......

... ...

•••

0.8709 0..... 0..... 0.'1896 0.810 0.saee 0.'1188 87m 0.'1420 0.811'11 0.8'1911 0.....

0..... 0""" 0..... 0..... 0.4161 0. _ Ull4 0. _ 0.31!'J' 0.... 0 _ 0.3893 O.JJOI 0.1979 0..... 0..... 0.3662 O.llt'1

p,

0..... O.IDeO O.JOO' 0."" 0."" 0"'88 0.....

0.9397 0.8622 0.'1182 0.'1102 0."" 0..... 0.81122 0.57" 0..... 0."" 0.4321 0..... 0.liIi79 0.4* 0....' O.llIH 0..... 0.4'1" 0.4861 0.4312 0..... 0..... 0.41'13 G.4276 0.3812 0.39" 0..... 0.3801 0.3'120 0.3820 0."10 0.3628 0. _ 0.3239 0..... 0..... 0..... 0.3198 0.3296

.....

· 31

.

IV. 2. V.Jut' 01"1 " . FuneUon 01 v and II .~ . 0.•

0.' 0.'

M 0.' ' .0

0. _ O.OOlS 0.0021 o,. " . 0..... .0033

..... .. U

...••U• """ 0.0038

....... u

'.0

0.'

0.'

•0

u

u

u

u

0.0030 0."" 0..... 0.0114. 0.0132 0.0184 0.0... 0.021' 0. _ O.OOCi! 0.00.. 0.01"1 0.0198 0..... O.om 0..... 0..... 0.001

..... ..."" .... , ..... ..... ..... ..... ..,.. .....

0.00'71 0 .0133 0 .0201 e.oen 0 _ 0..... G.D UI2 O.lJ33O 0..... O.olSli O.om 0 .0376 O.OlQJ 0..... 0 _ 0.041' ecere 0.0110 0.0121

O.OIU

0._ 0._

0.01%8 0.0130 o.oon 0.0 132 O.lXW.1 0.0132 O.OOU 0.0132 0..... 0.0 132

.......-

o.o:m o.om o.om 0..... a.out

O.067t 0_

.......

0..... 0....0 D.O.. 0.0247 0..... 0..... 0..... 0..... 0.00'1(\ o.02-n 0 .03'l" 0.....

.. .. ..

0.0016

0.....

o.oe31

......

0..... o.om 0..... O.(H,tl 0..... 0..... 0.0'1<18 0....' 0 _ 0."" 0..... 0.081. OJl"l' I" 0. _ O.08tI neeeo O.o1'l t 0..... 0 _ 0..... 0.01101 0._ 0..... 0.0'711 0..... O.IOU ' .07>6 0..... e. O.07n 0. _ '0Il'IJ ' um 0.07605 O.Oll1I 0..... . .087

..... . ... ...... . ..,

......

0.0030 0.0871 ' .0030 0.0'111' 0.0871 0 _ O.I0i6 0."'" 0.0'761 0.0875 0..... a.IOGl

IV. 3. ValuM 01fP l{e,. U IlFu.DeUoD 01'" and p,

~

.....

0.' 0.' 0.'

0.'

0.... 0.007

..

0.0.7

0-010 O.OUI 0.020

0.013 0.... 0....

0.00' 0.... 0.... 0.'"

0.... .." 0....

r.0 O.ol5

•••

..,. ..... 0.018

,.•

0.021

u

u

0.023 0....

0.'" O.0IS1

.....

.. 0.'" 0.... 0....

.... ..... . ....... .... ..... .. ••• ..... .... .. . ..... ......... .... .... .. .. .. . ... .." ....•• ........, ... - ......" ............ ...... I.... ...." . ..... u

0.0 10

O.OUl

0.011 0.0..

r.e

0.013 0.018

0_ 0.... 0.... 0....

(UIIII

U

' .00•

U

.....

U

0.... 0....

3.0

0....

0.001

ccee 0..,.

0.076

0.087 0.10& 0.111 0.117 0.123

0.031 0.... 0.00•

0.• " 0....

0.081

0.101 0.120

0.138

0.117

0.100 0.11. 0.128 0.138

O.OU 0....

0.130 0.161

u r..

0.160&

• 180

O. I~

0.166 o.J7.

0.153 0.100

0.186 Utili

O.17S

0.""

0.134 OJ M

G.l"

0.078 0.100

O.lU Uti O.let 0.111IO

0.108

G.134 O.IN 0.183

0.1 10

0.110

~ ..,

O.w 0.167

O.!M 0..280

0.211 O.!M

O.2llr6 0.308

0.""

0.s18

u te

0.... 0.112

0.' "

o.eee

0.... 0.318 0.33.

A,peDdlIV

Table of Time Factors The following table Iiata J.I, e", e- Al and l _ e- u where t y meuured in unit. of the hall·life T,; J. =

-~.! . Th_ 1

data are helpful in the evaluation

of foil meaaurementl (el Sec 11 1 I)

.',

......' .03.. ....•m....

0.01

." 1._

._A,

1

. -u

o.wn o.O.Me! 0.0138

"

...... ............ ...... ....... ....... ........,." ....... ...... ...... •...... ..... ...... .. - ....., ....... ....... ...... ....., ...... ...... ."'.. ...... ...... ..... .... ...... ..cse.. ..... ........ ....' .30 I."" . ...,., ...... ' .32 ...... ........ ...... ...... """ """" .....,8t3... ...... .... '" ....... ........ ....... .."'!87 ..,m ...... .-.. ...... ....." .... ...... .mo ...... ...... ""'" 008 008 0.10 0 11 0 .. 0 .. 0.14 0.16 O.le 0.17 016 0.11 • .20

otl

• .33 ' .33

I.OI~

1.0210 U 7H 1.0281 0.8728 1"'" 1..... 1..... 0 .... 1 _ O.MeI

. .0_

0.1138 1.1011 0.1IO'J5 I.IOM 0-11013 1.1173 1.12061 1.132lJ 1.10108 I.J'87 00708

...

l.lM'J l.lM7 J.I728 1.1810 1.18lt2 1.1875 • ....1 1..... U I42 ' .lI33O 0.8171t 1.2311 0.8122

• .36

• .37 0 .. Otl 0 41 0 41 0 .. 0 .. Ot7 0 .. 0.41

oro

0.8011 1.2870 0 _ 0 ,.., "'7" 0 '"" OTm

oms

O.GtI ClO

• .08238

1.0718

0.31

• .33

0.02'74 ' ''''1

0.00t193 0.01388 O.OOOl'D 0.027'73

...".

• ....,1 007&24 0.Clll318 0.09011 01....

0 . ... 0.11090

0.1112 0.1173 0.12340 0.12i5

0 1' "

0.1414

o.J"4

011133

0.11186

o.IU77

0.13170

0 '''''

0 1.... 0.15241 O.llI9U 0.10lS3e 0.173211 0.18022 0.1871(1 0.11M08 O.JOIO I

0.15ll1 O.IND O.J707 O.11M O.IHl 0.1878 UO'l'1t4 O.IDU 0.21488 0.1~ 0.22181 0.22814 0.2100

ot1"

0.-

."

."

................. ............ ...... " 'I

0.51

...

.........

1_.- A,

I.ill l 1.....

"

.......

0.7022 0.2978 0.6974 o,,.,. 0.3074 0.Cl8'18 0.3122 U&f7 e...,. 0.3110 U7U 0.0183 0.3217 I ..... 0.1I13e 0.3M4 U lHoD 0.3310 ....03 0 _ 1.6157

.

...a..

0.42975

0.3631H

0.38737 0.37430 0.38113 0.381111

...... ..3367 ....... ........ .... .""" ...,.. ...... ...... ...... ........ ....... ..,." ..•...,... .... e.ee . .... '.... .07 om .... ....,. I."" ......... .... """ ....... 1._ ...... ...... ..... .... , ........ .... ....... .., ...... ...... ....'" ........ I_ ...... ...... urn ...,. '" ........... ........,.. ...... ....... ........ ••"'" • ..... ...., • ...... .... .... ......,• ........ ,.-1."" ...... ...... .... 1._ ...... "'", '$1

U I ' .03 ' .03

1,M71 1_ I .....

010

.....1 IS H U021 UII33 1. _

0 '11

on

0 73 0" 0" 0.71 0.77 0.78 0.711

..--

0_ 0." 17 03683 0.&373 0"" ' ."'11 ""'16

o,

0""

UIDIt 0 3101 OJUH 0 _ o.ell3 03881 0.e071

U f.72 1..... I ...... U8J8 U ...

e.eece

1.7063 1.7171 0.4824 1.7291 0.8783 1.7411 0.57'" 1.7832 0.570(, 1.711M 1."'" 1.'JtlOl

0,.,1

0.~ 1 3

0 ....

0.4138 0.4178 0.4217 0.426(1

O.U &8D 0.42:::82

0.44381

.

0 .""

0 " "1 0.47134 0.4782'7 MIt2I3 0.41tllO'l

.

0.51ft3 0.51ll8e 0.52871t 0.53372

.. . . . . ... ..

I.3013 07_ o.t31O I.310(' 0.7&31 • .17032 I.3IM 0.7871 .....1 •.nne U llM OM" 0....11 0.'J4'J4 0..2ll112 I.3471 0 7U3 urn oml 1"'" 1..... 07330 G.3111n 1.37.. omo 0.31888 1.306' 01J3O 0 " " 0.3U7. 1.38067 o.nro 01830 0.33211 1..... G.7UO U IU 0 7071 .....7

0.81

nee

1.8Ia I

0.11 ' .02 0 02

1.8l133 1....1 1.8781 .....1 ...003

0 ..

U lllli 1.83111

0 ..

1.9725

• .00

aoooo

0,M71

I .....

0.4298

u.". 0.m8 0.," 14

O.,"DI 0 ....

0.4012 0.4M1

0.U 71l1 0.!IM52 0,(1614(1

0.(17531

0.689J8 O.5Mll

0 O.el

0 .."

0.6212

..-UI7• OJl If.I

0 ll3ln 0.4715 .83770 0.4711 0 0.4788 0'"

0.81011

• .8000

0.41l3O

0 . ...

0 .... 0 0 ll7828 0 ..." 0.031

Table or Time J'aewn 1/'1'1

."

. _ ~l

.-

I _ . - AI

I

"

...... ...........' .06 ...... ' .0l00 ...... •."'" •.scee ......" •. .... ...... e."'" •...... ."'" ....... ....... ,.. ...... ....... ...... ......'" ....... ...... • ."'" o.eoso ,.36 ...... ...... o,om • .02

•.om

!.l1:1li

0..&601

2.2181

n"'"

0.70'101 0.1%0117 0.127lK) 0.134014 0.14860 0.162" 0.17&32 U.111018 0.79111

2.29'7,(

0."'11 0.«13 0..(363

0.111191 0.83178

2.0Il48 2.11foO

1.10 1.12 1.1' 1.111 l.UI 1.111 1.00

' .22

• .26

'.26 ' .26 • .30 1.32 , .>0

1.35

2. 1 4~

0-4931 0.4tl3O O. fo'79G 0.4130

-

0.51ST Ul'J0 0.1I2Ofo 0.5270 0.63119

0.56041

...- .

0.,(293 0.6707 0.4234 0.6708 0.42Of 0 .41111 2,4284 0.4118 2,4623 O.oIOlI l 0.6939 2..(967 0.511911 2.11316 2,M01 ' .3923 0.8104 0.3IH2 O.t l61:1 '.6390 0.3789 0.6211 :1.3296 2.3.20

,.. •."'" 1.60

.."" 0.1llf723 0.90109 0.91496 0.92882 0.0311711 O.N 2&I O.lh:I&i4 0.17041

., ...... ...... •.- .,'" ... ,. ...... •."'" .,.,.,. ". ."

"'

I ."

• .00

3.t"8 ' .0000

. _ ~f

" . .. ...... •.ee» .,"" .....,.

' .02 ' .06 U 1240 ' .GO uno 0.24015 ' .06 U698 ' .2398

.

l_ . - U

-

0.11134 0.1588

•.37243

UOO18 1.'10&02 1..(2096 U2788 1.'0' 115 U &581 U -'1

" ...... '•..2300 "'" un • . • . " '" ...... ....' .26 ...... .,"" ....•..,.,.... ...... •....... "'" ,' .26.. ...... ...... ...... ...... ..... •. ...,., ."' •...... .,..,. ....• ' .36 ' .lI031l .... ' .36 ...... ........ ase 2.14 2.111 2.18 2.18 ' .20

' .22

• .30 2.32 • .>0

Ull7l

.(.40'16

U 382

.(." 111

4.ti3UI

c.-

' .2263 0.217t

.(.72M) 4.7569 4.7899

0.:110&4 0.2117 0.2102

• .203•

.." • .06» 1I.09ll2 11.1337

0.11711 0.1961 0. IN8 0.1821 0.1896

0.1'" 0.7700 0.1131 0.7'1'7 0.1162 0.7703 0.7824

0.7883 0 .7898 0.1912

e.-

1.<833<

U 902'7 80 Ull06

1.G24lt2

I.

1.6081 1.621

r.e

.- - ,. .- .......... .......... .-.................. .... ."" .... ... ... .... ..... ... ... • . ..... ...... ..... ....... ..... .. ...... ..... ...... ...... ........... ..........." •...... . ...... ... ...... ... ... ...... ..... ...... ...... . ... ...... ...... ....... •. ....... ..... • . ..... .."', ........ .... ..... .... .......... " .... ...... ..... .. ...... ... . ....... . ... .. . . ...... ... ...... .... ...... ..... ....

•."" ..' .6360 .... ...... " ... ...... .. ........,.66"" m, .......... . ....,... ..,..., .... ...... ...... '.3311, ...... U 2 I." I."

2.67611 2.7132 2.7321 2.7611

0.3737

0.&4 111

r.'"

2.8619

e.soso e,""

c."" n

3.001.(

I .II ~

3.07:JH

0.3263

1.66 1."

3.1161

U.3:!uy

' .72 I.,. 1.76 1.78 1.78 •.80

1.66 • .66 •.66 ' .86 1." 1.02

,I." ,

3.1Jll3 0.311:18 3.1602 "e.e... oe 00 .3121 .3078

03'"

3._ 3_ 3.387'

,,3<1

'3..... ''''

3.61101 3..... S.ll301 1 .... 21m 171142 3"'" 3.....

I.Cnus

". '" ."

11.31117 11.42&4 U3f1 .....1 11.11789

' .GO 2.6'

,

0.67.(7 0.67111 0 .8111.(

1.1:l'.!OO 1.I:JlI76 1.l ol389

:I.G11 l!.tltl

1.15062 l.lMf.8

1.1711311 1.111221

..,..,

0.21170

l.(l67411

...,eo

0 .8701

0.7008 • .2973 0.7027 0.7047 0.!811 o,0.7128

0.2717

1......

' .60

1.0IU:U IJ»611 U O\llI4;

c."'"

0.2832 0.2713 0.21"

0.911.(27 0.99ll13 • .00006 1.011" 1.1J3\l'l'2

0.6613 0.8M 1

0.3f1l1

' .60

t .'" 1.70

0.6263 0.63111

0.7188 0.7207 0.7228 0.7246

.,'"

,

O.7.(lt 0.7.(30

U I301 U IIl9f 1.33300 1.247l18

U lIl63 U 7l13t 1.28232 Ull926 1.30312 I.Sleee

1.:14.(71 1.34IN 1.3IIll67

IU lIliG

'"

..." IU l'19f

8.0tI2'il

0.18&11 0.1843 0.1830 0.1817 0.1792 0.1168 0 .1'7" O.l1 lt 0 .1708 0.11196

0.8062 0.8008 0.81011

I. I. "

0.8131 0.8157 0.8170 0.8183

1.8774 Ulilil l 28

...."

"""• r.- • •,• 1.70111 1.7190

1.7328

1.7'8'73

r.,

1.787

G.un3 o,0.11327 u.I&49

r.

0.11361

'.77

",•

1.711ll3 1.1lO21

e, '606 ,'"'" ""17 .,'" ...... , • • "'" 1.1"6

O. lm

0.8373

0.J582 0.1660 0 .1639 0.1618 0.).("

0.8U8

..".,

2.6. 2.70 2.72

6.3200

8..(980

I ." om. """ ". '" ~" 2.78

' .80

"730

""

7.3e18

2,82

0.81113

0.) "8

•

22 1.1081 1.11

,

•

1

c.....

7.()616 0.1.(111 'l'.1G03 O.llI9'7 7.!101 01387

". , ,,

0.8'81

...",

, .66, 1.871GO

""

1

lot1"

urn

•• ••

O.IS48 O.IUO 0.1321

I

,

2.01013

• •

28

AppeDd.ix V

",

."

2.M 2.96

7.87401 7.7276 7.7813 7.8898

3.00

o.J303 O.l!9ofo o.J28Ii 0.1267 8.0000 0.12IiO

0.8897 0.8708 6.8711 0.8733 0.871lO

2003786 2oOM78 2.06172 2.08M8 2.07'"

3.06 3.10 3.16 3.20

8.2!l21 8.67401 8.8786 lU89'J

0.12O'l'

0.8'793

2.l14ol0

0.1Ie8

3.%6

UI:n

3.30 3.36 3.400 U6 3.liO

9.8fo92 10.196 10.1l68

U8 2.88

10.928 11.3140

3.66 3.80

n .1I3 IU28 3.86 12.663 3.70 1 12.998

3.76 3.80 3.86 3.90

1"'" 13.929

3.96

lfo.tiO 1f0.921l IUM

40.00

18.000

UO

17.168 18.379

' .20 ' .30 ' .foO UO uo ' .70 UIO ' .90 11.00

IU911 21.112 22.028 2U62 25.992 2'7.867 29.867 3i.00lI

•

"

0.1I83fo

2.10&878

0.1127 0.8873 0.1088 0.8912 0.10111 0 .89fo9 0.10111 0.898li 0.0981 O.90UI 0.09f08 0.9062 0.0916 0.908Il 0 .0ll8f0 MII8

2.183fol UI80'7 2.28739 2.322Ofo 2.SIi6'70 2,39138 2.f02602

0.0llM 0.082Ii 0.0797 0.0770 0 .07403 0.0718 0.0893 0.0870 0.08"7 0.0826

0.91fol1 0.9176 0.9203 0.9230 0.9207 0.9282 0.930'7 0.t330 0.9353 0.9376

U8087 U9M3 201i2999 2.Mfo6fo 2,69930 2.83398 2.8lI88J 2.10327 2.73793 2.7721l9

0.0683

O.NI1 O.NM O.N" 0.9628 0.W8 0.9688 0.9816 0.9&fo1 0.968Ii 0.9888

2.8fo190

0.06U 0.0Ii08 0.0f074

600M2 O.Ofol! 6.038lS 0.03Ii9 6.033Ii 0.0312

2.!1273

2.91122

........... ....... UO

.20 ' .20

6.70

e.00

.... '.20

.... e.eo

7.00 7.20

7." 7." 7.80 '.00

.......... '.20

.... .... '.00 ' .20

aeo

2.98MS 3.Ofo98ll

10.00

3.l1916 3.l8868 3.!lim 3.32711 3.39612 U6li7fo

10.00 Il.OO 11.00 12.00 13.00

AI

1_.-U

"

.......

~~ 0.0292 0.9708 3......

0.0272 0.9728 0.0214 0.97f06 402. 0.0237 0.9763 foIi. 0.0221 0.9779 0&8. 0.0208 O.97N 1I1.98fo 0.0192 0.1l8Oll 66.11 0.0179 0.9821 0.0187 0.Il
39.

1l!''!J.i

7~.~~? 0.0136 ~.~ 0.OIl8

97.007 0.0103 111.403 0.0090 128.00 0.0078

3.87388 3.740299 3.81231

3.88UI1l 3......

.......

' .08967 UIi888

0._

'-29761

0.9882 0.9897 0 .9910

' .674077

'.~l4o

....... ....... ...... ....... 0."'" ...... 'c.-.lI932

1".03 0.0068 188.90 0.0069 0.99fo1

' .71MO

0.....

6.12929 6.28792

2Ii8.00 0.0039 0.9981

6.64618

....07 337.'79 0.0030 0.9970 3eaoe 0..... 0.99740 0roes O.llli7lI 612 .00 0.0020 0 .... 1188.1. 0.0011 0 .... 87U9 0.0016 0 ....

6.88381 6.82244-

1940.01 222.88

0.0062 0.00f0Ii

...."

....... ...... 0._ ...... ...... ...... ......

5.96107 6.09970

• .23332

'."' 98 UIM8

102f0.0

0.0010

0.....

UM21 U92ll4 8.931"

0.9993

7.2'7806

778.011

0.0013

8BU4o 0.0011

1448.2

0.0001

.....0 .....3

0.....

8192.0

0.0003 0.0001

o,-

7.82462

7.97119 8.31777 9.01091

A,peD4b: VI .

List of Symbols A

m&M num ber acti.,ity geometric buckling lU::ti.,.tion activatio n of a eadmium.covered foil un perturbed activation thermal ne utron acti.,.t ion epit herm al neutron activa tion diffuaion eooliDg con.Uant

.A. [cm-·.oo-l )

B'[cm-I ) C(ern-' aec-1) D [em-. eec' "] C. [em-· eee-r] ClA [em- I * , "1)

ce

C.,. (em - I aec-I)

C(em'sec"l) C c _ -D, - [em' ) c[em eec-e] D . D (E ) [em] D [em]

reduced diff uaioo cooling CODatant

8,8..[ev] or [Mev]

velocit y of light diffu llion coefficient a verage diffu llioo ooefficient for thermal neu t ron. diffuaion c:onatant eJ:trapolated end point t hicll.ntWI of ab.orber &lab neutron energy

Ef' ~ -2

i

D. [cm· lIeC-l ) d

[em)

.

d [em)

[eY]

En lev) Ee [ey) ECD [ey) E" [e..)

timee tbe mOllt probable energy of th ermal neutron.

tb ermal CDt -off ene rgy 10_ eut-off energy in re&Ona noe integ ral. eadmium cut-off energy r-:ln&ooe ene rgy aouroe neutron energy effective t~hold eneflj:r

.

E Q [Mev)

g:1f [Mev) 1; [ev em' l ]

aJOwing.dOWD power o( target . ubstanoe for ebarged particles B. lz) e xponential int.egra l F(r,D.. E) [cm..... aec·l.teradia o... e.,·I )} dillereD.waa " , neu ·wvu --flUJ: __ ~, _ F (r. O. v) [em.... aec- I at.eflOUlAU -1) F (r. 0) [em-' eec-! . teradiaD-1) vector fiux F,(r. v)[em.... &eO-I) ooeffioienta of th e { F(r . a, U)} in Legendre F, (r) [em.... _ -I) U panaioll of F (r.O) polynomial. JeD c.dmium cornoction fKtor Fa IE) [e.,-l) Horo witz·Tre tialtoff l peetnJ fun ction / (w. T) } Fouriert.raIM.formofthe aIowing -dow:udeMityarouoda /(w. E ) point aouroe in an infinite medium 9 .tatiltical factor ariling in the Breit.Wigner formula g. ,(T) WasTOOTT CO~tioD factor for DOD.l/l1..baorben

l

lIooebnt{III'lrU, II ..._

P'll7"'-

•

Appendis Vl

"[oveec] 1. [bam] 1.,. {barn] 1. [bam) 1; [bam] J, J [om""' tee -I]

..

K ... T [om-I]

} rev

("K) ~l]

L [om] L [om-I] L. [em] M [g]

probability that neutron energy ehangeB from E' to B in & collision I PLAl'OI:'8 coutant divided by 2" reeonanoo integral at infinite dilution effectiv e rMOn&noo integral rMOnan08 integral U088I reIOD&OOO integral neutron eurrent density

wan nnmbel' BoLTDUlflf'S constant diffuUon length thormalization operator tlowing-down length I l l ' " of a IC&ttering atom

·( ~.)I

t -·/af, {tV-I] equilibrinm thermal neutron 8peetrum in a modera tor with temperature To mean eqnered energy transfer in an eqnilibrium M, [bam] Muwellian spectrum m, m. (g] neutron D1888 N [om-I] number of atoma per cubic centimeter N (E) [Mev "I] fiaeion neutron spectrum n(r, n, B ) [em ' l steradian-I ev- l ] differential neut ron density A(r,.Q) [em -I Bteradian "l] vector density n(r), " [em .1] neutron denllity alE) [em - I e,, -I} neutron density per unit energy fl average occupation number of an oscillator p, p(E) , p(v) reeonence eecepe probs.bility Q [tee-I] 8OUfOe strength Q (Mev] reaction energy M {B) =

9(.I'I} (om ... .eo ·I]

q(v) !/,•• {r ) (em -1.ee"I)

tlo..nn,;-down denaity a t energy E (letha rgy v)

tlowing-d own density Jut above the th ermal energy range R [om] nuclear radiu. RoD cadmium ratio r .peetr&1 ind e••rising in the Welltcott convention iJ [oml ] mean squared tlowing-down diltance S(r, n, E) (em -I eec-I.teradian- I ev -I} diHerentiallOUfOO density S, 8(r) (om-l.eo- I ] ecurce dena.ity SIB) [em · 1 .eo· 1 ev· l ] 8OUfOe denaity per unit energy 8, effective p-lIe1f-abeorption factor • reduoed e:r.0ll88 resonance integral Z· .- t . . T '""-,..nen""vD raollllU881On

T

time factor

" ['K] T. [·K] T, [eee,miD,l. 4. , ] ;; [eec]

.,. [- J la, (lee]

•

Y (CIU 1&Cl-1j II (em eeo-1j til {em !leO- I ]

,

, .. -Vi'

VI' (cDuec·1j

neutroQ tempent.W'e

modentoOl' temperature half·life 1I1owing.do wn time to energy g temperature re1&nttoJl thn. thermalization time lethargy

_ttel'ins atom velocity neutron velocity mOlt probable velocity in .. thermal neutron Jlaxwell . pect.rum with T _293.8 -K moet probable ve&ocity iD a thermallleut.roD Kaxwell lpoetnma with tem perature T

e , 11. (.81) [600-1]

average nlocity in a tb enn&l neutron Mnwell. 1Jpect.rum with temperature T time decay ronatant of .. pcjeed neutron field

ll "" ~

ca pt ure .to. fiBeion ratio

ll - (~ ~~ r

marimum relative energy change in an elutic collWon

t1".1

Cl [em' g-I) ~ ""';Z: [aec- I ]

p .ray a beorption ooefficient

abeorption probability

p

a1bodo

r (ev)

r. lev)

:;trvn

· , - [!11M1O m ol)

reeolring power of .. time-of.flight .pootromet«

.,

r,. (ev] " IE/. " )

~ -e ·d[gcm -·J

•

• 6, ,If

(em-I ]

..

_, III (em-I] w.(r)

...diation

I

width of .. Mut ron r-mAnoe

joinina: f\UlctioD surface mau loading

oorroctJ on for neutroM incident on the edgN of .. foU

neutl'on yield per neutron abtorbed in .. fiMionable material ICI.tt.ering angle in the laboratory 1)'8tem &lymptotio nll. nOon ~nt of • neuttoD field ch.a.nr of the DeUtron ....... Dumber in a oollWoa. aet.i.....UOD oorreetioD Dux correction

). (_"I]

radio6cti.... decay OODIItant

A [=1 A,[=J A, [=1

mean free ptoth scattering abeorption

..1c.. [om]

uaneport

l,

fraction of abeorptiona in the i.rA I'NODItDOe of ..

, [=J

reduced d. Broglie ...... length of • DeUtroa

NlfIOD&h Ge

I

mean free path

detector thl.t leadl to the demed aetirii1

I

Appendix VI

;: I P~

total colliaion abaorption ffi ' t activation ece cien 8cattering absorption coefficient at th e peak of a resonanoo evceege number of eecondery neutrons in Ilsalcn average logarithmic energy 1088 slowing-down power density meeroecopic Cl'OIIIl eecnon (I = to tal, 0 = abaorption, act _ actiV&tion, , _ IC&ttering, " _tran,port) average macroeoopio abaorption el'Ollll lIOCtion for thermal lIElutrom.

[om' . -'J

P.

,

p.., [em1 g-i]

I ~

E. [em -I]

e [g . em-I]

E [em -I]

E. [em -I]

·x. --r;

moderating ratio

E. (0'

_n, E'_E) [em"l .teradian-1 IIlV-I]

(macro&OOpic) Cf088 eeetien for 8C&t. tering pr'OOellllell in which the neutron dirootion changee from A' to n. and the neutron entll'gy ehengee from E' to &1 E,tO' _n oU· _foil [em -I 8teradian-l} [macroeccpie] CI'OlliI section for &cattering proceeeee in wbich the neutron direction changes from n' to n and the lethargy changee from u' to u 1 E,(O' _0) (em -i llteradian- I] (mac1"08COpic) Cl'OfI8 aection for scattering prcceea in which the neutron direction changes from R ' to I

-I]}

_1t)}

n

E"I"" _u) [em coefficienta of the {I.{O' _n, u' in Legendre E" [em-s] expansioD of E,(O'_nj polynomials E.(E' _E) [em - l IIlV·I] (macroeoopic) C1'088 section for IC&ttering proce8lle8 in which the neutron en6rgy changet from E' to E· ruiercecopic Cr'083 section (t=total, .=scattering, a [bam] a = absorption. act=activation. m=non·elastic, tr _tranaport) peak absorption cl'OlIIl&Ction in a reeonance a.. {br.m] peak total CI'tlIII lIOOt1on in a ~nanoo a, [bam] [barn] f""" atom lCattering Cf'OlI8 eecucn bound atom lcattering Ortlllll aection aaa [br.m] average Cl'OfI8 lection in a fi.tIllion spectrum (J {bam] effective 0l'0U aectlon in WJ:8TOOTr'8 convention • (bun] eeoee eection for coherent acattering (looll [bam]

(I.,

(I, [bam]

a••• [bam]

plateau value of threshold detector Cf'088 eeeuon reaction CI'08II lIllCtion (ot = (IS, ot) prcceee ; y =radiative capture; n.=olastio _ttering; n' = inelastio scattering P =(IS , p) pl'OOMll ; 211.-(", 211) prooo88i !=fitlllion)

• lie.. ~"Iy. the d~ after the ooUWon ehould be in tho unitlOlid angle around n and th.- -flY in the unit eneI'IY ratIp around E (01' the lethargy In the unit lethargy range around .1. I preeedlng footnote.

a.

Litlt of Symbol.

CI'088 aection for an elastic scattering prooeu with the _ttering angle D. I Inelastic ucitation CI'088 eectlon a (E. E, >(barn] a (E·-+E) (bam Mev-I] CI"08ll section for an inelutio scattering proceea in which the neutron energy chaJi8e8 from E ' to E I a. (E' _E, 008 80 ) [barn ev-t] croea section for a scattering pecceee with th e eeattering angle {}. and an energy cbange from E ' to Ei-' croea secti on for a scattering proce811 with an energy change from E' to E 1, 1 a• . (21 Et [bam . (ev)"] energy tnnefer momenta of the scattering CI'06Il aection l

-.-

T. [em') T(E). T [emil

T; [em']

4>(r). 4> (em-I 8&0 - 1]

4>(E) [em-I eee-' ev · l] 4>,~ (em·' leO-I] 4l'. pl (em-I &eC - 1] f/J, [em-, sec-t] '1', (p.6)

.'(',Pl}

Fermi age nux age neutron flull: neutron flux per unit energy inte"al thermal neutron flux epithermal neutron flux per loglUithmio energy unit fut neutron nux abaorption probability of a foil in an isotropic neutron flux

fun ctioll8 describing the p.activities of foila

'h (ft. fJ )

intermedia te scattering function a bsorption probability of a oylindcr in an ieotropio neutron Onx oolliaion deMity, Le.,number of oollisiona per eml and

x(""' t) X, (E.R)

_00

•

, ,(E ) [cm - I sec- 1 (ev) -l] VI, (E.R)

Hun preolle1" the

-'.. I

scattering angl e in th e cente r-ai-lOaM system collision denaity per unit energy interval abeorptlon probability 01 a ephere in an isotropio neutron n w:

~

of the _tteriPj: angle mlilt be itt the unit mloenoJ atOlmd

• Mono ~,. the omergy aftl,r the _ u s ing prooeeI mUlt be In the unit toW1D' Int'loTN around Il• • Only uoed itt ron.oectloD with the moIeculu- iDe1utIc _ttering of lIIow DlIUtrow.. cf. IlKItion 10.1.

S. bJeel lu dex A.t.ohIt. -l1Dc at acrtt.,lt.ielI 2t8.eq. at-oIlIt- ~ of the Dttl~ fila

......

of ,. ~ It.nIlfth 303'"1. ~ ..... IIIlO&ioll 3 - - - . ~ b1 DOClIpan.o._ lbod. n,.-q. -

-

-

-

- .

~ucioD

tftIIYI

wi~

3t!.eq.

pU.d..._ . •

- - - . \able. to, -6OlIeeq. abeorptOoa pt'ObAbllit1 iD _ iIoocropM _ . troD fieoId for elat. UI - - - - - . for iIilinit.oI '11i ~ Sfoll - , (or .pberN !GO .cc.tn.t.ioD Z33 - 01 cyliDdric.l probN m - bl epid>cInna1 DllUWN 286eeq. - h, fM& _~~ .!8&eeq. - bl _ t z - iDoideat 011 IoiJ ede-.UG - b1 -uend _truw J43 - at __ HIlMI- d. ~ ~ U8

Mtintioa ~ 130181I- - , cUlala~bldiff1l1ioo1~W88Cl' - - . oalouJ.tlc-. . , v-pan \beorJ

...... pro_ ....""' ...

~

2M

--.~oI_tle l

-

- .1lX' _ 270 - . for tb.rm.tJ HIItroD11 Z68

.... 138" 148, 115

" - _ I..... v MId th...t - V a51 01. fIIIioe. _tnx. ill H.O UlI.eq. -. ~ oI • • ,_ t us....

-

-. ~ at

"'-.-

. ...k~....s _ \ I I

-r-, ~ 171.eq.

- - aDd . . tnn-pon eq1IMloa 168.eq. rJbMo llJ-.q. ~ di.wlblrtio:m iD. -u..itlI 14, 70.eq.

- """"'" " .-oc:iIWd

~ ~

303

aIJ1IIptot.io lOla.... at \be

w.. .....

ancle81, lIll.

1".310

Ioprithmio eoerv decnlmeat 118 III Hu;weU 1(*lVu.1ll " {.. - I _troa _ !e-.q. .~

Inns- ~1

."1-,-)1.."

1". 66 probability U l

~ llerinl

bon" Be-(". _jB" 60

.............

81". 00IlIll.er M bindinll: eMJ'gy of th e 1&ot neutron 22 B,..lppro",imltion 101 eq....tioo. _ tnnapoort eq ....tlO D. Bona Ipprosirnatioo 186 boroa blt.~ method 30lJ boroa piie 308 bot-.. . .t.iI1atoot 41 boaDd. aCom 18 Bolt&manD

-

-u.mc

II. 60, 61 BNi t.-Wla-·fomlw. a. 133-.q.

buok1iDc. _

v-pon eq_

.....

~

blKllilla(

" " 'yof the _troD ! ,.., ClOiDcidenoe method m ' ...u'.ablDrptioD 2311, Me _. e:
'act«

tadmlam oorreotion f7heq., S27 c.dmiam oul-ofl - V n4Mq. -.dmi.m diflwMloe method ns..q., m. 300. ,.. oeda&i... n.t.io ~

-

.,. daeory 1&6-.q. - - . tim fHchl _ •......tioa. IN - -r-, _UWXa ....u.oda 115O.eq. -

.nnp-meolth Cl _~

Br-a refIeetion

- , ~ at MO-.q.

- - . expmm.at.aJ

...

u ym ptotio ate in neutron tb(ll'malJ.u.Uon

m. !SI. 283

raoliati.... .. 711

- . 1J9-1&w 10

_~lo-fieeIoa ~

_my met.bod

'76eeq.

281

~6!,Slll

oobeo-t _u.;1lI 17 ClQId _trona III eoIllmat« 50, Sill aoIliaKm deoaity 121 llOIIlpuieoa methode for abeorpt.iog arc. 8IIlCltio.- 1174eeq. - - for _ ~ 307eeq.

CIOIIIpoud . 1MUo -tterirIt:

e

., oompolUld DlIcleui I'Mdioruo htq. eoUOOLD" IMtbod 20lI _ -'iou, ' TWap tb«ma1 iii - - . oWioitioa 3 aeq. - - , effect-h" .b.orption 210 - - , experiment.al rewltl 13aeq. - -r- , fialott 63.78 - _ , lnelutlo excitation 72 - _ , methoda of me&.lurement 8lheq. - - , Donelutic 69 -

- , retetlon 740 - . thermal, of th . elementa (table) 20 - , thermal,ofthe IaotoJlM (tablo)4oO'l' ~. - -e- , tn.n.port 88 cryNl filter III flr)',t.al , peotromoter III C4y'tem 117 llUnWlt denaity 81aeq. oat-off nt'I'IY 2CJ(l

DMler coefficient method lISl

.s. Brosl" . ,"*"p e. 18, 188 Deby~Waller·tlQt«

188

denait1 81 teq. detailed ~ 1840. l e3, 111i. tol deteetioa of_trona Mteq. DrEftIOl" method ZDfo diff~renta! derWt.1 82 differential noutron DUll 82 diffuaioto -'ficient 88 diffueioa oonttaat 102 - , IIleUUI"MDellt b1 themodulated8OW't"ll ~31Kteq . m~t with

, pw-d ~ S87teq. _ _ , t.empeN t ure depebdenoe in H.O 388 diff lUioa aooIin l 2111 teq. diffualon _ Ii na _tant 2111, 228 - - - , OIlperi.menttl det«min.tion 387teq. diffpion eq ....t ion 88 teq. -

-

- ,

-r- , energy-depetldeat I " -e-, ~lIt.1011 metbodt l07 wq.

- _ tor lhennal aeutrooa 101 - _ . tima-depeDdent 173.eq. diftuaion '-ting 217. 373 diffll&ioa kernel 108 diffu.ion Ieftatb 89. 106 - - . eomp .... 178 - - , de~ oa the H.O _trat.ioIl ill D,O 171 _ _• temperature deJ*>dep.oe ill ppltite

'"

iIl.1ter see diHu.ioD.length _ t a 380teq.. - bt aD iJIfirUta -m... 380 - ill t medlUDl IlIrTOUDded b1 1IUrl_ _ 3 6 l I.eq.

-

- . k1mpentllnl ~

JeDath i:D. . . . . . pi» W - ~ IlS8teq. c1iffuaioa. paramee.en for \benD&! _tt-.

diffuiot

~b1tbtJl'O'-:­

inc technique 372teq.

-- - -. ~with ~ ~S87teq.

-

-

-

~ion

- . tab" 106 theoty, e1em«lt.&r1 86

direot reactiou 8 Doppler tpprodmat.1on 193 Dopplet Bffeat 1M ~width 1305 doubJe ohopper llIi

Effectl.. poM1f-abaorption (aotor 300 effeoU.. _ _ IN, 208

effeot.i'" _troD teIIl pentllnl, _ aeu tn>a temp""'tIlnl .Ieoti"' ..-.nanoe integral In, 20lS .aeotln t.Ilrabokl-.:r t86-r aIeovio cbarp of the _troll 3 eIectroa liDear aooelentoor 3&, ..0.!!O Metroa. oeH.baorpUon!38, 1407, 300 e1ementuy ddfu.ion Illqll&tion, _ difflltoion

........

Z. tuacUona, _ ...pooenta! integral arodpoiJot CIllITlIeUoD 3M -s1 kl..- in t aeuttoa (Jeld 213, m - V ~ta ill paleed _&ron (1t01d.

""

epitbonual DelItro1ui 45 epith.mal ..u~ factor Ul' 0Il.- ~ce integral !SO, 282 U poatDtlal integral. definition 1401

-

- , tab"'" 4o%lleeq.

...tnpoltted end point N, 2%11. 3M, 3lI6 tlItnpolaUon leneth N, u s Faat coopt- 62 f..t neuUonllll, " fut neu kou pene tn.t iOll over large ditta.ooo.

'"

F_ I ... _age Fonoi 'ppro:o: imat.lon In Fowier uoaIyW of . pu l.ed neutron flOlld 38lS Fourier tl'uMfonn 1I1(). 1113. leo. la, 2U Fwa'ila.. 81, 103, 1406, 167. 173. 178 finl fiisb t llOn'OJct.ioII lU. 1407. 1M ~ OOWltel' 62

-

- . few_Von &emJ*'&tarI-..-..t

'"

_ _ ... . ~~28 1 fiIIIioa ~ Mlltiorl Cl3. 7' fiIIion _Von 1peCCnI.... m fiIIiaa width • f1llJ: 81 teq. fl u age, _ age

«,

." O....

~.~2Ml

U I deteet.or 68

-

- ._ t W tho_ depr-.ion 133

La" (4. ·I Bel'~

foilll m.-q.

1.l"(,.. . )8e' H limi'" for eigea Y&luet in t.b«maliud flClk1a

- . at-hat. OOWlting 01 ~

22h"q. JooaI. pile e»cillator 379

- . ~-U .• b.crpUon m ....:. ~1na.Ucxr. cd aoti...i ty m -r- , for ~Oll neutroM 268-eq.

b>1 CIOUot« 67

- , l ub.t&_ for abeoI ute ll>MoIUrO!Pllnt 297 - for t<>mptorat ure m_u~lMt 326

- fOJ' t hormal nout ron. 23"'oq, ._ , thrNboJd dotect0r8 2S6-eq. - , t .... o-foil me thod 279 fnil acti,..t.ioa. _ ad.intion foil pert.w'NoUon. 250Mq. free a tom _tterina ~ -.octiolUl 1'1 l und ......" ..... IIlOde In ~

IIACKLlIl ' e epheee 308 m&C1'08OOpio crou ....,tion. magnllt ie momont 3 m&!lg. _·br.th method 306 mathematical IDflthoda fllt thrMboJd detector .....lu lion 292 )laa...11 diatributioD.98aeq.

........

m....n ~p&th' mMft ~uared aIowing-down

I tTl-fact.or 100

-

~ b ..ek.linc 116

-

_. ,.,

- . detenaina.tioo1 witJl. pW.d _

Uoae

, ' - .ciDtlllaton (If ~ · G.-1i... a ppoultimalion 1)0, 158,

.....n.

sa

!rl,.• • )8 " S'l'

hN'deni na: of \be _ \roll l peotrum m H"(a. pIH" 14. e:I H"~e:I

Ippros:im.. t.i<m 197

e.

teriIla 11 mtepV pa. c:.nu..tor 380 IDtepsl tnIIlIpoR

~uatioa

85

1oiDiDa; fwacIUoa toll. ns in D.O )40 - - hi p pbh.13t -

-

I&

- . caJeuJ..t.ion 13lt. 142 - - , mMRl'Mlent :M3eeq. _h.wUcallllOllOdlromator 61 ID8thod 01 int.egrat.ed PIlUU- tlu 375 1Ilipt.ion.,. 163 migrWoo. Jeoath Ie. ki1ne prob1en:a M, ~ Kireille pile 374

H.o 338

'-

M1l1tnm OIllT'llDt 81 D8IItroa doMity 81 M1l1troDdotocton 6ti _\roll OIU 81

DOlItron m... t _troll probea, _ IoU. _ttoQ Ipin2 DMIttoQ width II _troll f>llldlllitoq. - - . pul.ed If'h eq ., 38300q. - - . thermI.Ibed t ltooq. _U'OI:l ~ !200q. - . (II, al 2e - ' , (" , a ) 38

a'a'

~ poI~ tI700q. lutaipt ~IU

--. (Y. - - , (p .

~pot,-iM87 ~Itl ooq.

- , JioId

Li"1.. _jH" 14.68

:w

_

N..l.k ln modol fur K.O 1a6 00q. nout.ron ba lanoe 113 _ Von #-de<».y 2 neutron binding Mergy 2:l lIO\IVon ob&l'go 3

bM v)' au model 20 1 HDnl.) 'K butloD 8 1 Horow itz -Tretiakoff method 20hoq., 201

inooh_ t _ ttoriDg 18 inoIutio n eitatloa _ .-et.ioo. 72 ineIMt.Io _ tteriP, 72 int8'fonaoo of potaa.t.ial IIoZId

dil t&nce 138

-

II><'ldeftklr 81 1DOderaw., ratio 121. l63 _entl rne\bod 1402 muJt igroap difflMiolD a-., 169 muh.ill"l'lP metbod 1tl8., M. 211

R' (I, . jHeI sa H',IIi', a)HeI

-

ID()ek r;.ioQ

(y. _ j r.eu trDG ~:11

In~t

L·ar-tem 117 Lutetium ..... neutron "thermometer" 327

31 3li

-

- , I'MdOl"I

~

~

)os-.:&. _

...

_ttolI IJ*Itn., uympc.oUc 403l1e
ill pphit. S37-.q.

-

-

-

-,kt.bpIU,406

-

-_... ~ ~

fOl''''

l'NOIutioa ~ 121

......

-, -.:I,._t. by ~ ~ 8I eeq. - , meullNlDell.t by t.hn-hoId ~

- , uteuunlment by the UmHlf·f1igbt method31t

_ _pe proMbilit.,. lUo, 147,

_ - -

--,~I3t

-e-, methoda of meuuremeot. 282 .... -e-, tab" '17Mq.

- , 1"Nu.lt. n ( lIMUIlnllIlente 3331llCJ.

-

- , time ·dependent oW llIMI.

l'MOnan"", palallltlten

-

-

_

In ...see 333

DWtroD temperature 2Of, 21', 22'l' - - iIlgnphiteUO - - , m...".,meot with foiJa 3261eq. - , ~t

_\roD. . . . . 119 - - •• ~I&I ot-n-atioa""-I.etioa

_

......pro'- 268

66

.-...-..... f'MC'tlona 8 _ _u..iDg8 rotatiD( Clpt&I ~ Ul8

22'l', 399

- -iIl • • w336 neutroa thenn.'iMtioa 180eeq. - -. ~pMdeot 200 Mq. - -,IJI6'»~t !OO8IIMI. - - . time-de~t 227eeq., 398eeq.

-= pi'." _

JO.5

iIlt.egra!l ll lllMl·,18O .t lDt.inite diJutoiM 132

-

-

_er- 361,163

r-d&~"-.q.

- -ln ~ " Mq.

-

...

eo

......po. f"Ob.bUity 166

kad.ich method 270 33 Sc-(P, ,,)Ti" 38 -ueriDg. upIu didribat.ioa Ia 16, 70

(Sb-- Be) _

-

b,. b8rJllium IN-.q. byehemically

-, ~t l1

boImd.-. 188-.q.

-e-,

-JlOUDd elutie 8

-

by a Deob,. did lin

-e-, dft:,& IlMticI 8

....... ~perature"

",

-

~tDodelll

by p-aphitoIllNI8CI. byaa--ie~II7-.q.

-, ~ I " I 88

- . ineIutie

e. 72

PU'U model for ppbit. IIlI pbMe ..locit,. oI_troa .UN ITt J'bot-Il'Oa _ 3111M1.

-,lDt«f_of fO'-tWMd _ 1 1 -e-, iIloVopkI16, II' - b,., _t<-R!al .... IU...q.

p&1I1 pile _i11ator 31a.,,'I. eoiaoidIlneIl matAod

- , "*In&II'''' 8

h""

40 JI ClOUnt« 288

-e-, potenU&l 8

2lI8

P1ao&ek (un otion, _ uponentla! lDt.esraJ PJacz.ek .alat.lon 126 Pr"ppro:dmatioD 116, 144

(P_ Bel Mluroll 30 poiaooJD& method 37211M1. (Pu- Bel _ 31 palled _ teohD.iquM 318, MfoMq.,

",,,.

- , Ibaclew II - b,._ter INIeq. _U..iDg _1eO\iOA 3 - - - , bowMl. atom 18 - - -e-, free atolD 18 - - - of eJemeuteltab") 20 _ - - of iIotopeI (table) ~1Ilq. &Iem(u~GoerbeI met.bod 167

1eH-ahWdioc ISS, m

1Ilf-CiIJdiDI fa.aIoor U3

.............. ee.

.........pirioII"IIl\llI.bodI for thrNhoId dIceeW

IR_ Be) _ 2h"q. - - . ekJwioI40- ill pphit.e)ll8 - -. ,wd ~ 3 10

Ilbadow ee:71 Ihado1r -UwiD( 13 "",,"'383

ndiat.ioa 1rick.II. • nd»t.l.. _ptuN 76 ~.. lndioaWn, _1OiJa ~"aeal.nxl_ _ H"""i. RADUWUY', ~pt.iaa 3elil, 373 I'MoticlIl. _ MotioQ ~t

.... ....... " _

_

-

-, ~t,. n P"ri-wJ

NllOiI JI"*IO

-

- , .~pmdlmt,. tMarJ

e.

_* 681014reoai! ~ w.oop. 81 redDOed _\roD. 1rick.II. 8

7.

Ilowiat: do-. ill .......,.~ U4~ ill

h~

......

121-.q.

, ,

II7~

-. ~t,. nprrimmI&I SM -.. -,~t,.tMarJll1-.q.

... ~-dowtll

deoooit1 Uh"q.

-

- . , yn1beUo UII lIlowiag-dowa IeD(t.h 138 -

-r- , Upeftmootal de....inat..i
lII.owinl~

JlO'"'" 26, 121

• lowi0l ·dOWll time 169, 112

- - . nperimeot&l determinatiM 366 alowiDg-do_ .time .pectromet« 3M' • Iow Dllllu.m. 6 SjV"PpnlJ:imaUoo In ~dfIflIIoit,.84

1p6tia1

~

hI puIloed eKperiment.l 386

~~I7.lJ6

ephMoaJ IbeU " ' - eg l pin 01 \ha _troll J •ponw - f. . . 34 •tand&rdiud on 01 neIIl.ruQ ................ t.I

...

_Ddud pile 303 ...ti.\iOIJ

. eipt fac:tor ea

Teepa pbet', eq...tioa 176 _ _ b'oa "",,pen~

lem~

komfll'"'UlN relqa,tM)Q ~me 228 - - - . elporimentaJ d&tennination 39t tt-m.J t'Olu mn 49 th.maliu.tion. _ aI80 DeUWoo ~iu,.

"""

t.hennaIiu.tioa open.loor 200 therma1iut.ion lime 2ft - - . ",peri _tal ob.erv.t.ion.oo lhf,muoJ oeutronll iT .-.q. t bidl t&rptI sg t h..-hold dateoto.. 2ll6..q. - - . ev aluUon IPetboda m - _ , materla!oo fOl' !ll{I1OIQ. thrNbokl. - V U . 28&1OIQ. time fKtof 2:W - _ (tabie) -dh eq . ti~.niJbt method 41, 72, 1M , 313. 321 timoHlf·flilht ~ 36 tot&! .c>tl~ 134 tl'wn_ i-ion t......uoa. al .. chopper J 16-.. tnM mjMjm met.hocl tor _e1nt ie _

-w._tot -

-

lor _\ron tem~ determiD&\ioa 3!6-r. 3518 for - . . ~ detenDin6-

""" ...

tn~n

IIl4lthod for ~ Integnl detennlNUOIi t84 - - fOl'to tal Cfl* IO'lOtiOQ _llI'IIment 66 u.n.pott _ llOCtion 88 tn.....port equation 83101Q. - - . nd age theory H 9...q. - - , uypmtolill-.o)uti on 89101Q. - - . BjV__ pprollimati Oll 161 - - , intA:ogral form 8fioeq• - , mOlDenla method of M uUon 14h eq. - - . p.... ,. ppn»imation 1MI - - . SN"approxima.Uon87 - - In ~ _tnla a.Jd
...

V" lp. _)(Jrf' 38 un de Craa ff aoceIen.tor 3.5 YeIltoI' denaity 82 vector fiul: 82 I/...la w for the <WIpWre

~ MlOtion

10

W. tar.b&th gold mllthod 30beq . WIllITOOTI" . oonvontion 27heq ., 328 WIOX'. method 87 width of OOD1 poun d at&teI 8 Wlo Ma a' , approIim&tioD. l27Mq. WIgnet .WiIkm..gu 201 Yield from (II, _ ) ~ 31

-- from from l"._j-..-40 photooI-_ _ 33 _ -

from {R.- 8e} _ tv from thick t.a.rgeCa 2e from thin t&rgetA !tI &om • m-ni1lJD tatpc. llDder e5ectroa bombanlmen\ 41

... V.neaU.ioll meLhod for .-.naaoe Inc.egtal detennlnt.tJon !M

eJowiol-dowa .s-ity lt:l M>ll' - - .lyntWk! Ull alowiOfl-doW1l IMgt.h 138

-

-

~pon_~88

- . l qoerimooW deknniDa&Mla W IIeQ.

.wine....

pow. !6. I t I m-iq-d0W1ll t.ime let, 172 - - • • ~w.I det«miDaboa 366 • wiDg.40_ .tiDM ~ S57 alo-Deuu - 6 SN-appn ",:imaUoa. In IIOUI"Oll

deo&itl N

IpaLial harmonic. ill puJ-S . qoeriQlolmW386 ~ b&r-x. et, M I pberio&l .wi tnunl""- ell epill cl the _tnla J

1"""_

-

1000t.otal el'Olll llllOt.ioo ~mtlnt 66

tn.M(Xft eq.... ti
-

- . B.w.. ppro:limatioo 161

-

- , inWgral lona ~.eq. - , mo men "" method of lOIution 142oeq. - , PN'lpproIimation 96

-

- , 8...-aPP"OJ;; ima.tion t1

-

-

-

"teq.

In

-

-

...

ltandard pUe 303 .w.tifiioal .."ight f...... ea

"'a.ution .._ m - . ..penm. ..... ~ination De

te1npentunl

-

th Mftlaliu.t ion . _

"""

thenn&l iu.~

I*' neutron lbermaIUA'

~_~t1 teq.

-

( \aIM)

m-..

~'DiFt~

_ _ 36

311. U I

tot&! "<:tin.UoD. %3fo tftn.... i·OoQ IlIDlltioo of. chopper 316-.. tnnamjMjoo mel.bod 101' _ ....t i\) _ ~ _t U

-

-

for _ \roQ t.Ma~ doM.enaiD&\ioa 3!6.eq.. 3818 for ~ ~ dec.ennin.a-

......

_ , detenn irw.Uon by me.:tul&told

-

- . del-mlnal.ion by the poiaon .

_

ing mothod 3721lq. - . d fltollnnlna t lon by t.hfl p ulaed

..,..,.. 396-..

triti u m tu'ptlo 38 - - . (lQUIloUng oI .·putidM 3lK

...

t wo-loil met.bod %18 tWlHJlf":U'1Inl InOtbod fur ~ int.clgt'&k

UTH J:'. method

~

un de Gruff ~ 35 Tedor denait182 ndoI' Dux 82

1,..... 101' the ..ptu.... _ _ -uon 10

t.h>o-Ir. wpw SIt thr.bolct u-1lWl-.q. - -e-, . ...l laation P>ethod. 29' - - . m. l«l.... for ueoeq. UuNbokI. -..r 24, 28eteq. time fKtoI' su

tilDlHll-ffilht 1Mt.hod ~II . on. I"

-

V"l (p._)Cr-I 38

<>pent« :!OO

~tioatiDMm - - . np!!ri_1&I ~ ~ 400

-

_ " - fleIde 22hlq.

pMh lOS

tee llniq.... 38'heq.

TNopoapW" eq-uon. n8 _ pentllre. _ _ tronlemperMw'e

t bermal o:oI umn 411

~

traMport mean free

f-'oa :M . tandardisatloft of neutrurl ID~'''''

-

83-..

IN>dar~ 14lleeq. - , MypPItot.io 'llIattoa

-

WltM.!l.th gold uwthod 30heq. WDTOOTT'S ClOO'Iontion 2181eQ•• 3Zl1 WWIIl:'. method t'1 width of oompouDd"'1a 8 W!ont:a' . lppn».iIII&tloG If7leq. Wipw.WiJkina.Il" 20 1 Yield &om (lI , _ ) ICMll'.- 31 _ from {d. _, ~ 40 _ from p ~t.roD. _

-

-

33

from IRa- Be} ~ 2\) rs- t.hick t&rgetIo 2S from thiIl t&tI"'tIo %G from .. wuUWIl t&lpl IlDder IIecVoo bombNd.meat

'J