IL NUOVO CIMENTO VOL. 23 A, N. 4 21 Ottobre 1974

Role of Tensor Forces in Neutron-Deuteron Elastic Scattering.

V.K. S~A~A (*) Department o/ Physics, Dharma Samaj College - Aligarh

(ricevuto il 18 Febbraio 1974)

S u m m a r y . - - With the aim of treating the elastic scattering of neutrons by deuterons at low energies in presence of tensor forces, the resonating- group method is considered, which gives a set of coupled integro- differential equations. The Hulthdn wave function is used for the s-state component of the deuteron ground state and it is shown that the coupled intcgro-diffcrential equations results essentially from the inclusion of the tensor forces.

1 . - I n t r o d u c t i o n .

In recent years, a number of theoretical investigations (1) have been made

to examine the properties of light nuclear systems using the method of the resonating-group structure (2-3). The results of these calculations have been

very encouraging, since for light nuclear systems the agreement between the

calculated and experimental results was found to be quite sat isfactory over

a wide range of energies. The resonating-group method has been found to be a powerful tool in t rea t ing

the three-body scattering problem as long as the break-up channel can be neglected. The results of these calculations (4) clearly indicated tha t the ef- fects introduced by the ant i symmetr iza t ion procedure are impor tan t and cannot

(*) Permanent address: 3/221, Adarsh Nagar, ~V[arris Road, Aligarh.202001. (1) Y. C. TANG: Proceedings o] the International Con]erence on Clustering Phenomena in Nuclei (Vienna, 1969). (~) J . A . W~EELER: Phys. Rev., 52, 1083 (1937). (3) K. WILOERMUTH and W. McCLuR]~: Cluster Representations o] Nuclei (Berlin, 1966). (4) J. SCHWAGER and E. W. SCH~IID: Nucl. Phys., 205A, 168 (1973).

679

~ V . K . SIIARM~

be omi t t ed if sa t i s fac tory agreement wi th the exper imenta l da ta is to be ob- ta ined.

A logical s tep is to app ly this procedure to more compl ica ted problems, which are difficult to handle wi th in tegra l -equat ion methods (~), such as neutron- deu te ron sca t te r ing wi th tensor forces. I t is of course well known t h a t s -wave forces are grossly inadequa te for the represen ta t ion of two-body data . The obvious answer is the inclusion of a t least tensor forces which are known to be i m p o r t a n t not only for t wo-body sca t ter ing bu t also for producing the re- quisite 4 ~o D- s t a t e in the deuteron.

I t should be more interest ing to look into the effects of the tensor force on the n-d problem, in so far as such a force has an accepted s ta tus in the two- b o d y potent ia l . The tensor force formal ism, wi th proper an t i symmet r izu t ion of t he th ree-nuc leon wave funct ion, has a l ready been applied to the calculation of several t r i t on pa ramete r s . The binding energy a l ready shows a good deal of i m p r o v e m e n t over the corresponding results of a pure s-wave theory . This fac t should w a r r a n t the expec ta t ion of a corresponding i m p r o v e m e n t in the resul ts of the n-d sca t ter ing problem.

W i t h the inclusion of tensor forces we expec t a significant modificat ion in the resul ts of a½, and negligible correct ions for a~, for the following reasons. The qua r t e t n-d ampl i tude is expec ted to show only a slow var ia t ion with changes in the t w o - b o d y potent ia l , typi f ied b y the rep lacement of an effective s-wave force b y a sum of s -wave and tensor forces. And, in so far as the s-wave force is m u c h bigger t h a n the tensor force, the modif icat ion due to the tensor force m u s t be regarded as a re la t ive ly small cont r ibu t ion to the two-body potential , a t leas t for low energies. Therefore the effect of the tensor force on the quar te t n-d sca t te r ing ampl i tude a t low energies m u s t be small. :Not so, however, for double t n-d sca t te r ing whose ampl i tude has a pole s t ruc ture near threshold, due to the exis tence of the n e a r b y t r i t on s t a te wi th the same q u a n t u m numbers T ~-J--=- 1. As a result , even small changes in the two-body po ten t ia l could make a significant a l t e rna t ion in its magn i tude . A second reason for the sensi t ivi ty of t he double t sca t te r ing length is t he smallness of the p a r a m e t e r a½ (0.7 fm).

I n the presen t p a p e r we h a v e considered the n-d elast ic-scat ter ing problem wi th t ensor forces in resonat ing-group approx imat ion . Section 2 provides the set of coupled integro-different ia l equat ions using the me thod of the angular- m o m e n t u m decomposi t ion. The Y u k a w a potent ia ls and the deuteron ground- s t a t e wave funct ion follow i~ Sect. 3. Wi th this choice of potent ia ls and the deu te ron wave funct ion, an expression of the di rect - in teract ion t e r m has been derived.

Then angu la r in tegra t ions and the m a t r i x e lement of the tensor operators are g iven in the Appendices.

(~) L. I). FADD•EV: SOY. _Phys. JETP, 12, 1014 (1961); Mathematical Aspects o] the Three-Body Problem in the Quantum Scattering Theory (Jerusalem, 1965).

ROLE OF TENSOR FORCES IN NEUTRON-DEUTERON ELASTIC SCATTERING- ~81

2 . - G e n e r a l f o r m a l i s m .

2"1. Derivation o] coupled radial integro-di]/erential equations. - The Schr6- dinger wave equat ion is

( r + 2: (ij) - Eo -- E J = 0 , \ /

where E a is the deuteron binding energy, Eo the inc ident -neut ron ene rgy in the centre-of-mass sys tem and T is the k ine t ic -energy opera to r

(2.2) T = -- ; ; (2V~a + ~V~.~),

where M is the nucleon mass. The nucleon-nucleon po ten t ia l used here is a m ix tu r e of cent ra l and tensor

t e rms (+~5):

(2.3) ~ ( i j ) ~- V(rij)(co -~ mPM<~) -~ --h ]c)<i/)s ~-..__~,hP(i¢h ~- (co'+ m' P~J))S(ij) U(rij )

where S(ij) is the tensor opera tor

[3(a~.r.)(~j. (2.4) S(ij) ----

t T i t

r.) (a~.a~)] ,

/)(~¢) P(~) and P(~J~ being Majorana, B a r t l e t t and Heisenberg exchange opera tors , M ~ - - ]B - - H

respect ively; and the constants co, m, b and h, co' and m ' sat isfy the equat ions

co -4 -m~-b~-h=- - 1 ,

(2.5) co ~- m -- b - - h = 0.63,

(e) L. M. DELVES and D. BROWN: .~uel. Phys., l l , 432 (1959). (~) A. DE SHALIT and I. TALMI: Nuclear Shell Theory (New York, N. Y., and London, 1963). (8) S. WATANABE: Nucl. Phys., 14, 429 (1959/60). (9) M. MOSHINSKY: Nncl. Phys., 13, 104 (1959). (lo) N. AUSTERN, R. M. DRISKO, E. C. HALBERT and G. R. SATCHLER: Phys. Rev., 133, B 3 (1964). (11) G. GOERTZEL and N. TRALLI: Some 2;[athematical Methods o] Physics (New York, •. Y., 1960). (12) H. HORIE, T. TAMURA and S. YOSHIDA: Progr. Theor. Phys., 8, 341 (1952). (la) A. R. EDMONDS: Angular Momentum in Quantum Mechanics (Princeton, N. J., 1957). (14) D. M. BRINK and G. R. SATCHLER: Angular ~omentum (Oxford, 1968). (15) B. H. BRANSDEN, K. SMITH a~nd C. TATE: Proc. Roy. Sou., A247, 73 (1958).

6 8 2 v . K . SHARMA

I f i is the (( incoming ~) neutron, and 2 and 3 are the neutron and proton in the deuteron, respectively, we use

(2.6) u = r 3 - - r 2 , r = ½ ( r 2 + r 3 ) - - r l .

According to Wheeler 's resonanting-group model, the wave function of the sys tem of three particles T, being completely ant isymmetr ic relative to the permutat ions of the particles, can be wri t ten as

(2.7) 1

T(123) = %/:~ (1 -- P~*')yJ(1, 23).

The wave funct ion ~p(l, 23) can be expanded into eigenfunctions of total angular momentum, its z-component and the par i ty:

(2.8) ~(1, 23) ~ TM = ~jM(1, 23) J M l ~

with

where

2

'" ~ (1, 23) ¢(u) , yJj~(1, 23) ---- ~ r -aF~(r ) ~ Jz~

(2.9) JZ~ ~M (1, 2 3 / = ~ (lmS~,m~,]lS~,JM> Yz~(0, ~ ) Z , ~ ( 1 , 23/,

Zs~,,~,,,(1, 2 3 ) f o r :¢= 1, 2 are, respectively, the quartet and the doublet spin function, symmetr ic in the two neutrons, ~b(u) is the s-state deuteron ground- s tate wave function.

The or thogonal i ty relation

fd3ufd (O, 23)]* (2.10) @(u) ~]~,~'~'~'(1, 23)] ---- 5jj, 5W(~, 5 ~ ,

and the equat ion

(2.7,:~) f d~u,Z'(u) r h~ 2 [-- ~ V ~ + ¢ " ( u ) - Ed]¢(u) = 0

are satisfied. I f we subst i tu te (2.7) and (2.8) into the wave equation (2.1), mult iply by

J Z ~ [ (u)~Yz~ (1, 23)]* and integrate over all co-ordinates except r and sum over

R O L E OF T E N S O R F O R C E S I N N E U T r O N - D E U T E r O N E L A S T I C S C A T T E R I N G 6 8 3 .

spins, a set of coupled integro-differential equations is obtained for the F~(r);

(2.12)

where

+ k 2 ~ ( r ) : ~ ~ d3u d~Q(O, cp) 3h 2

• r[~b(u)~d~ (1, 23)]*[{$/~(t) +$/ ' (V ) } r F j~ ( r ) ~ (1, 2 3 ) ~ b ( u ) - -

- - { T E o - - E a + ~ ( t ) + V ( u ) + 3 ¢ (V)}r F j~ ( r ) ~ ( , ,13)~b(V)]

1 ~ r = r 3 r l and t : r 2 r 1 r ' = ~ ( r 1 ~ - r s ) - - r 2 , - - - - •

I f the exchange operations, the spin summations and the angular integrations, indicated in eq. (2.12), are performed, the equations ma y be put into the form

( d ~ (2.13) ~ +

where

and

co

T r l ' k S l(1 r + l ) ) F ~ ( r ) = ~ [~ ~ , ( ) F ~ , ( r ) +fdr ' IC~, , ( r , r ' )F~: ( r ' ) ] ,

0

V~,(r) = V~,(r) + VL,(r)

K~,(r , r') K~,(r , r') + .t = K~,(r , r ' ) .

Here terms with superscript c are due to central interaction and those with superscript t to tensor forces.

2"2. Expressions o/ the direct interaction ~ t ~ , and exchange kernel K~ , . - In order to obtain the values of the potentials and kernels, we use the relations

T - - 2M .,V~ +~. ,

~ V~--E~ + ~(13)] ¢(Ir~-- r~1) = 0;

and expansions of V(l~-u/21), V(~lr--r'l), V(~lr'÷Orl), V(-~lr+2~'l), ~b(~12r'+ r]) and ~b(~lr' + 2rl) in terms of Legendre polynomials by tile Slater method (7), as e.g.

(o ) (2.14) V r - - ~ = Vk r,

k=0

where ~,)~ is the angle between r and u.

684 v . K . SgA~MX

The functions V~,(r), K~,(r, r') for central interaction in eq. (2.13)were obta ined by WATAI~ABE (8).

To evaluate V~,(r) and K~,(r, r'), which contain the tensor forces, the spin mat r ix elements of the tensor operator S(ij) ~re required.

In Appendix A it is shown tha t

(2.15) ( z ~ ( ~ , 2~)~(12)z.~,~,0, 2~)) = sl~ln

= ,,~, (-)'~ V--y < ~'( ' 2~)]lX~(¢,,, °5)i] s~,,(1, 2~) ) .

where 0~, ~ are the polar angles of t. Using the expansions (2.1~) and eq. (2.15), performing the exchange op-

erations and summing over spins, we obtain

(2.16) V~d,(r) : ~ ( )" (2(o '--m') ~ <lmS~m~llS~JM >. /~=--2 a l l m ' s

• ( l 'm'S~,m~,i l 'S~,gi}(2- #S~,m~,12,%,S~m~}(S~(1 , 23)l[X0,(~1 , ~)[] S~,(1, 23)}.

4 M / " ~ f • ~ J d u ¢*(,,) ¢(, ,) V(t) d~9(O, ~o) X':~(O, ~o) ~ , , ( 0 , , ~t) Y~,.,,(O, ~)

~nd

(2.17) K ~ , ( r , r ' ) : ~ ~ ~ (--)~.+~.+.~. rr'CP~. ~r ' ,~r •

, 4 2 2 "qSk. (~r ,~r) [(2m'-- o3')(--)~'U~ (~r,-~r') B~(~ct') •

• (S~(1, 23)I1X2(a,, a2)(1 _ ~]°(13) ]D(12)~a ,l[S~,(2, ]3)}

-- Uk. 5 r , S t B~(~g')(S~(1, 23)1[X2(~2, ~a)(1 Q-Pt~2a))t[S~,(2, 13)5--

where the B's are purely geometrical coefficients:

(2.18) ,1/4-~ 42 1) (lmS~m~ IIS, JM) • B~(o~o~')----~ ~ ( - - )V- f f (2k l+l ) (2k~+l ) (2k3+ /* $,11 m~a

i t t O • (1 m S~,m~,ll S ~ , J M } ( _ - #S~,m~,I2S~,S~,m~ }.

• fdg(O, ~)fd~(O', ~')y=(o, v)~'~,_m,(o, ~)~-~.m.(o, ~).

• y , y * m , * ' 7Y* tO' q~')t2:Y2~,(Ot, ~ t ) Yk...,(O, ~) v.~,(O , q~') k,-..,~. , q~') Y~,,..(O , q~') ~,m.~ ,

R O L E OF T E N S 0 R F O R C E S I N N E U T R O N - D E U T E R O N R L A S T I C S C A T T E R I N G ~ 5

Similar expressions for B.(~a') and Bo(aa') can be writ ten. In eq. (2.16), changing the variable of integration from u to t and expanding

/¢ffio

we can have

f d ~ u{C,~Cu)) V(t) :~,,(o,, q~,) (2.19) = s l a t q~,(r, t) u ( t ) t ~ 7Y,t,(O , cp) .

o

I f we subst i tu te expressions (2.19) in (2.16), the angular integrat ion and the sums over magnetic quan tum number are carried out (see Appendix B), giving

(2.20) 32M

V~.(r) = (2~ ' - m ' ) ( - ) ' - " -~ ~ [(2~ +1)(2S~ +1)]~-

• W(ll 'S~ S~,; 2J)<1020 ]12l'O><S~(1, 23)[] X~(a~, as)H S~.(1, 23)> ],(r) ,

where co

/,(r) -----fdt t, V(t) ¢,(r, t). 0

2"3. Evaluat ion of geometrical parts. - In order to evaluate B, the next task is to expose all of the implicit dependence on (0, ~0) and (0', ~0'), so tha t the angular integrations may be made. For example, Y2,(O., q~®) is expressed in terms of sphericM harmonics of (0, ~) and (0', ~0') by (9.,o)

~-o a=-, (ar)~(br')~-L (2.22) x2 Y2~'(O*' q~*) -- ~ ~2k --F1] 2k ]

• <k;~ 2 - - k ~ - - ~ lk 2 - - k 2/~> Y,a(0 , ~0) Y~-~.,,-a(O', ~'),

where ( : ) = x ! / ( y ! ( x - - y ) ! ) , and 0,, ~% are the polar angles of vector x such

tha t x : ar -F br' , the spherical harmonics Y,t,(Ot, q)t), Y,~,(O., ~ ) and Y~,(O,, ~,) are expressed in terms of (r, 0, ~0) and (r', 0', ~0').

Subst i tu t ing expressions (2.22) in (2.18) and performing angular integrations (see Appendix B), we obtain

(2.23a)

(2.23b)

(2.23c)

4 4 4 'B B,(o~') = ~ r 2 B l + ~ r ' * B ~ - - ~ rr 8,

4 16 8 , B~(otct') = ~ r~B1 ~--~- r'2B2 -F~ rr B3 ,

45 - I1 Nuovo Cimento A.

686 v. r:. s ~ M x

where

(2.24a) ~, " " 2 1 ' ~ 1 5

• <lOl 'O]l l '20}W(U'S~S~, , ; 2J )<k 'OkaOIk ' k s l 'O> 2 ,

(2.24b) B z = ~ ( _ ) J - s ~ + H ' 3~ ,/(~_7~(21+1)(21'+1)(2S~,+1)<k~Ok~OIk~k~ k'O>2" 7,' 2 ~ - . t 5

• <lOl'Olll' 20} W(ll 'S~,S~,,; 2 J ) ( k ' O k s O [ k ' k s l O > ~ ,

(2.2~e) B~ = ~ ~ (--)~-~ 2k"+~ :~' /¢~

• <lOl'O Ill '20> W(ll 'S~,S~, , ; 2J)(k'OkaO [k 'kak"O>ZB(l l 'k");

here B ( l l ' k " ) denotes

(2.25) B ( l l ' k " ) ~-- (--)~'" V /150 ( l O l O ] l l k " O } ( l O l ' O l l l ' k " O > W ( l l l l ' ; 2k") . <lOl'Olll' 20>

We thus see t ha t the coupled integro-differentiM equations results essentially f rom the inclusion of the tensor forces. The direct potent ia l is a single t e rm and the exchange kernels are the admixture of three terms depending on the quar te t and doublet spin states. The exchange kernels are due to the Pauli exclusion principle•

3. - Express ion o f the d irec t - in terac t ion term w i t h Y u k a w a potent ia l .

3"1. T h e Y u k a w a p o t e n t i a l a n d the H u l t h ~ n ' s wave ]unc t ion o/ deuteron. -

The potent ia l shape is assumed to be of the ¥ u k a w a form for bo th central and tensor forces, say

exp [-- ~rij] (3.1) V(r.) = Vo ~r.

and

exp [-- y r , ] ( 3 . 2 ) U ( r i j ) = U T

~r~¢

F or the deute ron ground state, the Hul th6n 's wave funct ion is given by

(3.3) 65(u) ---- N u - l [ e x p [-- flu] - - exp [-- 7flu]],

where the N is the normal izat ion constant such tha t

(3.4) N = \ 9 ~ 1 "

ROLE OF TENSOR FORCES IN NEUTRON-DEUTERON ELASTIC SCATTERING ~ 7

3"2. Evaluation oj J,(r) in the direct-interaction term with Yukawa potential. - We use the following expansions (n):

exp[-- 2lr-- tl] = ~ ( 2 Z + 1 ) (3.5) Jr-- t[

I~+½(Xr<) Kz+t (2r>)/)~(eos 0,t)

(3.6) exp[-- 2 I t - - t[] I~+½(Xr<) Kz+½(kr>)t~(cosO,t) ( I r - tt)' = ~E~ (~L + ~) r< r>

and

(3.7) exp [-- V [r -- r ' [] I,~+½(~r<) K~+½ (vr>) Pz(cos 0,,, ) ( i r _ r , I ) a : ~ (2Z,-F 1) r<~/~< r> y'~>

where r< and r> mean the smaller and the greater of r and r', respectively in eq. (3.7), and I~+½ and Kz+ ½ are modified Bessel function and modified Hankel function respectively.

Thus using the above expansions in eqs. (2.21) with Yukawa interaction yields

8 V 2 (3.8) ~-U~N- ]4(r)=

_ lexp[0~r] 1 (1 3 -[- 3 3Za(r~r) [ -~- (71r)t -- ~ ~;,-2)(-~1(71 r) -~ 7101 + ~ +

1 ( 3 3 ) ( 30o( ,r) Ol (r;r)~ l +~lr + ~ r ~ G~(w r ) - , - + , ~ , ~ 7101 ~)1 ~1 ]

exp[-- 01r]~ ½ (1 ~_ 3 3 1 6 4

~exp[02r] 1 (1-- 3 + 3 ~ (~ l (y2 r ) , 3Ea(y2r) 3Es(y2r)~ -"° / 0~ (7~r)~ ~ ~ / ~ - ~;;~ + r~o--~ ) +

720~ 72 b~ ]

exp [-- 02r]~½ ( l _{_ 3 _~_ 3 ~(1 6

fexp [0ar] 1 (1-- 3 + [ ~)3 (yar)' -~ar-t-~Tr2)( El(7~r)-~

3Ea(73r) 3Es(yar)~

r~o~ - ~,;~o~ !

7aOa 7aOa '

6 8 8

where

(3.9a)

(3.9b)

and

(3.1o)

G J x ) = ; e x p [y.] d y , y~l 2

0

co

( exp [-- y] ~n(x)=3 y~ dy, $$

(~, : 4fl, (5, ---- 16fl, 63 : 2Sfl ;

7, = 4fl -[- r/, 73 : 16fl -~ r], 73 =- 28fl -J- U ; t ! !

r , : 4 f l - - U, 7,---- 16f l - - V, 7 , : 2Sfl-- n .

V . K. SHARMA

4 . - C o n c l u s i o n .

W e have described a m e t h o d for solving the th ree-body scattering problem with tensor forces in the resonat ing-group approximat ion. Thus we find t h a t angu la r -momentum decomposi t ion of the Schr6dinger wave equat ion for two- body in terac t ion with tensor forces results in a set of single-variable coupled integro-differential equations. Fur ther , an expression of the direct in teract ion t e r m with ¥ u k a w a potent ia ls has been obtained.

Final ly , we notice t ha t , in the con tex t of the resonating-group method, one can get the numerical solution of th ree -body scat ter ing problem with tensor forces, for which a more e laborate computa t ion can be done using modern computers .

The au thor is pleased to t h a n k Dr. J . H. NAQVI for his st imulating interest in the subject of this work.

A P P E N D I X A

I n this Appendix the ma t r i x elements of the tensor operators are derived. The m e t h o d of ]~[ORIE et al. (lz), where the tensor operators were split into the p roduc ts of two irreducible teasors , and the Wiguer -Eckar t theorem (7.13) are employed.

The spin ma t r ix elements are eva lua ted b y expanding the tensor operator as

(A.I) /J=--2

ttOL]~ OF T]~NSOtt POliCIeS I1~ N]~UTI~ON-D]~UT]~RON ]~LASTIC SCATTERING 6 ~ 9

where

(A.2)

X2.~ 2 ~ v ~ , ~ u j

X~o = 3a,,¢~,- (a,.aj),

and ~ the Y2#(~¢) is spherical harmonics of the second degree. The Wigner-:Eckart theorem enables us to separate the magnet ic q u a n t u m

number dependence of the ma t r ix element in the form of the Clebsch-Gordan coefficient. Thus we have

(A.3) (Zs~,,~,(:l, 23)S(12)ZB~,~,(1 , 23)) = tp|n

-<S~(1, 23/]] X2(a,, a2)]} S@, (1, 23)>.

The values of reduce ma t r ix elements are given in a table b y I~ORIE et al. (12).

. A P P E N D I X B

(]~.1)

We have from eqs. (2.19) and (2.16) the t e rm

• <2-/~&,m~,12G,S~m~>fdf2(O, q~)Y~m(O, q~)~._.(0, q~)G.~,(0, ~) •

The angular in tegrat ion (is) is

(B.2) (d[2(O, of) Y*m(O, q~) Y2-~,(O, q~) :Yrm'(O, ~) = d

= ¢ ~ [2t + 1 ]*<ira2_ L2t' + 1.1

[12z'm'> <z02o lt2t'o>.

I f we apply the formulae for the sum of four Clebseh-Gordan coefficients (14)

(B.3) ~, <lm2 m ' - - mll21'm" > <I'm'S~, ,M-- m' i l 'S~, ,JM><lmS~,M-- m l I S , , J M >. t

• (2m' -- m S~,,M -- m' 12S~,,S~,m~,~> = (--)~*J-~'-~W(W S~,S~,,; 2J)[ (2/'A- 1)(23~-t-1)]~,

and combine (B.2) and (B.3), (B.1) becomes

(BA) (--)J-¢-s~[(2l -t-1)(2S~ + I)]~W(ll'S~,S~,,; 2J)<1020ll2l'O > .

6 9 0 v . Ko SHA~MA

F r o m eqs. (2.22) a n d (2.18) we h a v e

(B.5) B3 = ~ ~ ( l m S : , m ~ , l l S ~ , J M ) .

• ' ' m l ' M S ~ , m ~ , 1 2 8 ~ , S ~ m ~ ) ( 1 2 1 ~ I I 1 2 # ) A.F . ( l m S ~ , ~'t S ~ , J ) ( 2 - - #

w h e r e

(B.6)

f g¢ t i 7 t .

T h e in t eg ra l s over t he spher ica l h a r m o n i e s are done fol lowing EBMO~])S (13)*

+a)Iek3 + )le ~m~ r ~ r L 647t3 l

- - ,~ "Fgt/, - - Tgtl Tf~2 rf~k qqbk - - ~P~3 m]~

: 1

Simi l a r express ions for B1 a n d B 2 c a n be wr i t t en .

• R I A S S U N T 0 (*)

A1 fine di t ra t ta re lo scattering elastico di ncutroni da parte di deutoni a basse cncrgie in presenza di forze tensoriali, si prende in considerazione il metodo del gruppo risonante, che fornisce un insicmc di equazioni integrodifferenziali accoppiate. Si usa la funzione d 'onda di Hulth~n per la componente dello stato s dello stato fondamentale dci deu- toni e mostr~ the le equazioni integrodifferenziali accoppiate dcrivano csscnzialmente dalla inclusione delle forze tensoriali.

(*) Traduz ione a cura della Redaz ione .

P o a b TeH30pHblX CHa B ynpyr0M pacce~HHH He~Tp0HOB Ha ~efiTpOHax.

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(*) Hepeeec)eno pec)atcqueS.