r

I L NUOVO C I ~ E N T O

V o L X X V I I I , 1~. 6

16 Giugno 1963

On the Reaction n + d -+ 2 n + p and the Nucleon-Nucleon Interaction. F. FEI~RO~I I s t i t u t o di F i s i c a dell' Universit~ - Torino I s t i t u t o iVazionale di JFisica Nucleare - Sezione di Torino

and V. WX~AGm~ (*) I s t i t u t o di F i s i e a dell'Universit5 - Bologna I s t i t u t o Nazionale di F i s i e a N u c l e a t e - Sezione di Bologna

(ricevuto il 21 Gennaio 1963)

S u m m a r y . - - A field theoretical model is applied to the reaction n ~ - d ~ 2 n - ~ - p . The final state interaction between a pair of nucleons at a time is taken into account. The fit with the experiment indicates t h a t there is no bound singlet state of two neutrons. Some new measurements are suggested in order to improve the determination of the singlet neutron-neutron scattering length.

I. -

Introduction.

ILAKOVA~ et al. (t) h a v e r e c e n t l y m e a s u r e d t h e c r o s s - s e c t i o n for t h e r e a c t i o n n+d-~2n§ as a f u n c t i o n of t h e p r o t o n k i n e t i c e n e r g y for s m a l l a n g l e s of t h e e m i t t e d p r o t o n . T h e s e a u t h o r s h a v e t r i e d also t o fit t h e i r e x p e r i m e n t a l r e s u l t s w i t h a c a l c u l a t i o n u s i n g a B o r n a p p r o x i m a t i o n (~). T h e y t a k e i n t o a c c o u n t t h e i n t e r a c t i o n b e t w e e n t h e final p r o t o n a n d e i t h e r

(*) Now at the Istituto di Fisica dell'Universit~ di Torino and I s t i t u t o IqazionMe di Fisica •ucleare, Sezione di Torino. (1) K. ILAKOVAC, L. CT. KvO, ~ . P]~TRAVIC, I. SLAUS and P. To~A~: P h y s . Rev. Lett., 6, 356 (1961). (2) K. ILAKOVAC,L. CT. K u o , ~r PETRAVIC and I. SLAVS: P h y s . Rev., 124, 1923 (1961).

ON

THE

REACTION

n+d-* 2n+p

neutron. In this approximation part of the proton spectrum. In the present work we have as proposed b y CHEW and L o w This model assumes t h a t in (1)

AND

THE

NUCLEON-NUCLEON

INTERACTION

1343

t h e y could only fit the shape of the upper applied to this problem the peripheral model (2) ((see also ref. (~)). a scattering process of the t y p e

A+a~B+b+...,

the target A m a y split virtually into two particles one of which interacts with the incident particle while the other acts as a spectator. The Scattering m a t r i x for this process is ~ function of the square of the m o m e n t u m transfer and has a pole in the unphysical region of this variable. If the binding energy of the virtual particle in the target is small compared to the incident particle's kinetic energy, this pole is near to the physical region and the virtual particle is almost on the mass shell. Then it is possible to relate the m a t r i x element for process (1) to the probability amplitude for the scattering of the incident particle with the (~virtual )) particle now on the mass shell. One assumes t h a t this physical process gives the main contribution to the cross-section. In this model one is also able to use the methods of relativistic q u a n t u m field t h e o r y and obtain the correct numerical and kinematical factors in the expression for the cross-section. The use of relativistic kinematics simplifies considerably the analysis of the reaction. The interaction between the incident neutron and each of the nucleons which make the deuteron is described b y means of the effective range approximation while the interaction between the spectator particle and the two interacting nucleons is neglected. I n Section 2 we describe the reaction and the model. I n Section 3 we compare our results with the experimental data. 2" - In the peripheral approximation for the d(n,2n)p reaction one takes into account only the two processes represented in Fig. l a and lb. Pl, P~ are the m o m e n t a of the incoming neutron and deuteron; P'I, P'~, and p the m o m e n t a of the two outgoing neutrons and recoil proton respectively (*).

p,

(3) G. F. CHEW and F. Low: Phys. t~ev., 113, 1640 (1959). (4) I. S. SHAPIRO: )7~Ct. Phys., 38, 327 (1962). (*) We use the metric goo=--gn=--g2~=--g3a=l.

~'

p,

Fig. 1.

p

1344

s. FERRONI and v. WATAGttIN

I t is convenient for w h a t follows to define the i n v a r i a n t s : W ~ = (p~ + p~)~ !

(1)

w~ = (pl + p'~)~

; w'~ = (p + p'~)~,

a ~ =

;

3 '~ =

(p', -

,

--

~'~

(p,--pj

~'

(p -

p~)~

=(p_p,)~

'

!

p~)~, 2

.

I t is clear f r o m the a b o v e definitions t h a t W 2 is the square of the total center-of-mass energy; w ~ is the square of the t o t a l energy of the two outgoing neutrons in their baricentric s y s t e m ; A ~ is the square of the ~.o difference of f o u r - m o m e n t a for the deuteron and proton. w '2 and A '~ are correspone : ~ o cling quantities for the case l b 0.8 in which t h e spectator particle : ~ is the neutron. Because of the law of con:30. servation of energy and m o m e n t u m , the t o t a l barieentric energy 06 of the two outgoing neutrons %~ is k n o w n once one measures ~ the energy and angle of emission of the recoiling p r o t o n : 0.4i

(2)

w ~ = (Md + T1L-- To~)~ - -

--(P~r-- 2 IPILI]PLI c ~ 02

0

0.1

0.2 2 2 q 2 + c(/p,

Fig. 2.

0

+P~) 9

H e r e M d is the mass of the deuteron, TI~ and T , r are respectively the kinetic energy of the incident n e u t r o n and outgoing p r o t o n in the laborat o r y s y s t e m ; # is the l a b o r a t o r y angle between the incident neutron and outgoing p r o t o n (*).

(*) In this formula and in the rest of t h i s w o r k we neglect the neutron-proton mass difference.

ON T H E tCEACTION

nd-d-+ 2n-4-p

AND THE NUCLEON-NUCLEON INTERACTION

1345

I n Fig. 2 are plotted (*) the curves p~ vs. q2§ obtained from eq. (2) for different fixed values of 0. All the cm~ves lie inside the region bounded b y the curves v~ = 0 ~ and v~ --~180 ~ the upper ones corresponding to small angles, the lower ones to large (backward) angles for the emitted proton. We notice t h a t for protons emitted in the forward direction small values of w 2 are only compatible with large values of p2, while in the backward direction only small values of p2 are p e r m i t t e d and they are compatible with small values of w 2. This observation is relevant to the relative importance of the contributions of diagrams l a and lb to the cross-section and we shall come back to it when discussing our final results. Using well known rules we can write the m a t r i x element corresponding to the F e y n m a n diagrams l a and lb: M 2

S ~ = - i(2u)-t

(3)

V2p2op~op'~oplopo M 2

V2p~op~oP'~op'~oPo Here pf and Pi stand for the total final and initial four-momenta respectively. Fa is the neutron-proton-deuteron v e r t e x function; J t '(n~) and ~ ( ' ) arc proportional to the neutron-neutron and neutron-proton scattering amplitudes. u~l; ... ; u,. are nucleon spinors; u'_~ is the charge conjugate spinor of u~, C is the charge conjugation operator. In order to evaluate the differential cross-section with Iespect to proton energy and angle we must sum the amplitudes corresponding to the diagrams l a and lb, square them, sum over the spins in the final state, average over spins in the initial state and integrate over the final neutron momenta. The effect of the Pauli principle on the final state interaction is important. W e classify the possible final states according to the total spin J (upper left index in ~ which can be either ~ or 89 the n a t u r e and spin state of the two strongly interacting particles, ( u p p e r right index) and the spin s y m m e t r y of the two neutrons wave functions (lower right index). Because of the small energy available we m a y safely assume t h a t all pairs of final particles are in relative s-states (**). The only contribution then comes from 89 n~)" for diagram la, and 89 89 for diagram lb. Because of the Pauli principle only the interference between 89 and

(*) In units of p]. M is the nucleon mass and -- B is the deuteron binding energy. (**) Then the t~'(nm~ does not contribute to the ~o~1 cross-section.

1346 89

(4)

s. FERRONI and v. WATAGtnN contributes to the cross-section. We obtain in this way:

d T d f 2 - - ~s

Mdp~w(2z)s'~

(COS~9") 9'" 2 I ~ / / ' : " l s

--1

(As_ ;gs)(A,s

0

~s)

( A '~ __ l f s ) s

where ~'~ and ~'~ are respectively the scattering and azimuthal angle of P'I I/O /~0 with respect to P l in the system P in which Px -~ P2 ----0. The minus sign of the interference t e r m arises from the phase difference in isotopic spin space between the diagrams l a and lb (or equivalently from exchanging a neutron with a p r o t o n in order to obtain lb from la). The factor 89in front of the integral stems from the fact t h a t because of the identity of the two neutrons one should integrate only over half the solid angle. In the derivation of the expression (4) for the cross-section we h a v e made use of the nonrelativistie approximation in the evaluation of traces. In particular we have used the nonrelativistic form of the n-p-d v e r t e x function for the case when all three particles are on the mass-shell. Since the kinetic and binding energies involved in our problem are of the order or less than 1 % we expect t h a t the error due to the nonrelativistic approximation is not larger t h a n 1%. In expression (4)~ 89; ~ (nn) s and 89~ff](nP) s , ~J Z (nP)t represent the amplitude for neutron-neutron and neutron-proton scattering respectively and are related to the scattering phase-shifts through the formula:

(5)

w exp [id] sin ~ = (2~)~ MS q

,

where q is the relative m o m e n t u m of the two interacting nucleons. ~(~) is the nucleon-nucleon phase-shift and in the effective range approximation (*) is given b y

(6)

qctg6=

- ~ 1 + 1 ~ro qS .

Substituting (5) and (6) into (4) and performing the integrations with the aid of an electronic computer we obtained the curves of Fig. 3 and 4.

(*) In the actual calculations we have neglected ~he smM1 correction 89 s.

o N T H E REACTION n - { - d - > 2 n ~ - p

AND T H E I~[UCL~,O~!-NLTCLEOlq I N T E R A C T I O N

1347

3. - Comparison with experiment and conclusions.

The agreement between our theoretical results and the experimental data Of ILAKOVA~ et al. for incident neutron energy of 14.~ MeV and ~ 4 ~ and 10 ~ is reasonably good as can be seen from Fig. 3 and 4.

=4 ~

//~

=theory

10

E~ Q

5

I

4

I

__

I

6

__l

Tpin

l

8

I

I

10

[

),

12

MeV

Fig. 3. At both angles the maximum near the upper end of the proton spectrum is due to the n , n interaction while the maximum at lower proton energy is due to the n , p interaction. The agreement between theory and experiment is within 25 % all through the spectrum, except near the upper maximum where the agreement is worse. This is to be expected because in the forward direction and at high proton energies the momentum transfer is large and the intermediate particle is far from being on the mass shell. These are the conditions where the approximations of our model are the least likely to be valid. As the angle of recoil of the proton becomes larger, the energy of the maximum due to the ( n , n ) interaction becomes smaller (see Fig. 2) and therefore one expects a better agreement with experiment also at the top of the spectrum. At backward angles (for instance ~ = 1 3 5 ~ our calculations confirm that only the maximum due to the n , n interaction remains and is displaced

1348

s. FERRONI ~nd v. WATAGHIN

In towards lower energy; these are the best conditions for measuring (a.). ~n this ease b o t h w ~ and p~ can be small and the contribution of d i a g r a m lb will be negligible c o m p a r e d to t h a t of d i a g r a m l a . The best e x p e r i m e n t a l eondi15 'A

i

10

dl :>

E~ .S

~5

/

i tt t

t I

t I

I 4

~

I 6

I

Tp in

I 8

I

I 10

I

I 12

t

MeV

~Fig. 4. tions to d e t e r m i n e the sign of a~n is p r o b a b l y to m e a s u r e the p r o t o n s p e c t r u m at angles around 90 ~, where the interference t e r m should be large relatively to the other terms. One m i g h t be surprised t h a t the peripheral m o d e l which is usually applied to high energy reactions, should give a r e a s o n a b l y good a g r e e m e n t with e x p e r i m e n t in the ease of a low energy process. W e believe t h a t this is due to the fact t h a t the binding energy of the deuteron is small and t h a t for low proton energies the exchanged nucleon is almost on t h e mass-shell. Therefore the extrapolation of ~/~=) and J/d ~=') to a m p l i t u d e s for scattering of a real nucleon on a n e u t r o n is well justified. Our theoretical curves were obtained with the following values for t h e scattering (*): a,. . . =. a

= - - 23.7 fermi;

atan = 5.37 f e r m i ,

the equality between a2~ and a~." beeing suggested b y the hypothesis of charge independence. (*) 1 fermi

10-13 eiIl.

ON THE

n+d-+ 2n+p

REACTION

AND

ThE

NUCLEON-NUCLEON

INTERACTION

1349

The interference t e r m contribution is approximately constant, its absolute value v a r y i n g between 3.5 and 4.2 mb. The sign of the interference t e r m is negative with the above choice of a: ~. The existing experimental data at low energies give the order of magnitude to the possible violation of charge independence. If we t r y to estimate how different a~~ and a~" can be without worsening our fit, we find t h a t a positive sign for a~~ is excluded while a small difference of absolute value between the two singlet scattering lengths is not excluded. The fact t h a t the negative sign of a~~ gives a b e t t e r agTeement with experiment m a y be interpreted as an indication t h a t there is no singlet tow energy bound state of two neutrons.

The authors would like to t h a n k Prof. S. FUBn~I for having suggested to t h e m this problem and for several useful discussions. Thanks are due also to Prof. M. VEI~DE'and to Dr. F. SELImEr for useful suggestions and to Mr. I~IA~ANZANI of Bologna's Centre Calcolo del C N E N for the numerical computation with the IBIv[ 704.

A d d e d in proof, When this paper was already finished we received a <~preprint ~>of ILAXOVA0 et al. where they have ex~ended their measurements to larger angles. They have fitted their experimental data by treating with a Born approximation both (np) and (nn) interactions and introducing three arbitrary parameters besides the scattering lenght.

APPENDIX

I n this Appendix we shall give the expressions of the primed invariants defined in (1) as functions of vq'p and ~p'P. These formulae were used in order to carry out the integrations in expression (4) and t h e y can be obtained using the conservation of energy and m o m e n t u m .

w'2= W~- M~--2M ~+ An+ j n , =

M~+M

~'~ =

2M z-

A '~

~-

" " + 2 tP~' { ]P~l cos zP'P, 2p~op2o

~, p 2p~op,o

+ 2 Ip~, l Ip~leos oe,

cos ~P = cos ~9~ cos zP'"+ sin ~9~ sin z9'p cos 9'P cosec'2 = M ~ + M

2 + 2pxop~o ~ ~ 2lp~[ IP~[

-

-

W~

S, FE~aONI and V, WATAGHIN

1350 and w

p'l~ = p's~ = ~ ;

Ip~i = Ipl ~ I =

2w P~

--

P~0--

W s -

M

s -

. w s '

w~+ M ~- A s 2w ;

[(M--w)S- W~]~E(i + w,~-- W~]~

ippl =

]P~] - -

w s + M~' -- g2 P~~

2w

_ Ms ;

2w

;

[ ( M - - w ' ~ - AsJ~[(M + w ' s - As] ~, 2w

[(M~-- w)~-- ~sJ~[(M~ + w'~-- gs]~ ;

Ip~l =

2w

RIASSUNTO Un modello basato sulla teoria dei campi ~ applicato alla reazione n + d - + 2 n + p . Si tiene eonto hello state finale dell'interazione fra due nueleoni alia volta. I1 confronto con l'esperienza indiea che non e'~ uno state legato di singoletto di due neutroni. Si suggeriscono nuove misure allo scope di determinare meglio la lunghezza di scattering di singoletto neutrone-neutrone.