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-' 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.
' 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 (-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 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 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. 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~
'" 14){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 .!._ 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, 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 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, 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
_'_= 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 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 '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.. 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,(') 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 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
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- o'/Y3 &ad for r ~ _ 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(&') 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.
."_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), 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'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.) 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 [;~:'. { kT.\ "",
I
!X (",,). -.--
af
(lO.I.36a)
tI,.
(lO.I.36b)
-CO
+-
j
(. - r) (
a' =fx (I' ,f )' - . -
-...
:1-.')-:(-. - (.- i;;)).
r-
II) w;
(10.\.3 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 Boei: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+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 (I+a)+ ~
Il/I;" .
.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. (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. ' 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!'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 (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.
[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 -....
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')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 ) ( . ~,. - -
- 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) 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 . (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)) < 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-.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.'
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 ., .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,
"'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,.RL . 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 , "" - 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..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 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 (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•• 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 _ ..... ,.:,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&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\
..
'" 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 •
•-,,
•
. . - . . . . .. ~ . . . "'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 abt1 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. 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 . 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 ( 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 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(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,.
•. . . . 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,;, 0• • lIllCl Y OI1lfO
Dlpt..nyl
C. " .ll85x U..... IVJII H I 3.s211 x
>,..
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 uped
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 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
....
... " (,) - 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!_->
........... ,'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 )- ~ ./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
,""
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. LooglI 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 ol1_, ......
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 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 . • , - -,----,,---,- --.= 20 peo. Th. if th e diHerenoe of both Cl1II'l'N 11. fanned, the fund. mental mode ,..
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 Nb" n CO)
NbN(U x l 0071
. ±I
H '
IN_
,...."
.....,
I.IS ± O.02
11.7 --SU
)a
b
US ±O.08
Ol'
-
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'
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. ...± .... "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'"
~
' 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
,!
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).
Appennanoe 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
.: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 '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.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: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