VOLUME 16, NUMBER 3 P H Y S I C A L R E V I E W L E T T E R S 17 JANUARY 1966

DETERMINATION OF THE NEUTRON-NEUTRON SCATTERING LENGTH FROM THE REACTION T(d, He3)2rct

E. Baumgar tner ,* H. E. Conzett, E. Shield, and R. J. Slobodrian

Lawrence Radiation Laboratory, University of California, Berkeley, California (Received 13 December 1965)

Recent p a p e r s 1 have r e - e m p h a s i z e d the need for an accu ra t e de te rmina t ion of the neu t ron-neutron lS0 s ca t t e r ing length ari9 s ince c o m ­par i son with the pp ZLiidnp s ca t t e r ing lengths, ap and lanp, p rovides a t e s t of charge s y m m e ­t ry and charge independence in the nucleon-nucleon interact ion.

Severa l de te rmina t ions of an have recent ly been repor ted . 2 " 5 The reac t ions studied exper ­imental ly fall into two ca tegor ies : (1) the r e ­action 7r~ +d — 2n +y, with only two st rongly in terac t ing pa r t i c l e s in the final s ta te , and (2) the reac t ions D(n9p)2n and T(n,d)2n which have th ree pa r t i c l e s that may in te rac t s t r ong ­ly in the final s t a t e . The most r ecen t i nves ­tigation4 of the Tr~~ + d reac t ion has resu l ted in the p r e l im ina ry value

a = - 1 6 . 4 ± 1 . 3 F ,

where the sign was as sumed and the e r r o r introduced by the theore t ica l uncer ta in t i es 6

was not included in the quoted probable e r r o r . In cont ras t , ana lyses of the data f rom the r e ­action D(n9p)2n at 14 MeV have given the v a l ­ues2*3

and

a = - 2 1 ± 2 F n

a = - 2 3 . 6 i ? ' ° F ,

which a r e consis tent within themse lves but which do not a g r e e with the u" +d r e su l t . F i ­nally, the reac t ion T(n,d)2n has provided the value 5

a = - 1 8 ± 3 F . n

Neither the D(n,p)2n nor the T(n,d)2n s p e c ­t r a could be in t e rp re ted in t e r m s of the final-s ta te neu t ron-neut ron in teract ion alone.7"9

Therefore , t r e a t m e n t s were used that includ­ed in te rac t ions with the th i rd f ina l - s ta te p a r ­t ic le 3 o r effects produced by the reac t ion mech ­an i sm itself.5 The d i sc repanc i e s among the values obtained for an c lea r ly indicate the need

for a t es t of the theory used to de te rmine an

from exper iments other than that of neu t ron-neutron sca t t e r ing .

We repor t he re a de terminat ion of an f rom analys is of He3 s p e c t r a from the reac t ion T(d9

He3)2n. Also, t r i ton s p e c t r a w e r e obtained from the m i r r o r reac t ion Ke3(d9t)2p. Analy­s i s of these spec t r a for a de terminat ion of ap9

which i s well establ ished1 0 (ap= - 7 . 7 1 9 ± 0 . 0 0 8 F) from low-energy pro ton-proton sca t t e r ing da­ta , provides a d i r ec t t e s t of the theory used to deduce an.

The react ion Re3(d9t)2t> has been invest igated by s eve ra l groups, 1 1" 1 6 and it has been shown that t he re i s a good semiquant i ta t ive ag reemen t between the f ina l - s t a te - in te rac t ion calculat ions and the measu red t r i ton spectra . 1 2? 1 5 The r e ­action T(d, He3)2w was f i r s t studied s e v e r a l y e a r s ago.1 7

The p resen t exper iment was c a r r i e d out at the Berkeley 88-inch va r i ab l e - ene rgy cyclo­t ron. The react ion T(d9 He3)2n was studied at deuteron energ ies (E^) of 32.5 and 40.2 MeV. The He3 s pec t r a w e r e obtained with a r e s o l u ­tion of 240 keV at s e v e r a l l abora tory angles between 6° and 25°. The He3(d, t)2p was s tud­ied 2XE(X- 29.8 MeV (to match the f ina l - s ta te c m . energy of the previous reac t ion at 32.5 MeV) with a resolut ion of 140 keV. The expe r ­imental techniques were essent ia l ly the s a m e a s desc r ibed in an e a r l i e r paper . 1 2

Figure 1(a) contains the t r i ton s p e c t r u m m e a ­su red at a labora tory angle of 8°, and Fig. 1(b) exhibits the He3 spec t rum taken at 6°. The f i t ­ted curves we re calculated from the Watson-Migdal theory. At these angles , no effects ofp-T o r n - H e 3 f ina l - s ta te in te rac t ions w e r e seen. It has been es tabl i shed that both r e a c ­tions proceed through a d i r ec t reac t ion m e c h ­an i sm at s m a l l angles , 1 1 ' 1 3 ' 1 4 ' 1 7 consis tent with an I = 0 nucleon t r ans f e r . The pe r iphe ra l na ­tu re of the reac t ion then impl ies a sma l l o v e r ­lap of the outgoing T o r He3 wave function with the wave function of the nucleon pa i r , and thus a sui table descr ip t ion should be provided by a dominant nucleon-nucleon f ina l - s ta te i n t e r ­action.

In the studied r e a r r a n g e m e n t col l is ions the

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VOLUME 16, NUMBER 3 P H Y S I C A L R E V I E W L E T T E R S 17 JANUARY 1966

6 0 0

5 0 0

4 0 0

3 0 0

2 0 0

100

— i — r ~ - - " i i !

11 268 events

i.,,

events/; •// /

/ l A / / '

[—

(a)

1 i 1 i 1

1

He:

£ * ] • Elab =

•

, .!__

i 1

5(d,T)2p 29.8 MeV

& 0|ab-8deg J

\ \ 1 \ '$ 1 •4 H

\u 1 1 1 1 |

1

-

i *

\ •

24.5 25.0 25.5 26.0

Energy (MeV)

500

4 0 0

300

200

100

H 3 (d ,He 3 )2n

E lab= 32.5 MeV

0 | a b = 6deg

26.5 27.0 27.5 28.0

Energy (MeV)

28.5

FIG0 1. (a) Triton spectrum at 8° lab of the reaction H.e*{d,t)2p at 29.8 MeV with theoretical fits calculated for 11268 events and for 21 798 events. The dots are experimental points.. The solid line is the best fit for 11268 events with ap =-7.69 F. The dashed line is the best fit for 21 798 events with ap = -7.41 F. The dash-dot line is obtained with ap =-6.90 F, for 11 268 events, and the dash-double dot line is obtained with ap =-8.33 F; they in­dicate the sensitivity of the theoretical curve to variation of ap. (b) He3 spectrum at 6° lab of the reaction T(d, He3)2w at 32.5 MeV with theoretical fits calculated for 17 782 events. The solid line is the best fit for an = -16.1 Fo The dashed line is obtained with an =-14.0 F and the dash-dot line corresponds to an =-18.0 F,

differential c r o s s sect ion can be expres sed a s

d2a 2?r 1 dEdQ, K v , if v "

d J (1)

where Tzy is the t rans i t ion ma t r ix e lement and p{E) is the phase - space fac tor . The genera l form of Tif i s Tif= Styf^Vjipidr, where Vj i s the in teract ion causing the r e a r r a n g e m e n t . F o r the react ion T(d9

Ee3)2n at sma l l angles , we can wr i te 7 ipf=^fte3<Pjif(r)(ei6sin5)/k, and therefore

-id —J\f{r)*^JvfldT =g(9)

-id sin6 (2)

The t e r m (e"^sind)/k comes in s imply through the S wave function of the neutron pai r in the final s ta te , 6 is the singlet S-wave phase shift, Hk i s the re la t ive nucleon-nucleon momentum, * H e 3 i s the internal wave function of He3 , and cp^ d e s c r i b e s the re la t ive motion between the He3 nucleus and the neutron pa i r . The function g(0) depends on the reac t ion mechanism, 9 being the c m . angle of the ob-

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VOLUME 16, NUMBER 3 P H Y S I C A L R E V I E W L E T T E R S 17 JANUARY 1966

served particle (i.e., He3). For the reaction T(d,YLe3)2n, we have used

IT l2= \g(6) I2

J 2n v ' nn ' n rn 2nf

and for the reaction8 He3(d, t)2p,

1 V*= lg{6) |2 cHv)E2p + (^pl-i/^-Mr?)/* +ypz~^2

where C(T]) = 27rr//(£27rT?-l), #£ is the scatter­ing length, r) = e2/Kv, h(ri) = Re[r'(-iTi)/r(-ir}j\ -lnr?, R = H2/mpe2, y n = 3.19X1011 MeV"1 c m ' 1 , corresponding to an effective range re = 2.65 F, y£ = 3.40xiOn MeV"1 cm*1, a n d £ 2 n (E2/>) is the relative nucleon-nucleon energy. In both cases p(E) = C £ 1 / 2 ( £ m a x - £ ) 1 / 2 , where £ is the c m . energy of the observed particle with a maximum value E m a x -

Using expressions (3) and (4) converted to the laboratory system with the Jacobian t rans­formation, we have obtained fits for the spec­t ra of both reactions. Experimental values for \g(0)\2 were used and the instrumental r e s ­olution was folded in. The section of the spec­trum of the reaction He3(dft)2p shown in Fig. 1(a) contains 21798 events. It is rough­ly equivalent to the section of the T(d, He3)2n spectrum shown in Fig. 1(b) (E2 w = 0 to 1.57 MeV) which was used to extract an. From the reaction Ke3(d,t)2p, we obtained, for 11268 events, the value

and for 21798 events

a = - 7 41+0.39 F

using a minimum-x2 criterion. The er rors are the probable e r rors determined from a normal x2 distribution. Calculations were al­so done with changes of all experimental input parameters, such as \g(0)\2, Erf, and the en­ergy resolution, within their respective lim­its of error , and no significant changes result­ed for the scattering length. The 0.3-F differ­ence between the value for 21 798 events and the value obtained from proton-proton scatter­ing is still within our probable error .

For the reaction T(d, He3)2w at £^ = 32.5 M e V ,

the best fit for 17 782 events yields

a =-16.1±1.0 F. n

Also, the value of an is stable to within 0.2 F irrespective of the section of the spectrum (up to E 2 n = 1.57 MeV) employed for the fit. Even though we estimate from the comparison between our ap result and its established val­ue that 0.6 F is the upper limit of er ror due to theoretical uncertainties, we consider this stability of an to be a good indication that this er ror is in fact smaller. A fit with a positive scattering length yielded an - 18.4± 1.1 F.

Since our spectra were fitted over an ener­gy range from zero to 1.57 MeV in the neutron-neutron c m . energy, we were able to obtain a x2 fit for the effective-range parameter re. The value used in the fits for an was re - 2.65 F; then, for the best value an= -16.1 F, var­iation of re gave a best fit for re = 3.2± 1.6 F.

We have made a determination of the sign of <%, assuming that in both reactions \g(0)\2

in expressions (3) and (4) are the same to with­in a few percent. Then the reaction He3(d, t)2p can be used to determine \g(6) I2 which in turn can be used to predict the T(d, He3)2n absolute yield for both signs of an. Fig. 2(a) contains the pertinent plots, and it is seen that the com­parison with the experimental spectrum is con­sistent with the negative sign. The spectrum calculated with the positive scattering length does not include the contribution3'8 from the resulting bound state. This contribution would increase the discrepancy between the calcu­lated and experimental spectra. In fact, a bound-state contribution given by {do/d^)Dound ^ 0AS(d2(j/dQdE)dE would make it impossible to fit the shape of the measured spectrum. This independent evidence for negative an

would provide support for the assumption of

107

V O L U M E 16, N U M B E R 3 P H Y S I C A L R E V I E W L E T T E R S 17 JANUARY 1966

o u u

4 0 0

3 0 0

2 0 0

100

-^

3̂ • •

1 1

(a)

. / • ^

^ ^"^

• •

i 1

/

•

1 1

/ /

y y

y y

' *yC • y^m

i 1

i

\ \ \

" \ \

Aw "\\\

% \\ \ V

V

i . \

-

*•• ' 1 26.5 27.0 27.5

Energy (MeV)

28.0

1 1

(b)

• -v~—' •

• • *

1 1

1

• • y •^y^ •

1

T r —

* ~y

* •

i I

— i —

* \

i

~~] '

I

\ -34.5 35 .0

Energy (MeV)

35.5

FIGo 2, (a) He spec t rum at 8° lab of the reac t ion T(d,He3)2w at 32.5 MeV together with the theore t ica l fit for an = - 1 6 . 0 F , shown with solid l ine . The fit is consis tent with the value de te rmined for the 6° s p e c ­t r u m . The dashed l ine i s a predic t ion for an = - 1 6 . 0 F using the |£"(0)|2 de te rmined from the reac t ion He3(d, t)2p, and the dash-dot l ine for an = +18.4 F . (b) He3

spec t rum at 8° lab of the reac t ion T(d,He3)2w at 40.2 MeV together with the curve calculated for aw = - 1 6 . 0 F . It i s consis tent with the value de te rmined at 32.5 MeV.

near equivalence of \g(0) I2 for the mirror r e ­actions.

Our determination of

a =-16.1±1.0 F n

compares well with the 7r~ +d result and with the value -16.4 to -16.9 F predicted by Hel­ler, Signell, and Yoder1 from ap on the bas­

is of a charge-symmetric nucleon-nucleon in­teraction.

We wish to thank J. Meneghetti, J. Downen, D. Landis, and R. Lothrop for their consider­able assistance with the experimental equip­ment.

t T h i s work was pe r fo rmed under the auspices of the U. S. Atomic Energy Commiss ion .

*On leave from the Univers i ty of Base l . M i c h a e l J . Moravcs ik , Phys . Rev. 136, B624 (1964);

L. Hel le r , P . Signell, and N„ R. Yoder , Phys . Rev. L e t t e r s 13., 577 (1964).

2M. Cer ineo , K. I lakovac, I. Slaus , P . T o m a s , and V. Valkovic, P h y s . Rev. JL33, B948 (1964).

3V. K. Voitovetskii , I. L. Korsunsk i i , and Y. F . Pazhin, Phys . L e t t e r s 10, 109 (1964); V. K. Voitovet-sk i , I. L. Korsunsk i i , and Yu0 F . Pazhin , Nucl. Phys . 64, 513 (1965).

4R. P . Haddock, R. M. Sa l te r , J r . , M. Ze l l e r , J . B . C z i r r , and D. R, Nygren, Phys . Rev. L e t t e r s J14, 318 (1965), and r e f e rences the re in .

5V. Ajdacic , M. Cer ineo , B . Lalovic , G. P a i c , I. S laus , and P . TomaS, P h y s . Rev. L e t t e r s 14, 442 (1965).

6Myron Bander , Phys . Rev. 134, B1052 (1964). His e s t ima te of ~ 1 F is cons idered to be somewhat op­t imis t i c by Moravcs ik , Ref. 1.

7KennethM 0 Watson, Phys . Rev. 88, 1163 (1952). 8A„ Bo Migdal, Zh. E k s p e r i m . i T e o r . F i z . 28, 3

(1955) [ t ransla t ion: Soviet P h y s . - J E T P 1, 2 (1955)]. 9R. J . N. Phi l l ips , Nucl. Phys . 31 , 643 (1962); 53 ,

650 (1964). 10R. J . Slobodrian, to be published. The quoted value

is obtained from ve ry accu ra t e phase shifts at five low ene rg i e s , excluding the vacuum polar izat ion c o r ­rec t ion .

! 1 0 . M. Bilaniuk and R0 J . Slobodrian, Phys . Le t t e r s 1 , 77 (1963).

12H. E . Conzett , E . Shield, R. J . Slobodrian, and S. Yamabe, Phys . Rev. L e t t e r s .13, 625 (1964).

13M. Jakobson, J . H. Manley, and R. H. Stokes, Nucl. Phys . 70, 97 (1965).

1 4E. Mo Henley, F . C. R i c h a r d s , and D. U. L. Yu, Phys . Le t t e r s 1 5 , 331 (1965).

15To A. Tombre l lo and A8 D. Bacher , Phys . Le t t e r s J/7, 37 (1965).

16R. W. Zurmiihle , Nucl. Phys . 72, 225 (1965), r e p o r t s an apparent d iscrepancy with Refs . 11 and 12 in an ex ­per iment detecting the two protons in coincidence. His published spec t rum cor responds to (0Ctm) = 38° for low re la t ive momenta of the pp s y s t e m , whereas the re levant spec t r a r epo r t ed h e r e and in Ref. 12 c o r r e ­spond to 0 c . m o = 12° and 9 c m # = 16° and the re fore a m a r k e d difference is to be expected (see Refs . 13 and 14).

1 7J . E . Brol ley , J r . , W. S. Hall , L. Rosen, and L. Stewart , Phys0 Rev. 109, 1277 (1958). Although these au thors did not ex t rac t a value for an f rom the i r data , it i s in te res t ing to note that fits to the i r h i s to ­g r a m s at 10° lab give 12 F < - a w < 18 F0

108