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A distorted wave analysis of quasi-free proton-proton scattering in Li6 and Li7
V¢i~:~e
I, numl~r 7
PHYSICS
DISTORTED PROTON-:PROTON
A
WAVE
LETTERS
ANALYSIS SCATTERING T. B E R G G R E N
I
July 1962
OF QUASI-FREE I N LI 6 A N D LI 7 *
NORDITA, C~penhagen and
G, JACOB ** Insl~futo for Theoretical Physics, University of Copenhagen, Demn~mrk Re~elveCi 8 June 1962
a recent note !) it was stated that the a r g u ments put forward by Inglis 2) to explain the e x p e r imental re~mlts of Garron et aL ~ on the reaction ~!6(p,2p}He ~ seemed to be r a t h e r difficult to r e ~'~neile with these experimental remLlts. At the ~ n e time, a m o r e detailed analysis el this reac~on was promised. While thin a~lysle was being performed, we received the resuhs of Tibell et al. 4) prtor to publication ***. In ref. 4), improved a n g u l ~ ~nd energ7 resolution m ~ l e possible the experimental verification of ¢. mintmum at the angle corresponding- to zero m e " ment~m t r a n s f e r to the recoiling HeS~mcle~,s in the r,eaetion Li6(p,2p)He~ a s prediet~:'~ under quite general assumplions I). These iraproved experimentalresults show the following features." a) Very .arrow peaks in the energy ~ucctrmn corresponding to separation ,energies of 22.4 Me~/e~d 4.8 McV of the two proton, group~; b) Two peaks at 38.50 and 46.5° , separat~ by a ~'n~r~a~attmat ~ 43 ° in, the ang~.dar correlation dlstribu~ion o~ the two oufgofng protons, in the e~tse of the m o r e wealdy bound proton group~ and one pezJ::~/~ 39 o in the a~guhtr eorreladm~ disLribution in the case of the m o r e Strongly bo,~d proton group. These remllts thus agree -¢rlth the sheli modc£ assi~ments for the '/.woproton groups, Le., %hm':the mo;e weakly be,rod proton is a p-proton -~m_dthe m o r e ~rc~ngly bound an s-one. it ~s the purpose of the p r e s ~ note to repot~ and bri~ly dlseuss the rein/Its of a distorted wave ealcu'~ttlon performed for Li 6 r~r~dL i ~ anC to e~mpare fl~ese resnlts with experiment, The i m pulse approximation 5) has been used and the dis* On ~eav¢of :~h.senvefrom the ~tRut~ for T.heore;!cal Physics, UrAverei~y of Uppsala, Sweden. ** Oz~leave of absence from Uni~eraidade do R.~o Gr~mde do SUl, Pt)z~o A]egre, Brazil *** We are ~et~y thankful to G.TibelI for making ~he remilts of his group available to us.
2S8
tortion has been taken into account in the semichssical approxlm~tlon. 6); both approximations should be w:ry good at the e n e r g i e s involved, As has ~een done in the experiment in question, only the eoplanar s y m m e t r i c case ]ms been treated, i.s~, the outgoing proton~ h~Te ~vt~ ! ~-ergie~, a r e in the s a m e plane a s the incoming beam and f o r m equal angles with it. Under these condition~, it h~s ~een shown by M a r i e 7) that the c r o s s seetton f o r the angular correlation dis~ibutlon of the two outgoLug protons 'in a (~,2p) p r o c e s s i s ¢~3~ 4~t ~ "c 2 k l2 s i n 2 o t + M 2c ~
dn I d~ 2 dE 1 - h% 2k2 c 2 k12 sin2el + M2 c 4 x
N
d~fr
'~
(i)
where k ! i s ~he wave Jmmher of both ou~'oLr~g p r o tons and 01 the angle they f o r m with the incoming bear~; vo is the velocity of the incomin,~; proton and N the number ~ protons in the she)] with angular momenb~m L dcfr/d~' i s the f r e e proton-proton ~ e r e n t i a l c r o s s sectlori in the c~n~-e of m a s s s y s t e m at 90 ° , taken at z kinetic energy T = ~_2 c 2 k2 sin2O M c2 in the rest system of one of th~ protons. This cro~s sen'don (in rob/st) ~ n be fated wi~in 5% in the energy intor~l 20 M e V < T (in M e V ) < 8~5 M e V to the exp~rlme~tat ealue~ ~) by the e .)tension
dtfr
230 4;.~9 t'P ~: = 1,9 + ~ + ~ +
-
~mr~. __,.~. - ~~o-) ( T - t40) 'tO 00o + 0.4 T 3 "
(2) Using ~ shugie-partlele wave function %~('~) for the nuclear proton, the distorted moraenttun distribu-
Volume I, number 7
PHYSICS
v ~ z~Ig~m{ 2 m (1)is ~ v ~ by l
z ~ ( ~ ) ~ (2~)"~ y e~p (- i ~. ~) "~ being the momentum of the nuclear prOtOn ~ d Dj(7) the distorting functions
The integrations in (3) a r e to be taken over the c~aesical paths of the incoming and outgoing protons; these paths a r e assumed to be straight lines, thus neglecting reflection and refra..~tion. The d i s totting Fc~.estial h a s been chosen to be of the f o r m and the p a r a m e t e r s V], Wj and bj have been d e t e r mined f r o m the nucie0n-nucieon scatte.-lng a m p l i tudes u~ing the methods given by K e r m a n st aL 9). In o r d e r to be able to compare tbe r e s u l t s with experiments done at different laboratories, an e~.ergy oZ 170 MeV for the incoming protons 1ms been used; thee p ~ r a m e t e r s of the optical potential for the outgoing protons have all been calcuIated at an avez.age energy of 75 MeV. The relevant p a r a m e t e r s a r e l i n e d in table L Teble I Parameters of the optical potential.
~e~°°~ i <~m} {~e~, l ~'~' ~ {~'°~ I {~°~' I t Use h a s been m a d e of single-panicle wave lyricfinns of ~ e forn~ ~,~"(~) = .4 i, Jr/exp (- ~ r) r ~ ( o , ~ ) ,
~l~l
(5)
so a s to give p r e f e r weigh~,: to the low momen~am CO~tpO~ler~s~ Or~ ~t other v,grds, to have a :;orrest asy~p[o~Ic bshaviour ]~ s~ce. In (5), B l is ~.m se~r~tlon energy of the t~rgtons in the different shells and t ~helr an,~lar momenta. The separation e, m r g i o s u ~ 4 a r e 4) Bo(L~.~) = 22.4 MeV ,
BI(Li6) = 4.8 M e ¥ ,
Be(LIT) = Z5.5 M e V ,
BI(LIT) = 10.5 M e V .
%VI~/~these va~%~es the m e a n square radius 10) of L i 7 f s v e r y well reproduced, w h e r e a s a value 25% too large i s obtained f o r Li 5. F o r comlmrtson, harmonic oscilintor wave runetions have also been used, with p a r a m e t e r s adjusted
LETTERS
I July 1962
so a s to f i t the electron scattering data. In the case of Lt 6 the values suggested by Elton 11), who has used different Im~monic oscillator wells for the ~and p-prot~Tn~, w e r e employed. Formulae. (1), (3) and (4) cannot be evab~ted analytically and have therefore been ,. ~Iculated with an electronic computer. The r e s u l t s obtained with the exponential wave functions (5), multiptied by a suitable scale factor so a s to adjust them to the experimental differential c r o s s sections, a r e given by the solid lines in figs. 1 and 2, together with the experimental points of Tlbe]l et al. and G~rron et al. It is seen f~,~ the calculated absolve ma,~ttudes a r e all too l a r g e , ~ t the general shapes of the angular correlation dfsh,~tmtion~ a r e well reproduced, the peaks falling at the correct angles. O n the other hand, using harmor~e os~Itafor wave functions, the oppcsite is true; the peak~ of the auguIRr correlation distribution in the case of the p-protons fall at angles corresponding to too large momenta both for L i 5 and Li 7, but the a b s o lute: magnitudes a r e better reproduced, the factors being 1.72 and 1.32 Instead of 0.25 and 0.50 r e s p e c tively. ~ the c a s e of the s-protons, the shapes of the ~gular correlation distribution for b e ~ types of wave functions a r e ~sout the s a m e , ~ the a b solute magnitudes of the c r o s s sections a r e w o r s e In the e a s e of %be h~--~m-.o."!cosetilator ones; the factors being 0.56 and 0.25 Instead of ONO and 0.80 f o r L i B and Lt 7, respectively. The followh~g r e m a r k s apply to the comparison between the r e s u l t s obtained with the two wave ~.ucdons in question: a) The peaks in the angular correlation distributions fall at the wrong angles in the ease of harmonic oscillator wave funct!ons owing to the fact that the potenttal, beIng infinite, r e duces the amount of low momen ,turn component. b) The absolute magnitudes come out too large in the e a s e of the p - s t a t e s f o r e ~ o n e s t ~ l wave functions partly because the mean square radius of the distorting potential i s too smaP. a s Compared to the m e a n square radius of the p-shell calculated with these wave functions. As ~ r e s u ~ , the dtstor~on has too s m a l l an effect for :heae p-wave func ~ tions, It would certainly be posstble to improve the agreement by increasing the extent of the potential. Such an i n c r e a s e would have the' effect ~ reducing considerably the c r o s s section f o r the p-~protons, probably without affecting too strongly ~aat f o r the s-protons. It i s also seen that the m i n i m u m at z e r o m o mentum t r a n s f e r i s filled in v e r y little. In o r d e r to s e e to what extent the finite angular resolution f i l l s in the dip, the angu]s:," resolution u ~ d in the experLments of Tibell et al. has been folded in *. * We are very greatful to A.Jehanssvu for performing this calealatlen for u~ with his programme.
~59
Volume 1, ~ a b e r 7
PHi'/3ICS L E T T E R S
~3oo ~ . .
0,~0 x cotcul~ted
½0rs~y
: lioo,
ref 3)
t July 1~2
~
~
0 25 x catculnled
,oo;Ig
, 'tl
1
% ,
/
.r---~,
.
tlts
, '\'\
,
Fig. 1, Co,~parlson of the ¢,xperh~6eatal and the calculated angular correlation dlstrllmtton~ for S- and p-@roton$ H Li . The absolute values in the Uppsala results uomstltate a p~ulimlx~ry estimate d ~'Ibell et al. The ~Mshed curve result~ from folding in the finl~ angular re~k'tk~n quoted in r~f. 4).
300 ~
0 ~0 x CO~UlOt~d
} Orsay, re'l, 31.
--
0,50 x coEul~ted
~-
~ l with O~I~UIGP rec~MiOn
i~OO
folded in
200 ~
{ ~t=_ .
b fi'
;oo
l
3g
1,0
5G
60
e~
30
40
50
60
i
Fi~. 2. The same a~ ~ig. 1 for Li~. The r e s u l t s a r e Hdica~ed by the dashed curves figs. I and 2. it i s ~ee.~ ~ha% there s t i l l r e m ~ s a p~rt of the f i l l H g Ln of ~he ~ip $~ ~he ease e.~ Li 6 t o be explained. ~ might be that ~ere ~-s a smal~ con~ibu~ion d s - s t ~ e In the overlap lute~ral of Li6 and He ~ which would be sd[inlsn~ to f i l l in the dip so ~s to agree v l t h the expsriment~I r e s i t BeBides, i t Should be berne H mind t h a t t h e refraction of t/~e ~ o t o u paths in the mlelear pct~utinl, which has bea~ negle:ted in these c:~l,.~ul~ttons0 cos/d also have a s m e a r i n g out effect en the angul~r eorre]atton ~,li~rlbuflon. Ther~o~,e no d~fi'dte conclusions can be drawn yet a s to the reason for the h~"{fe f i l l i n g in o~ the dip. Flm%lly, a r e m a r k shc,~hi be made regarding the 4 i a g r e s m e ~ t d absolute values. When a proton i s knock~d out of ~ nucleus, the residual nucleus i s not left ~tth unit probebili~y In its ground e'~t~, but.. with ~mt~ prgbabillt~, i n many other st~tes; Using : as c~verla~ integrals betwee~ i n i t i a l and fiuak ~'.~te~ ~J~ ~7~ve ~t~Jol,s (~)~ i t has been assumed t.hat the ~ 1 ~ t h s were fruity in. each case. Iu the p a r 260
ticuinr case of L l 6 , this p a r t i a l ddf~h zhoutd be rather small, because i t invo1~,'es the o v e r ~ p lnt e g r ~ of ~ e initinl and f H a l sb~tes; m t h e Luit~al state the neatrori i s bound (its wave funetiun therefore having an expon@tinl tall), whereas in the final state i t i s unbolmd (thus having an osci~intory wave fur,.~tion). Evidently the outgoh~g_protons cortes.pending %o e~eats in which the He nacleus i s ~ot left in the ~- resonance state will, expertu ext~ll~', constitute a smooth background s p r e ~ t over a comparatively large range of energies. Slrni~2r r e m a r k s @ply to the case of L i ~ , exe@t that, due to tJ~e h o t that the r e , I d o l lie ~ nucleus is bound, ells would expect '.his effect to be much smaller. Thanks are @as te G. Tihell and A. Johanss~u for m a n y helpful discussions on the U p p s a h eAg)erLmeninl results° "~peelal t b ~ k s a r e due t o PerE r i k Pe~sson for his kind assis~mce in testing and rum:tug the compvter programme with the Mercur)"
Volume I, n~mber 7
PIIYSICS
1 July 1962
3) J. P. Garron et al., Phys, Rev. I.~tters 7 (1961) 25':. a~d preprint, to he published in Nuclear Physics. 4) G. Ttbell, O.Stmdberg and U,Miklav~16, Physics Letters 1 (1962) 172. 5) O, F, Chew, Phys. Rev. 80 (1950) 196. 6) G. P. McCauley and G.E. Brown, Proc. Phys. Soc. 71
Of AD Atomenergi, Stockholr~. Correspondence with Th.A, J, Marls and conversations with G. E, Brown were extremely helpful t~ ~.~s. O~e of u s (G,J.) gratefully acknowledges a felh~w~;hlp from the Ford Fonndatinn and a t r a v e l l i n g grant ~rorn Campunha Nac~onal de Aperfeisoamento dc P e ~ = stral de Nfvel Supe~inr (CAPES), BraeiL
(1958)
l~ferences 1) T. Ber£grell, G.E. Browr~ and G. J~cob. Physics Letters 1 (19~3) 88. 2), D.R. Iaglfs, Prec. of the Rutherford Jubilee Int. Conf. (Heywo~d ~ Co. )r)r~,, Leyden, 1961), p, 637 and preprl~, to be pabl:'shed fn :Phys. Roy.
ENERGY
LETTERS
893.
7) Th.A.J.Merls, .~clearPhysics 9 (1958/59) 577, 8) V.S. Barasheak~v an~ V. M. Maltscv, Forlschrltte der Phyelk 9 (1951) 549. 9) A.K.Kerman, H.MeManus and R.M.Thaler, A~n. of l)hys. 8 (19~) 551. 10) R.Herm~.a ~ ~LHofstadicr, High-energ 7 electron scettering f~blc~ (Sta~ford University Press, Stanford, California, 1960), p. ~2. 11) L.R.B. Elton, N~wlear size~ (O~dord Uvlvernity Prc~5~ London, 19~1), p. ~5.
LEVELS
O F 0 19
S, K. ~ A H Physical Researoh Laboratory, ~medabad, Tndia * Re~eJved 7 J u ~ 1962
In ~ prevlous paper 1) w e h~ve discussed the n~%ure of the effective n~ clear interaction in T = I ~tates of nuclei 018, T I S 0 etc. The parameters of the in~eractlon in singiet even and triplet odd states of the two nucleons outside the closed shells were determined under t~/o assumptions (a) the even s l a t e h~terac~fons are non-ineal and are effective in ! = O {-~ r~srS to re,T~c;,e o r ~ t a l augular m c m ~ n "turn) s-states unly, and (b) the even state interactions ..... 912 4 .... L ' a l ~ are the same in a l t states ~ = 0, 2 etc. For 0 18 the ~ r/S~2zS/S. . . . . . : S12 ~ 9 1 2 lrara~aeters of the even interactions (assumed to be Z 7/2 ~712 Gaus~L~tn e ~ e ~ o e x p ( . y 2 / v ~ ) ) were determined 3 to be. (a) V o ~ -25 M e V , ~ = Vo/7/= 1.0 end (b) V o = "9/2 ....... ;~2 • ,SIS -40 l~:eV, X ~ 0.65, and s e v e r a l sets of correspond~ $ / S - ing odd state interactions which give a good fit to 2 the O 18 speotr~m were given. In this note w e apply - - I12 ---112 these different sets of ~arameters to calculate the I - 112 energy level spectrum ~ 019 , to see ff thqse addi. . . . . . !112 . . . . . . . . . . I I S ttonal data c~u h e l p to distinguish b e t w e e n t h e d f f * ~; _ _ ~ ~1/2 .. _ 31;I O -.~12 = :: : 5 1 2 ~ 512 : SI~I ferent sets, T = 3/2 states of O1~ arising from the conflgV(a~ t~) cc~ Cd~ rations (d~/2)2, (d~/2) '~ ( e l / 2 ) aud (dsl2) (S1/~):,2 are considered. T h e resuRs a r e sho~m i~ fig. I. Ffg. 1. The energy levels of O19: (a) c~lculated W e f~rd that iz~ case (a) the odd st~e interachons Vo = -25 MeV, ~ = loO (Is-state inmrantlo~ on!y) and V 1 = - 6.3 M e W )~= l.O (~tS-BtateInterac.~ppc~r .% ~ v e only a s m a l l effect on the ener~;ies tions) ; O~ s a ! c ~ l wl~ v o ~ - 4 o MsV, ~ ffi O . 6 5 of ~ e dff~vren~ levels, ~ud a l l the dlffez-ent sets Of ( s- and ~d-staP~/nteractiozm) and ~. = - 25.3 odd interaction given i u r e f , I) give essentially the MeV, ~.= 0.5 (~p-siate lntersctiams); (e) calcusame results. The level scheme for one t n t e r ~ ; lated by Talmi ar~ Uaas; (d) e~pertmestal level tion iS shown in fig. i. spectrum. The state J = 9,J2 ~ ner~A.~Jisad u * Work supported by M.S. Univerni~:]of Baroda, Baroda. ground staff.
i
I, numl~r 7
PHYSICS
DISTORTED PROTON-:PROTON
A
WAVE
LETTERS
ANALYSIS SCATTERING T. B E R G G R E N
I
July 1962
OF QUASI-FREE I N LI 6 A N D LI 7 *
NORDITA, C~penhagen and
G, JACOB ** Insl~futo for Theoretical Physics, University of Copenhagen, Demn~mrk Re~elveCi 8 June 1962
a recent note !) it was stated that the a r g u ments put forward by Inglis 2) to explain the e x p e r imental re~mlts of Garron et aL ~ on the reaction ~!6(p,2p}He ~ seemed to be r a t h e r difficult to r e ~'~neile with these experimental remLlts. At the ~ n e time, a m o r e detailed analysis el this reac~on was promised. While thin a~lysle was being performed, we received the resuhs of Tibell et al. 4) prtor to publication ***. In ref. 4), improved a n g u l ~ ~nd energ7 resolution m ~ l e possible the experimental verification of ¢. mintmum at the angle corresponding- to zero m e " ment~m t r a n s f e r to the recoiling HeS~mcle~,s in the r,eaetion Li6(p,2p)He~ a s prediet~:'~ under quite general assumplions I). These iraproved experimentalresults show the following features." a) Very .arrow peaks in the energy ~ucctrmn corresponding to separation ,energies of 22.4 Me~/e~d 4.8 McV of the two proton, group~; b) Two peaks at 38.50 and 46.5° , separat~ by a ~'n~r~a~attmat ~ 43 ° in, the ang~.dar correlation dlstribu~ion o~ the two oufgofng protons, in the e~tse of the m o r e wealdy bound proton group~ and one pezJ::~/~ 39 o in the a~guhtr eorreladm~ disLribution in the case of the m o r e Strongly bo,~d proton group. These remllts thus agree -¢rlth the sheli modc£ assi~ments for the '/.woproton groups, Le., %hm':the mo;e weakly be,rod proton is a p-proton -~m_dthe m o r e ~rc~ngly bound an s-one. it ~s the purpose of the p r e s ~ note to repot~ and bri~ly dlseuss the rein/Its of a distorted wave ealcu'~ttlon performed for Li 6 r~r~dL i ~ anC to e~mpare fl~ese resnlts with experiment, The i m pulse approximation 5) has been used and the dis* On ~eav¢of :~h.senvefrom the ~tRut~ for T.heore;!cal Physics, UrAverei~y of Uppsala, Sweden. ** Oz~leave of absence from Uni~eraidade do R.~o Gr~mde do SUl, Pt)z~o A]egre, Brazil *** We are ~et~y thankful to G.TibelI for making ~he remilts of his group available to us.
2S8
tortion has been taken into account in the semichssical approxlm~tlon. 6); both approximations should be w:ry good at the e n e r g i e s involved, As has ~een done in the experiment in question, only the eoplanar s y m m e t r i c case ]ms been treated, i.s~, the outgoing proton~ h~Te ~vt~ ! ~-ergie~, a r e in the s a m e plane a s the incoming beam and f o r m equal angles with it. Under these condition~, it h~s ~een shown by M a r i e 7) that the c r o s s seetton f o r the angular correlation dis~ibutlon of the two outgoLug protons 'in a (~,2p) p r o c e s s i s ¢~3~ 4~t ~ "c 2 k l2 s i n 2 o t + M 2c ~
dn I d~ 2 dE 1 - h% 2k2 c 2 k12 sin2el + M2 c 4 x
N
d~fr
'~
(i)
where k ! i s ~he wave Jmmher of both ou~'oLr~g p r o tons and 01 the angle they f o r m with the incoming bear~; vo is the velocity of the incomin,~; proton and N the number ~ protons in the she)] with angular momenb~m L dcfr/d~' i s the f r e e proton-proton ~ e r e n t i a l c r o s s sectlori in the c~n~-e of m a s s s y s t e m at 90 ° , taken at z kinetic energy T = ~_2 c 2 k2 sin2O M c2 in the rest system of one of th~ protons. This cro~s sen'don (in rob/st) ~ n be fated wi~in 5% in the energy intor~l 20 M e V < T (in M e V ) < 8~5 M e V to the exp~rlme~tat ealue~ ~) by the e .)tension
dtfr
230 4;.~9 t'P ~: = 1,9 + ~ + ~ +
-
~mr~. __,.~. - ~~o-) ( T - t40) 'tO 00o + 0.4 T 3 "
(2) Using ~ shugie-partlele wave function %~('~) for the nuclear proton, the distorted moraenttun distribu-
Volume I, number 7
PHYSICS
v ~ z~Ig~m{ 2 m (1)is ~ v ~ by l
z ~ ( ~ ) ~ (2~)"~ y e~p (- i ~. ~) "~ being the momentum of the nuclear prOtOn ~ d Dj(7) the distorting functions
The integrations in (3) a r e to be taken over the c~aesical paths of the incoming and outgoing protons; these paths a r e assumed to be straight lines, thus neglecting reflection and refra..~tion. The d i s totting Fc~.estial h a s been chosen to be of the f o r m and the p a r a m e t e r s V], Wj and bj have been d e t e r mined f r o m the nucie0n-nucieon scatte.-lng a m p l i tudes u~ing the methods given by K e r m a n st aL 9). In o r d e r to be able to compare tbe r e s u l t s with experiments done at different laboratories, an e~.ergy oZ 170 MeV for the incoming protons 1ms been used; thee p ~ r a m e t e r s of the optical potential for the outgoing protons have all been calcuIated at an avez.age energy of 75 MeV. The relevant p a r a m e t e r s a r e l i n e d in table L Teble I Parameters of the optical potential.
~e~°°~ i <~m} {~e~, l ~'~' ~ {~'°~ I {~°~' I t Use h a s been m a d e of single-panicle wave lyricfinns of ~ e forn~ ~,~"(~) = .4 i, Jr/exp (- ~ r) r ~ ( o , ~ ) ,
~l~l
(5)
so a s to give p r e f e r weigh~,: to the low momen~am CO~tpO~ler~s~ Or~ ~t other v,grds, to have a :;orrest asy~p[o~Ic bshaviour ]~ s~ce. In (5), B l is ~.m se~r~tlon energy of the t~rgtons in the different shells and t ~helr an,~lar momenta. The separation e, m r g i o s u ~ 4 a r e 4) Bo(L~.~) = 22.4 MeV ,
BI(Li6) = 4.8 M e ¥ ,
Be(LIT) = Z5.5 M e V ,
BI(LIT) = 10.5 M e V .
%VI~/~these va~%~es the m e a n square radius 10) of L i 7 f s v e r y well reproduced, w h e r e a s a value 25% too large i s obtained f o r Li 5. F o r comlmrtson, harmonic oscilintor wave runetions have also been used, with p a r a m e t e r s adjusted
LETTERS
I July 1962
so a s to f i t the electron scattering data. In the case of Lt 6 the values suggested by Elton 11), who has used different Im~monic oscillator wells for the ~and p-prot~Tn~, w e r e employed. Formulae. (1), (3) and (4) cannot be evab~ted analytically and have therefore been ,. ~Iculated with an electronic computer. The r e s u l t s obtained with the exponential wave functions (5), multiptied by a suitable scale factor so a s to adjust them to the experimental differential c r o s s sections, a r e given by the solid lines in figs. 1 and 2, together with the experimental points of Tlbe]l et al. and G~rron et al. It is seen f~,~ the calculated absolve ma,~ttudes a r e all too l a r g e , ~ t the general shapes of the angular correlation dfsh,~tmtion~ a r e well reproduced, the peaks falling at the correct angles. O n the other hand, using harmor~e os~Itafor wave functions, the oppcsite is true; the peak~ of the auguIRr correlation distribution in the case of the p-protons fall at angles corresponding to too large momenta both for L i 5 and Li 7, but the a b s o lute: magnitudes a r e better reproduced, the factors being 1.72 and 1.32 Instead of 0.25 and 0.50 r e s p e c tively. ~ the c a s e of the s-protons, the shapes of the ~gular correlation distribution for b e ~ types of wave functions a r e ~sout the s a m e , ~ the a b solute magnitudes of the c r o s s sections a r e w o r s e In the e a s e of %be h~--~m-.o."!cosetilator ones; the factors being 0.56 and 0.25 Instead of ONO and 0.80 f o r L i B and Lt 7, respectively. The followh~g r e m a r k s apply to the comparison between the r e s u l t s obtained with the two wave ~.ucdons in question: a) The peaks in the angular correlation distributions fall at the wrong angles in the ease of harmonic oscillator wave funct!ons owing to the fact that the potenttal, beIng infinite, r e duces the amount of low momen ,turn component. b) The absolute magnitudes come out too large in the e a s e of the p - s t a t e s f o r e ~ o n e s t ~ l wave functions partly because the mean square radius of the distorting potential i s too smaP. a s Compared to the m e a n square radius of the p-shell calculated with these wave functions. As ~ r e s u ~ , the dtstor~on has too s m a l l an effect for :heae p-wave func ~ tions, It would certainly be posstble to improve the agreement by increasing the extent of the potential. Such an i n c r e a s e would have the' effect ~ reducing considerably the c r o s s section f o r the p-~protons, probably without affecting too strongly ~aat f o r the s-protons. It i s also seen that the m i n i m u m at z e r o m o mentum t r a n s f e r i s filled in v e r y little. In o r d e r to s e e to what extent the finite angular resolution f i l l s in the dip, the angu]s:," resolution u ~ d in the experLments of Tibell et al. has been folded in *. * We are very greatful to A.Jehanssvu for performing this calealatlen for u~ with his programme.
~59
Volume 1, ~ a b e r 7
PHi'/3ICS L E T T E R S
~3oo ~ . .
0,~0 x cotcul~ted
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~
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/
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.
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,
Fig. 1, Co,~parlson of the ¢,xperh~6eatal and the calculated angular correlation dlstrllmtton~ for S- and p-@roton$ H Li . The absolute values in the Uppsala results uomstltate a p~ulimlx~ry estimate d ~'Ibell et al. The ~Mshed curve result~ from folding in the finl~ angular re~k'tk~n quoted in r~f. 4).
300 ~
0 ~0 x CO~UlOt~d
} Orsay, re'l, 31.
--
0,50 x coEul~ted
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b fi'
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60
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30
40
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Fi~. 2. The same a~ ~ig. 1 for Li~. The r e s u l t s a r e Hdica~ed by the dashed curves figs. I and 2. it i s ~ee.~ ~ha% there s t i l l r e m ~ s a p~rt of the f i l l H g Ln of ~he ~ip $~ ~he ease e.~ Li 6 t o be explained. ~ might be that ~ere ~-s a smal~ con~ibu~ion d s - s t ~ e In the overlap lute~ral of Li6 and He ~ which would be sd[inlsn~ to f i l l in the dip so ~s to agree v l t h the expsriment~I r e s i t BeBides, i t Should be berne H mind t h a t t h e refraction of t/~e ~ o t o u paths in the mlelear pct~utinl, which has bea~ negle:ted in these c:~l,.~ul~ttons0 cos/d also have a s m e a r i n g out effect en the angul~r eorre]atton ~,li~rlbuflon. Ther~o~,e no d~fi'dte conclusions can be drawn yet a s to the reason for the h~"{fe f i l l i n g in o~ the dip. Flm%lly, a r e m a r k shc,~hi be made regarding the 4 i a g r e s m e ~ t d absolute values. When a proton i s knock~d out of ~ nucleus, the residual nucleus i s not left ~tth unit probebili~y In its ground e'~t~, but.. with ~mt~ prgbabillt~, i n many other st~tes; Using : as c~verla~ integrals betwee~ i n i t i a l and fiuak ~'.~te~ ~J~ ~7~ve ~t~Jol,s (~)~ i t has been assumed t.hat the ~ 1 ~ t h s were fruity in. each case. Iu the p a r 260
ticuinr case of L l 6 , this p a r t i a l ddf~h zhoutd be rather small, because i t invo1~,'es the o v e r ~ p lnt e g r ~ of ~ e initinl and f H a l sb~tes; m t h e Luit~al state the neatrori i s bound (its wave funetiun therefore having an expon@tinl tall), whereas in the final state i t i s unbolmd (thus having an osci~intory wave fur,.~tion). Evidently the outgoh~g_protons cortes.pending %o e~eats in which the He nacleus i s ~ot left in the ~- resonance state will, expertu ext~ll~', constitute a smooth background s p r e ~ t over a comparatively large range of energies. Slrni~2r r e m a r k s @ply to the case of L i ~ , exe@t that, due to tJ~e h o t that the r e , I d o l lie ~ nucleus is bound, ells would expect '.his effect to be much smaller. Thanks are @as te G. Tihell and A. Johanss~u for m a n y helpful discussions on the U p p s a h eAg)erLmeninl results° "~peelal t b ~ k s a r e due t o PerE r i k Pe~sson for his kind assis~mce in testing and rum:tug the compvter programme with the Mercur)"
Volume I, n~mber 7
PIIYSICS
1 July 1962
3) J. P. Garron et al., Phys, Rev. I.~tters 7 (1961) 25':. a~d preprint, to he published in Nuclear Physics. 4) G. Ttbell, O.Stmdberg and U,Miklav~16, Physics Letters 1 (1962) 172. 5) O, F, Chew, Phys. Rev. 80 (1950) 196. 6) G. P. McCauley and G.E. Brown, Proc. Phys. Soc. 71
Of AD Atomenergi, Stockholr~. Correspondence with Th.A, J, Marls and conversations with G. E, Brown were extremely helpful t~ ~.~s. O~e of u s (G,J.) gratefully acknowledges a felh~w~;hlp from the Ford Fonndatinn and a t r a v e l l i n g grant ~rorn Campunha Nac~onal de Aperfeisoamento dc P e ~ = stral de Nfvel Supe~inr (CAPES), BraeiL
(1958)
l~ferences 1) T. Ber£grell, G.E. Browr~ and G. J~cob. Physics Letters 1 (19~3) 88. 2), D.R. Iaglfs, Prec. of the Rutherford Jubilee Int. Conf. (Heywo~d ~ Co. )r)r~,, Leyden, 1961), p, 637 and preprl~, to be pabl:'shed fn :Phys. Roy.
ENERGY
LETTERS
893.
7) Th.A.J.Merls, .~clearPhysics 9 (1958/59) 577, 8) V.S. Barasheak~v an~ V. M. Maltscv, Forlschrltte der Phyelk 9 (1951) 549. 9) A.K.Kerman, H.MeManus and R.M.Thaler, A~n. of l)hys. 8 (19~) 551. 10) R.Herm~.a ~ ~LHofstadicr, High-energ 7 electron scettering f~blc~ (Sta~ford University Press, Stanford, California, 1960), p. ~2. 11) L.R.B. Elton, N~wlear size~ (O~dord Uvlvernity Prc~5~ London, 19~1), p. ~5.
LEVELS
O F 0 19
S, K. ~ A H Physical Researoh Laboratory, ~medabad, Tndia * Re~eJved 7 J u ~ 1962
In ~ prevlous paper 1) w e h~ve discussed the n~%ure of the effective n~ clear interaction in T = I ~tates of nuclei 018, T I S 0 etc. The parameters of the in~eractlon in singiet even and triplet odd states of the two nucleons outside the closed shells were determined under t~/o assumptions (a) the even s l a t e h~terac~fons are non-ineal and are effective in ! = O {-~ r~srS to re,T~c;,e o r ~ t a l augular m c m ~ n "turn) s-states unly, and (b) the even state interactions ..... 912 4 .... L ' a l ~ are the same in a l t states ~ = 0, 2 etc. For 0 18 the ~ r/S~2zS/S. . . . . . : S12 ~ 9 1 2 lrara~aeters of the even interactions (assumed to be Z 7/2 ~712 Gaus~L~tn e ~ e ~ o e x p ( . y 2 / v ~ ) ) were determined 3 to be. (a) V o ~ -25 M e V , ~ = Vo/7/= 1.0 end (b) V o = "9/2 ....... ;~2 • ,SIS -40 l~:eV, X ~ 0.65, and s e v e r a l sets of correspond~ $ / S - ing odd state interactions which give a good fit to 2 the O 18 speotr~m were given. In this note w e apply - - I12 ---112 these different sets of ~arameters to calculate the I - 112 energy level spectrum ~ 019 , to see ff thqse addi. . . . . . !112 . . . . . . . . . . I I S ttonal data c~u h e l p to distinguish b e t w e e n t h e d f f * ~; _ _ ~ ~1/2 .. _ 31;I O -.~12 = :: : 5 1 2 ~ 512 : SI~I ferent sets, T = 3/2 states of O1~ arising from the conflgV(a~ t~) cc~ Cd~ rations (d~/2)2, (d~/2) '~ ( e l / 2 ) aud (dsl2) (S1/~):,2 are considered. T h e resuRs a r e sho~m i~ fig. I. Ffg. 1. The energy levels of O19: (a) c~lculated W e f~rd that iz~ case (a) the odd st~e interachons Vo = -25 MeV, ~ = loO (Is-state inmrantlo~ on!y) and V 1 = - 6.3 M e W )~= l.O (~tS-BtateInterac.~ppc~r .% ~ v e only a s m a l l effect on the ener~;ies tions) ; O~ s a ! c ~ l wl~ v o ~ - 4 o MsV, ~ ffi O . 6 5 of ~ e dff~vren~ levels, ~ud a l l the dlffez-ent sets Of ( s- and ~d-staP~/nteractiozm) and ~. = - 25.3 odd interaction given i u r e f , I) give essentially the MeV, ~.= 0.5 (~p-siate lntersctiams); (e) calcusame results. The level scheme for one t n t e r ~ ; lated by Talmi ar~ Uaas; (d) e~pertmestal level tion iS shown in fig. i. spectrum. The state J = 9,J2 ~ ner~A.~Jisad u * Work supported by M.S. Univerni~:]of Baroda, Baroda. ground staff.
i