Surface absorption of low-energy neutrons

Nucd~ar I'~:y: ics 38 (196Z) 68+~----69~! ïVOt to be regroaiuced br

~'or$>xa-1~Iallear~ri

b~sshi~g Co., .4

writte~a perttàï~i phatoprint ar nsicrofilan wïthaut

1-1 . FIEI)EL~I)E~ and w. E. ERAt1N Departrrsent v~Ph}~sirs, (infversí'ty of S'tellrrr6vsefa, Ca~ ~rwirrre, ä'oret

sSer~a~ez

Fs+a~ ihe~ pset;rlisher

lro

Received 2s Idíay 196 ~l~tract: ~.n attempt is made to settle the question concerning the effects of surface absorption on the S-wave neutron strength function. Earlier arguments based on an analytic treatment of the S-~"a~°e scattering prot~lem are extended and confirmed by num ' 1 calculations of T~°' ; i7 and R'ç~R. It is shown (i) that surface concentration of thr imaginary aptical ps~tcntial indeed results in qualitative changes in the behaviour of the strength function, and (ü) that fairly large ant with the surfert®-d°plume absorption ratios are required in order to obtain over-all a A C 130. expesünentai data of Tmm',.'D in the mass number region 9ü

.Int

o®

many part of the ín the optical anodel i }, absorption is represented by the coanple~ potential which describes the removal of the incident nucleon from tlae entrance channel . Front a anany-body point of view this arises from eolli~ions c f the incon~ng particle with the nucleons bound in the target nucleus . For tow incident ertergàes tha collision probability in the nuclear interior is greatly reduced because the fault principle severely limits the available final states of the scattering . Yn the nuclear surfa~~. ho~~e~~er- the nuclear en~sBty, an;~i hence the Fermi ever , decreases and the e~elusion principle m~s less ef%ctive . At the saine time the effective nucleí~n mass increases to ïts normal `~alue . If these enacts outweigh the deccrease of the density fac r, e may expect. that the nucleon mean free path is considerably shorter in the surface t in the ïnterior of the nucleus, resulting in a surface-poeaked shams of tht: ° i;, .~ potential. t~'itit increasing incident enfrrgica a growing nurn~r of tïüal sues Mme av ° ble far scattering in the nuclear interior, and the ratio of surface to ~-eluane absorption ~=í:1i decrease until, for energies of the order of the Ferrrti energy° at the centre of the nucleus, the radïal distribution of the imaginary potential roughly foil®~~ that ol° the nuclear density. ~#tis simple picture is henna out b~' calculations of the i~xtaginary part ofthe optica! p¬~terttial from the a.~°erage iwoabod.y dross sections in nuclear matter ~~') . ~f~e resultïng distribsatïon has a ~a~irnuxn in the n leer surface which is two to thé~:e times higher than the approxianately constant vaïue in the nuc:Year interior . Thr surf yak recedes with increasing energy relative to the growing ~r®Yams absc~e ~tion ~~~~, and vanishes above 2 l ei. there re ins, however, Borne uncer 'ncy as to the validity of th results cat1~L of the way the surface ïs treated in these calculations . Át is twen rote ~ount rath~:r fi~fi

RP'rTOti C~~ LCiür'-Et~ERGY :ti'~li~ROtiS

~gi

plter seaming that the shape of the imaginary potential is deterues mirted of the nuclear density, an assumption wrhich is particularl Y uestton the very surface raglan ~~-h ere the: density is changing ra Pidl y . It is le to test the theorttict~l conclnsic~n by theref~ e obtaining direct experimental ~vt , c ör t e e ïstence of enhanced surf ~c~e absorption from optical model analyses tterjng data. of lo~~p-ener t he e l ° analysis of total neutro~~ cross sections by means of a complex square rwell tentia$ already required a decrease of the abtclrption parameter with increasing mass nu t° , , which could be avoided if the imaginary potential tivere

concentrated in a surface layer of constant thickness '~ $ ) . rift higher energies, satisfactory agreernent with the differential scattering cross sections could be obtained ~° 1 °) with pure surface a o ton potentials of Gaussian shape . The latter analyses in fact prompted the a ve-Mentioned theoretical calculations of the radial imaginary potential distribution . scantly, enhanced surface absorpt icon has lien suggested ` ~ ) as a means to resolve the diacre nc tween theoretical and experimental S-wave strength function data in the Miss nuxnl~r region 90 < ~ < 130, `~~here the optical model calculations with vc~ un°de a sorpi on fail to reproduce the lo~v experimental values of I'n° ~/D. Subsequent calculations with pure surface absorption potentials 12 ° 13) support this cort, cture, and similar results ha`9e lean obtained with a potential combining volume and surface abso flan r~) . In a previous r i s) (deno ted by ~ I )) `ve have loco°estigated the effects ofcombined ~®olunte and surface absorption ors the lo`v~-energy scattering parameters by means of a tenti 1 for which the S-w®av` pr~.)hle~t~ ~~an he studied analytically . I~ was shotijrn that surface coneentration of the i ag¢~tary potential has the desired effect of lowering the siren h function values b°`tween resonances . Gn the other hand, ~,lagin ~t al. ' b ) contradict the conclusions of ref.u =} by assetirng that surface abäorptic~n gia~~es nc~ new results beyond those for volume absorption, and is unable to r~:sol~°~.D the discrepancy, between theory and experiment

in the radon 90 < ~ < 130, The pu se cif the present paper is tt~ Llarif~' the° situation by extending the analytic ireatxnent ~is~en in {~) to include the 4~ects ~~f trt~n surf~d' absorption, and to present the results c~f numerical calculations tl~ ~'~,~~' ~ ; nd R' -R for different surface-to-volume ~ . absorption ratios in comparison ~~°itl~ thv ~ ~~~~i itn~~rtt:~l data . and Scattering ,rinn ®® the S_~ Ï a`e ~tr~ength Funct>io® Iden : th

tart an

from the patentief '4~É'i~~{) ""

y. < Ra p . > Ro

°sre r s e ~ ts y n s erical nuclei w ic a ües essi s r ~e s in®®rhit c®u ling, we erive in { ~ closed~ d lin e~ c . le .g an t s tteri cti~n ~°

ti ~ ~ ~ ve stre , t r ti t ese

)j ex ressi~ns red ce t®

--

7t

~ _ t aCi~ CtDS2

1r

c~S .~ 1

°~..

~~_~_:

g .~-

S4n

1

°..

° _? ?

C2~

sin i ßs2 +__ _ _ct®S2_~ sin ï where

® nuclear radius = r® =

i c®sh' 1.J2

-~- r g =

® -- ~ e s " ~

,

2In2,

sin 2 _ --° c®s x ~ C c®s2 -® `

1

° c~s ~~ °

2

~

c®s2

~' + (

~- ~ 1 + Ya'

..~ ~)2

2

n t e regi®n

®dg ? 1 .5 we ~

2

C)

a = ln ®di + X1 -where

R~l ® {

uler's c®ns

ve

nt =

1

6 1

-~- 31n + ~, ® 1} ® i s

. 77

~2

n

~ llllveStigate the ~ e~~S t e 5 ~ (~ t$Cti! ~~ . . selves, as ~ C }' t isc Ss~ s~ l , 1 ®

'~ ®)/

~~~r

.

®u re5$ri~ . .ââ~

~,tr~iét~5

O~d

°,/ r® i )~~g =

(

or

--1-')~ (

~(

Low-~t~EiZGit I~EUTROhfS

occur for

(n integer), (n - )~ n~ + arct g~o~

ir}~ ;~ r

-- .) - 0.1

o~

3ko d i ~- â~a-F-( ;6kodi)~'(a-~-)~~

.1 ~3k o d 1

+ ~~~-$-(16k®di)-'(~-

)~+arctg ~o

(n-~- 8)~-~- ~akodi(~a/~i}

ï)~~a

ott °t~o ~ for the strength function minima differs from that given in (i) by the a ora l ~ et ®, which is small far moderate surf~.ce absorption, but becomes i r surface-to-volume absorption ratios ~a/~i . It describes a dis t f t e stren h function rninin~a from their positions midway between l the so s ion the NC O -scale] to smelter mass numbers, tending to 1 - (0.3719/kod i)

o - -

(9)

.2273 + ln k o d i

in the ' 't of une surface absorption (~ i = 0). This shift gives rise to additional te s ~ . ~ . fia} for the ininnum values of the strength function and modifies the idea ~ .Sb~ and (L57), which are no longer sufficie~ttly accurate in re uct~o fo the ~res of strong surface absorption, ~ues Of ~°~/ at the SitlOns (6) and (7) are t j~ ~ ko

a

+

the f

r,°~

~

(

o

I

~ ko

a arct

o

i+~âi - ~~ akodi(1-5o tg

.~

C4S X i

.

~~ i ( ra ° ~n + â~a ko di x

Yi

} -~~~i

-l -

(10)

i .1684~a)ttg ~i+~o)}~ ( 11 )

widths at half magnum of the res :~nances are given by ~,~ -- 2(~X~)~~ - ~i~n°~)~t+-~~~kod~x .

ßi12)

indicated by lFor itions of the eattren~es of ' (with n~a . .~¬ima and minima t:~e sv ~ r~ ts -~- nd --, res ctively) ~~~ obtain _,

t _rr;~lll

tt:

~~ :5 -

~t )~

~ pr~

rtionl to

t~~t a wrong sign in ~ c~rrr

â~+~~2k®di~ .

~.

in (I with the assumption made _~., ds in accordance facto:ns numerical .~ta~, afgectin~ some cps. ~I .50a~ and ~I

below. ted ïn the forr~iulae liven

~~~~ ~ ~

ide

he values of

. Ra

~

' at these positions are l ..t-

etry wit respect to

s owing a as

® ®~. 1 ®

~( ~

of the a aunt

2Ô~ d i ~ si

~

x~

g-~

z o ~ (

®

(1S~

decreases As can be seen from eq. (IO), the inclusi®n of a surface a sorption to e cte from the ener~l the height of the resonances, which is of course to of 'n° ~/I~ "d ping" eï~ect of imaginary potentials, t the sa e t° e the ° i Mite ` e for elaare raised, but to a considerably smaller extent. n or er to give tion of the effect of surface absorption on the strength function, it i~ therefore necessary first. to define precisely hove the two cases of pure vol e a o tion ( ) and -~ ~ are to co , red, rocs optical coz~hined volume ph~s surface absorption potentials with pure volume absorption are known to `ve t sfacl~ory a eexrtent with the strength function data in the vicinity of the = 3 resonance ~ ~ 5~ ), it appears to be most natural to require that t e inclusion o a s dace a sorpt ion to should not deteriorate the fit in this region. ~ t er ore ti to t at the parameters ~~ and ~z be adjusted such as to give the s e h i L width of the = 3 ~~sonance as pure volume absorption with the ete ~a w ile atl cither potential parameters arP left unchanged . The in e ~ct of s f e a orpt ion g i l l then be reflected in the ratio of ~1r°n°~h)~~~ for the t o ses -~ ) . nc~tin an the nth ma:xixnu and minixne,~rn of I'n°,/L~ by ~"~ and ~~, res eti~el , e obtain fr~ the requirement of equal heights of the r. = 3 resonance ~( -~ j ~~~ ) ; Inserting this into eq . (11 ~, we get s

where k®dig ~z (~Zl~~, k® d~~ = 2~ 1 ~ ~r -- , $g

- 1 .16~4

k0 d1 a

~g

ig ~~ + ~ --. tg x'

(

zf'

_ 1 . Î6~4

&a i «

~ , c ~~ ~.

~dt

j~~ ~~~ ~ ~~ ~~ =~` ~

y

es (t e e

lity

oils only for ~Z -_ 0 , the ratio

(17) is ' ~~.~ ~ ~~ ~ th one end deereaaes with increasing surface-to-volume absoxption ' ~~ t~ a~~i ~ ~~ t o

_ .

1_ _ _

1.168 odi

tgx~-

~ a

tg~1_

1 .168 Q di ~

a 18

s face absorption . of e other résonances relative to that of the n = 3 resoc~ance we (19) ®f t e relatio (20) the re wne

ire eat (16) of equal heights also implies equal width of the n = 3 resonance, the hi her- resonanncs ave broadened unless ~ ! = 0. In the Latter case, O.e . ltt t: ~t of pure surface absorption, all features of the strength function become tndepe~n~ nt of , which aneans equal heights and widths of the resonances, and equal ~ralues rel five sitions ®f (.~n°'i'Di~, ;m . t

' e®tal

st

In fi .1 e cow re the calculated strength functions ~~~ith experimental data "_ a ~) ®f ,~~~ i the region 4ti <~ .~ ~ 160. the curves are computed from the exact expressions 'ven in (I), for three va~ues of the surface-tc~®volume absorption ratio: ~/ ~ ~ his ~.7, end ~, but otherwise unchanged parameters . A similar comparison is de for ' in fig. 2, where the experimental data are obtained from refs . ~' -~3) . X11 e emits o the surface absorption di"cussed in the preceding section are clearly visible : lowering of the strength functi~~n nvn$m1.~m with increasing ~a/~1, = de ï' b eqs . (I7~ and (1 ~ . f`~~~.m eq . X18) we obtain for Y® 5.~ l~eV . frn the reduction ~"~t ~ ~ ~(~') i Q.28 in agreement with the cornpofe .. u

v ~ es.

minimum to lower e shift of the position of the strength function (9). From eq. (9} with increasing ~a,~ ¬ , as d4scribed by eqs . (8) and the n~inia~um ..~~. ~~ct '~ = - °0 .2 5 for r~ -~ ~, and ~~i`~. .~m ~~~~} -- 9~ = 90, in agreefor ~re surface absorption is predicted at ~lm;Q(Sj co used position .

,

_

,,~.~;

t~tit~l ~~ ~, ~o er~. (

i

r

cox~~s eq

~ ~ai of t ~

t

~

~

or

nre

ro

Fi~. ~ . ave tr~n stren~th fuQCt~n a~ a fu~tïen of num : fro refs. 2~°~~. curves arc culat fro~ ~xp ict ~sf v~lues Yg = 52 ~~, r~ _ ) .1~ fm, r; _ +Q.32 fm, d, ~ 3.~ fm, d~ 1~, (duc

~'

~.

data are

~ ~~'~ f~r

~~Il

~v~n

p~~rameter ab~orption

~U ~AGE ~BSORP~ION OF LOW-ENERGY NEL .

ONs

6S3

ced as etry of the maxima and xn.inima of R'/R with res "ct g vvgth ~aj~i as descra~d bY e9" tl5) . ` lculation of the strength function curWes from, the exact exp ressi ons de ~n order to test the accuaac of the approximate forn~u_ae i ~2) Y and. ~ ). in th:e results agree within a few ~rcent, the simple explicit expressions ~2) and j su~`tciently accurate for all practical purposes . Thereby it is possible to le Y e ~°~/ as a function of A by hand or desk calculation wit han ' a few hours .

v

4

6

a

1 V

d

2t~

4C3

60

ß0

~

.00

~

A

.20

t

i40

16C

Fib. 2. Ratio ~f scatttrin,s length R' to nuclear radius R as a function of mass number. The experimiental data ate from refs. ~-~~. The curses are calculated from the exact expressions of ref. !b) for the parameter values Tl~ = 52 31~eV, r~ = 1 .15 fm, r, = 0.32 fm, dl = 3.0 fm, dE A, and the foltc~vlf ng aâ~~tiu>a parameters: ~, = O.Ofi, y~ = 0 (dashed curve) ; as = 0.01, ~s = 0.027 (solid =

curve) ; ~, -= 0, ~$ = O.ß32 (dot-dash curve) .

s the agreement with the ex, riment=~1 data, it can be seen that the eurves of . ~`~~,,,~ for strong surface absr~rption 've indeed the low values required by ex< 13~. because of the considerable scatter of the p~~i~n+~xtt ixt the region 9th ~ ex r ~nentt~l to in this region, it is di cult to single out a particular r*slue of ~ Ïa~ i which yes an t~ptinturn over-all fit, i~~ioreover, the strength function becomes incre s ~ insensitive to the value of 52,'`a5 ~ in the region of strong surface absorption . ~t is~ +~ ~~ ~b~îQtas that the data require a ratio ~;~/~i of at least two, and ~aÎ~a = _ absorp~ "° , c~' ted y theory ~ ~ ), dyes gaol over-all agreerlxent . Pure volume ti~n ~. definitely excluded . For pure volume absorption, low rnirimum s t~ ~s re

fry. square easy try ~ val es of the req ' B order of a 'lade are o course = comes + t °s case the width of the awever, tential . ell f *:gin curve th:e sire uc too small. tr action of surface di~useness raises e~ ' ~ tal cl ta, but ole, thereby increasing the wilt i~, the re °on. of the eorreci hei ht end width ~.i and nünima co e too ig~r . o obtain the _ t enlarge: ~ °ï' ïs, however, of the resonance, the absorption parameter ~ i has to ciently hrt~a re~o ance and e requirements of s raises the ° i a still fu er . s ciently low 'nimum values are therefore irrecancilable for pure valu e absorp-

m

tion . t should noted that our conclusions are not affected by tire disregard in potential {1) of deformation, spin and spin-orbit effects, which account far the deviations of ~< ~ < 8tl and the e~ ri ental data from the calculated curves in the regions ants eri~°e, < 1 ~0, from where our ar < < 1 ~. n the region 90 < respansible defo tlon ~s esseiiltlally absent, a.nd spin and spin-orbit efl~cts may a4) . for e individ 1 variations of the data ~ ~~

ur results clearly contradict the assertions of la ' et al. r~), but are in agreemen ne et _~a~. ~ ~ ~. with the conclusions of refs. ~Z t i~), con 'ng the conjecture of haviour of I~~°~jD, t surface absorption does in fact change t~,e qualitatïve

can be seen without any calculation by observing that for pure surface absorption all its features must be independent of mass nutn r, since the i~nag:in potential is then concentrated in a surface layer of constant thickness independent ofthe nuclear radius, e lowering of the strength function minima is, however, not so obvious an effect . ~s mentioned in section 2, the inclusion of surface absorption ,,fr~r c gad taodu~ic~

absorption even increases the minima of ° )/I~. T`he reduction erect comes about only b~ adjusting the volume absorption parameter so as to leave the height and width of the n = 3 resonance unchanged. this may be one of the reasans ~.~~. lrlagin et al. 1 ~) arrive at conclusions different from, sure .

®ne could of course object that the effects of surface absorption might different for di~f'erent potential s s. owever, previous cor~nputer calculatians of .l~,°}d~.I~ i a i~), with surface abso lion employing various _real and i ° ry potential distributions, also indicate the presence of the efl`+~cts (i)"{ivy clis~rassed in sect. ~. s su ests t t these e acts are at least natively the s t~entiai +~r all q fo s w °ch are e uivalen` as far as the -wave siren h funetïon is caneernec~ . ( ince t e latter is rather sensitive to the "tail" of the real ten~ïal, the condition: ~.~~ "e ~~alence" wIt re °rl irnplies a ce in relatï n tw n ~he dïl~fusen~~> Sara stars of the tentials w ch are to co or e . inst~nce, n order ~~~ o lain a ee ant wit t e siren function cu e calculate b e h c et ~1 . ° ~~ fr a 0o sa on tential it ure volu e a o lïon, c have fou~ld i~

~A

A

R~iïON O~ LOW-ENERGif NELiTRONS

69â

~ ss ~ to i~lerease the value of dl in fYgs . 1 and 2 to abaut twice the value d y = 1 .53 e ared f~°anx the requirement of equal 91~ ~-10 ~ fail-off distances is }, whereas eters remain essentially unchanged . This means that the of ~ the 90 ~-10 'tic~n is not the correct criterion to compare the finite-range fall-old + potential (l~

lth the " ite tail" Moods-Saxon form . In order to give comparable results for Icy miner scattering, the outer slopes of the potentials should be similar, which for , nt al. ~l} i~ achieved by increasing the range R ® at fixed half-fall-offradius R~ . e thecef¬~re conclude that the

experimental values of r~°)j D in the region ~ 130 can % regarded as optical model evidence for low-energy neutrons uch Muse strongly absorbed in the surface than in the nuclear interior, in iu accordance with theoretical consic!erations . Large surface-to-volume absorption rgtios ate also consistent with recent analyses of P-wave neutron strength function data z~, a~~ la`~r

e should lil~e to thank the South African Mutual Life Insurance Society for the generous offer to use their Ferranti Perseus Computer, and in particular 1Vir. E . Bieber and . . l..iprini for th~:ir assistance with the computations . ûne of us {I-I.F. ) is indebrted to the South African Atomic Energy Board for a research bursary . References 1~eShbach, Pe~rier and ~YeisskoRf, Phys. Rev . 96 (i 954) 448 ~;. . Lane and C. 1F. N~andel, Phys. Rev . 9~ (~9ä5) i524 K . i~iarada and ~ . Oda, Progr. ~hec~r . Ptays. ~1 (1959) 260 1~.. ~kuehi, ~uclear Physics 17 (!~)59) 345 t~. L . Shar~r, Ann . of Phys. ~ (1959 ) 1t39 L. ~, domcs, Phys. Rev°. 116 ( ! 959.a 12'_6 V~. Bteni~, ~lucfear Physics 13 (1959) 33 (19SEr) 16~6 .'. Atnster, P:~ys. Rer . 1 Sherman, Phys . Re~,. l~i {1916i i832 Bjorktund, 1~embach and Phys. Re~~. it}9 t l958) l'_95 F. B;jtar lund and S. Fernhach, Lane, Lynn, etkanaff and Rae, Ph~~s . Re~ . Let t . 2 { ! 9 "9 ) 424 1~. ~. hanna and ~. +~. Tan~, Nuelear Ph~-sies 15 { 19591 337 ilets, unpublished : pri~ ate communication to h. K. Seth. See, for inD. là~ . and L. (l959), p . i î4 s~anCe, Proc. intern. Cc~nf. c~n the ~uclear C?Ptic«i t~~Lldel (Tall~ahassee 578 2S {1961) 14j~ T: . t¬rueger and B. il~lar~olis, itiuclear Ph~°sics ~S~ i~, Fiiedeldey and V~. E . Frahn, Ann. of Ph`-s. t% dl96 .1 387 682 1f) ~~ag n, Lyul'~a and t`lemiro~'skit, Soviet Physks TETP ~4 (l962) 461 i'~ ~:x~:hes, ~ixnmertrtan and C~hrien . Phys. Re~~. Lett . 1 i.1g581 28s t$I ~at+~, Bo~itn t and Le Blarac . Pliys. R~:~u . r 1 t s; l958) 258 l ~ Saplakoglu, Botli?n~er and Cot~ . Ph~'ç. it~.~ . 1~ { 1958) ! !'0 Le ianc, Cot and Bolïin~er, ~uclear Physics 14 (l959) . of Phys. 14 ( !«6! ) 3R7 r~itn ~e"son, Smith ancl 2t) Bïipt~~h, th, Bown~tan, Talaony . l i4 ( l~~18 ) 6~î2 2~~ th, u , ~i~n~nerman and C°ïarth . Phys. Re~ . 1 ~ ) ~. t.. ~a tor, Phys. Rev . 91 ( i 913 ) ~~~ . ~.. ~etJ~, uet u Physics 2~ t 19b t ) i 69 Tvew by F. ~~zenber~-Selo~e (Academic Press, ~' ~ ~h F c~, in ~ucle~ar Spectroscopy. edited ï~ ~~ 3a 4~ 5~ 6) ?) $) 9) 1~~ t 1~ 1~~ l ~)

iP"~+~~., 1~ patt ~, P . ltî33 1, ., to i~e ~~ublis)aed . ' 2~~ ~dey and `~. L. ~rahaa, Ann. of Phys