Resonating group model for few-nucleon problems

Acta Physica Academiae Scientiarum Hungaricae, Tomus 42 (2), pp. 111--125 (1977)

RESONATING GROUP MODEL FOR FEW-NUCLEON PROBLEMS*

By

V. K. S~ARMA** DEPARTMENT OF PHYSICS, DHARMA SAMAJ COLLEGE, ALIGARH-202001, INDIA

(Received in revised forro 16. XII. 1976)

A simplified procedure based on the X~esonating Group Approximation is proposed to obtain the integral equations for three- and four-nucleon problems. FXVDE]~V three-nueleon approach has been extended to obtain the FADDEEV--YAKUBOVSKY (FY) model of four nucleons taking into account of their spin and isospin in two-channel resonating group approximation. In this approximation we consider a completely antisymmetric wave function whieh can be written as the clustering of d -4- d and n -4- He 3 (of p + H 3) systems with antisymmetrie spin-isospin states. The two nucleon interaction is assumed to be of the separable YAMAGUCHI forro in Ss I and ls 0 states. The equations for the states with quantum numbers S = 2, 1, 0; T = 0 are obtained. It is shown that the FY equations reduce to a set of one-dimensional coupled integral equations with the kernels containing the functions of two- and three-nudeon problems. By this conjecture one can treat few-body problems involving any number of bound subsystems.

I. Introduction

Few body problems have been s tudied with rising interest during the

last sixteen years. I t is now becoming an impor t an t tool in studies of nuclear

reactions. This is pr imar i ly due to the fact t ha t the invest igations on few- nucleon problems gi re valuable informat ion about the nucleon-nucleon inter-

act ion whieh is basic in nuclear physics. The method of the exact integral

equat ions [1] is used almost exclusively at the present t ime in a t t empts at car ry ing out numerical solutions of the th ree -body problem. Techniques based

on FADDEEV equat ions have been used successfully for s tudy ing th ree -nuc leon

bound state [2], elastic and inelastic n - - d scat tcr ing [3]. After the pioneering work of FADDEV on the formula t ion and solution

o f the th ree -body problems, several a t t empts have been made to obta in FXD-

DEEV-type equat ions for N-particles. A m o n g thesc, the YAKUBOVSKY equations

are one of the best as their homogcneous equat ions are free f rom non-physical so]utions [4]. I n reeent years the FADDEEV--YAKuBOVSKY cquat ions for the

four-part icle problems with separable pair interact ions are widely used to invest igate the propcrt ies of the bound and scat ter ing states for four-nucleon

* Work supported in part by the University Grants Commission, India. ** Permanent address: 3/221, Adarsh ~Tagar, Marris Road, Aligarh-202001, India.

Present address: Department of Theoretical Physics, University of Manchester, Manchester M139PL, UK.

Acta Phydr Academiae Sc.ientiarum Hunga¡162 42, 1977

112 ~ K. SHARM.A

systems [5, 6]. To obtain the integral equations for four particles, various effective approximation methods [7, 8, 9] have been developed. In particular [9], progress has been achieved in the direction based on the resonating group approximation which reduces the four-partiele problem in one dimension.

In this paper we apply the systematic method [10] for the three- and four-nucleon problems, based on using exact integral equations of the FADD~.EV-type, in the resonating group approximation to obtain one-dimensional coupled integral equations for various quantum states. As ir is elear that t he kernels of FADD~~V three-particle equations contain one delta function which can be removed by using nonlocal separable potential to obtain Equivalent Two Body Problemas in MITRA'S formalista [11]. This formalism provides a concrete concept of WHE~LER Resonating Group Substructures [12]. Thus ir can be extended to more than three-body problems with nonlocal separable potential. However, in contrast to the three-body problem, the kernels of the FY four body equations contain two delta functions, these delta functions disappear after taking two iterations of the kernels. The exact integral equations for four-body problems with nonlocal separable potential lead to a set of two-dimensional coupled integral equations. In our view, the resonating group structure with nonlocal separable potential temores these delta-functions to obtain a set of one-dimensional coupled integral equations of Fredholm-type.

In Section II, the kinematics is fixed and the separable two-nucleon interactions used in the following are given. The technique to be followed is described in Section I I I in case of three-nucleon problems, including bound and scattering states. The integral equations for two subsystems of four-nucleon problems are given in Seetion IV. In Section V, the one-dimensional coupled integral equations for four-nucleon problems (i.e. d ~ - d and n - { - H e 3 of p ~- H 3 scattering and bound state of four-nucleon systems) with different quantum states ate obtained using the method of Section III . Section VI dis- cusses briefly the possibilities of using this framework for applications to few- body problems involving any number of bound subsystems.

II. K i n e m a t i c s a n d t w o - n u c l e o n i n t e r a c t i o n s

I t is convenient to use the relative momenta in the normalization of LOVELACE [13] and WEYERS [14]. In the four-nucleon system, we have [9]

kt j = (k i - - k j ) ,

Pq.k ---~ 2(3¡ (kt -4- kj -- 2kk),

Acta Physir Academiae Sr Hungarir 42, 1977

RESONATING GROIIP MODEL 113

[1] q,jk,t = 2(6m)1/2 (k, -4- kj + k k -- 3kl) ; (2.1)

1] kkt ----- 2Cm) 1/2 (kk - - kt) ,

[ i i Sij.k t = 2(2¡ ( k i + k j - - k k -

(m is the nueleon mass).

kl) ,

The normalizat ion is chosen such tha t the four-nucleon c.m. kinetic energy is given by

Ÿ o k2j + k~t + 2 (2.2) = 8ij,kl = k2y + p2j,k -4- qij,2 kl �9

Similarly, the kinematical description of three-nucleon sys tem is identical with the kij, Pij,k and the three-nucleon e.m. kinetie energy is given by

H 0 k~j + 2 (2.3) = Pq.k.

Here, kq represents the relative m o m e n t u m in the two-nueleon subsystem. Let us consider the s-wave separable potential model which ineludes

nei ther hard-core nor tensor forces (for simplicity) as,

< k I V[ k ' > = -- ~ ~t~gt~ (k)gt~ (k') Pt~, t~=O,l

(2.4)

where Pt~; t~ = 0, 1 ate the spin-isospin projection operators of deuteron and vir tual singlet bound state, respectively. The ansatz (2.4) leads to the following form of the two-nucleon t-matr ix as

where

t (k, k ' ; z) = ~ " g,~(k) gt~(k') ~,~(z) Pt~ , tot=0,1

(2.5)

f gt2~ (q) (2.6) ~ ~ l ( z ) = _ 2 t ~ 1 - d q ( z _ q 2) '

and YAMAGUCHI [15] form factors as

1 gro(k) ---= ; t~ = 0, 1 . (2.7)

(k 2 + #L)

The parameters ~0, ~t, #o and/~1 are de te rmined by the binding energy of the deuteron, the singlet and tr iplet scat ter ing lengths, and the singlet effective range.

Acta Physica Aeademiae Sdentiarum Hungarir 42, 1977

114 v.K. SHARMA

HI. Integral equations for three-nucleon system in resonating group approximation

In th is Section we describe a new method [9, 10] for solving the FADDEEV [1] equations for three nucleons taking into account their spin and isospin. The FADDEEV equations for three-particle transition operators are given as

T (ii,k) (z) = Tq ( z ) + Ti i ( z ) Go(z)[T (jk,i) (z) -}- T(ki , i ) (z)] , (3.1)

with Go(z ) = (z - - H0) -1, is the free three-particle Green function and T= is the two-body transition operator in three-particle space. Here H o denotes three-particle kinetic energy (see Eq. (2.3)), and z ---- E -4- i~ is the energy of the system with a small positive imaginary part. (Later, however, we will reinterpret 90, ~, ~E0, g= and $ as the corresponding four-nucleon quantities.) Using the relation between the operator T and total Green function

G(z) = (z - - H o -- V) .1, G(z) -~ Go(z ) -4- Go(z)T(z)Go(z);

one can obtained the set of equations determining the corresponding Green function as

G(q,k)(z) = Gij(z ) - - Go(z ) + Go(z ) T,2 (z) [G(Jk'i)(z) "Jf- G(ki'J)(z)] , (3.2)

where the following notations are used

and

Gij(z ) - - Go(z ) = Go(z)T~j(z)Go(z), G(iJ,k)(z) = Go(z)T(q,k)(z)Go(z),

G(z) = Go(z ) 2 I- 2 G ( i J ' k ) ( z ) �9 ( 3 . 3 ) (ª

The antisymmetric total wave function of three-nucleon system ~o~ is defined by the expression

~o~ lira ieG (E + le) ~~~, (3.4) t;~0

where q~~r is an antisymmetric asymptotic function with account of spin and isospin 7. Using operator Eqs. (3.2) with Eqs. (3.3) and (3.4), we get a set of equations for three-nucleon wave functions as

v, tu,*~o, = ~~J,~~o, + Co (=) %(=) [Fc~,,,Oo, + V't~,,J~~'],

where ~(o iJ'k)~r is the initial wave function.

(3.5)

Acta Phydr Academiae Sr Hungaricae 42, 1977

RESONATING GROUP MODEL 115

According to WH~.EL~.R'S resonating group method [12, 16], the wave function of the system of three-nucleon ~~~, being eompletely antisymmetric relative to the permutat ions of the nucleons, can be writ ten as

= -m, , (kq, pq,~) I ~t,, (q, k ) > ; (3.6) (il, k) 1lt~f

= .~' .~ ~o,,,(k,j) F ~ (p,j,~) I ~,,, (q, k ) > , (U,k) ~t,~

where qt~ (k) is the space part of the normalized wave-function for two-nucleon bound s t a t e ; F~(p) is the Spectator Function [11] of the corresponding argument with pari ty quan tum n u m b e r / 7 ; and I CŸ is the antisymmetric spin-isospin states with conserving quan tum numbers a and ~ of three-nudeon system.

The invariance of two-nucleon interaction under space inversion and the symmetry of the total wave function of three-nucleon system with antisymmetric spin-isospin states lead to the following pari ty relations for the functions ~~ta(k, p) and qt~(k) as

~~, ( - - k , p) = ~ ~ , a (k, r ) , and (3.7)

~,~(- k) = ~ , , (k) �9

Substi tuting the form (3.6) of three-nucleon wave function ~~ into the FADDE~V equation (3.5), following the resonating group method [9, 10] with account of Section I I and pari ty relations (3.7), we get a set of one- dimensional integral equations as

F Ÿ z) = F£ (p; z) -F . ~ f d p ' U~t:~t~" (P' p , ;p ,2 _ b) ,~,. ( f - z - b) F;~'" (p ' ; ~),

where (3.8)

At~t~ . (3.9) UŸ p ; z) = 2 4Ÿ (p2 _~_ p . p i .Jf_ p '2) __ z

3

Here the spin-isospin recoupling coefficients AŸ ate given by the expressions [13]

At~~Ÿ " = (-- 1)~ ~/(2s~ -4- 1)(2ta -f- 1)(2s~ + 1)(2tp + 1) • 11//11 ff~ $a y 7;~ t a

(3 .10)

3 Acta Phydca Atademiws Scientiarum Hunsaritas 42, 1~77

116 v.K. SHARMA

The scattering of a nucleon on two-nucleon bound state

For three-nucleon seat ter ing problem, the inhomogeneous t e rm F‰ and energy parameter z of Eqs. (3.8) are given by

F‰237 (p; z) = (27t) 3 �91 -- Po); z = p~ - - b -~- i 0 , (3.11)

where Po is the relative m o m e n t u m of initial asymptot ic motion and b i s the binding energy of the two-nucleon system.

In analogy with the two-nucleon problem one should put the outgoing wave bounda ry condition as

A ¡ ~" (P, Po) (3.12) FŸ "~~ (p; z) ---- (2zt) 3 �91 -- Po) + 4zt [p2 _ po~ _ / e l ,

where the value of ~ p2~_ -//11 (P, P0) for P~0 is the exaet ampli tude for three- nucleon scattering. Subst i tut ion of (3.12) with (3.11) in Eqs. (3.8) and partial- wave decomposit ion leads to the following set of integral equations

oa T a l O ~176 -~t ai/% (P, PO) : 2~2 Ktli (p, Po) -4- 4zt

where

dp' p'~ OaTa t KŸ A,. (P,Po) (p'2 __ p2 _ ii)

(3.13)

K ~ " (p ,p ' ) =

8 ~:* dxPt(x) 3]/- ~ 2 At=t~ t:tl# 1 [-~(f~ + 4p. p' + al) + b]

x - - P'P~" (3.14) [PI IP'l

The solution of this equat ion leads directly to the phase shift ~t given by

e i~z s in �91 A?h(p, po) = (3.15) Po

Since different values of ( / , / I ) are not coupled, superscript on -//t~ and kernels can be dropped for simplicity.

.4eta Physir Academiae Scientiarum Htmt$aricae 42, 1977

RESONATING GROUP MODEL 1 l~

The bound state o f three-nucleon system

The homogeneous integral equations (3.8) for F U " (p; z) with F~¡ = 0 and z = -- B q- iO ( B i s the binding energy of three-nucleon system) de- sc¡ the three-nucleon bound state.

IV. Integral equations for two subsystems of four-nueleon problem

Here we consider the systems obtained by switching off all the interactions except those that ate internal in the two channels of four-nucleon problem. For the four-particle case, the components of the transition operators of the system of three particles and two non-interacting pairs of particles satisfy FADD~.v.V-type equations [4] as

M~,~(~) = $~(~) �91 q- g~(~)~0(~) .~ M~,~($5) ;

and

No,g~) = ~~(~) a~~ + ~~(~) qo(~) ~ N,,p(~) ;

ot, ti, ~ = ij , j k , k i (4.la)

ot, ti, 7 = ii , kl, (4.1b)

respectively; and other symbols have their usual meaning analogous to the three-nucleon problem (see Section III).

With aceount of spin and isospin the amplitudes M~,~, and N~,~ can be expanded in the spin-isospin space as

and

M~,tj ~ ~S2sT~T - -tttlfra% ~ S~~T2T = ~t~~,~, > M%p < ~tpo,,, (4.2a) STaa%

te�91

I (--1) s+T, a # fl STtett- ~ tDt ~

respectively. The amplitudes M~,~ can have nonphysical singula¡ in particular, the poles on the negative real ayas not responding to the real bound states of the three-nucleon system. These nonphysical singularities cancel out when one proceeds to the following combination as

~ (k, k ' , p'; Z) = ~t(+gY"* (k, k ' ,p ' Z) -k M:~tp p; "~u,~J P; ;

-~- .Lvzij, j k t,~, p; k ' , p'; Z) + ~,~q, ki p; k', p' ; (4.3a)

3* Acta Physiea Ar Sr Htmgarieae 42, 1977

118 v. ~. s r ~ ~ ~ ~

Similarly, i t is more convenient , however , to eonsider the eombinat ions for t he t rans i t ion opera tor wi th the defini te p a r i t y / 7 of the s ta te as

"~U,r t g, x; k ' , x ' ; Z) -4- I'tlI'4"S'FT Art~t~lmtPll- lp' Z ) .

.3f- (__ ~l ~~lj, k i ~~, x ; k ' , ;

(g.3b)

With t he separable forro of pai r - in teract ion (2.4) t he ampl i tudes M and N (*) can be represented in the forro

M7:~: (k, p; k', p'; z) = g,,(k) g,,(k') ~,,(Z -- f ) ~,,(Z -- , '~) x

X [�91 ~~' (Z -- p,2) ~(p _ p,) -4- Xt~'t~(p, p ' ; Z)] (4.4a)

a n d

Nt(~~tt~ (k, x; k ' , x ' ; Z) = gt.(k) gr, (k') ~t~ (Z -- x 2) ~t, (Z -- x '~) •

(2) • [�91 zT l ( z - - x '2) �91 -- x') + Yi~t$,q (x, x ' ; Z)] ; (4 .ab)

2) where the funct ions Xt.t~"~Ÿ and Yl~~t~ti satisfy the fol lowing integral equat ions :

Xt~tp ( p , p , Z) ---- UŸ (p, p ' ; Z) -4- ~t~ .J dp" U~'~~ (p,p"; Z)rt~,(Z - p "2)

x x,~;; (p-, p,; z) ,

and

(4.5~)

v,'~~,,,~~2' (~, ~' ; z) = ( - 1)" w,~,z~,~ (~, ~'; z) + ( - 1),,+s+~ _ y ( d~"• t~,~ .1 (4.5b)

• IVt~t~t,t~ (x, x"; Z) ~t~(Z x "z) (-~) " Z) - - Yt7t~tptff (x , x ' ; ;

with

U~ats (p, p t ; Z) = 2

and

2 1

4 2 p, 3 (p + p" + ~,5) _ z

ffttT a At~ t~, (4.6a)

w,,t~,,,~ (x, x'; z) = g~ (x) g,,(x') ~t~~ ~t;t,. x ~ + x '2 -- Z

(4.6b)

AotQ Ph~,s~ Ar .StiMtiarum Hungadea# 42, 1977

RE8ONATINO GROUP MODEL 119

V. Integral equations for four-nueleon system in resonafing group approximation

In the previous Sections we have discussed what is essentiaUy the problem of the interna] mot ion in the two types of two-body channel (i.e. one nucleon -~ three-nuc]eon type, and bound pair -~- bound pair type), and we have writ ten various integral equations related to the ecattering in the corresponding subsystems. I t is now necessary to consider the FADDEEV--YAKUBOVSKY (FY) [4] equations so tha t we can Ÿ them into the four-nuc]eon prob|em.

The FY equations for four-nucleon wave functions ate obtained by following the method of Section I I I as [4, 7]

~[qijk,~~T ~_ r + ~0(~ ) Mtj, lj(~)[~ffqt.OST ..}_ ~[J(iIaj)ST _]_

+ ~,u~,iosr + ~,(k~JOST] + ~o(:~)M,~,~~ (:~) [~,.~�91 + (5.h)

+ ~~.,~)ST + ~(k~JO st + ~(iJ,~t) ST] + ~0(~) M~~,~~ (~) X [t[/Uii,k)ST ~ t[.f(kji, O'.ST _~ ~/(ij, k l )ST .~_ t[l(jk,il) ST] ,

~,,~,~osr = ~~~~,ko s t + ~o(~) ~r~::~(~)[~(~~,osr + ~,(~.j)sT] + (5.~b)

+ ~o(~) N~~.~~ (~) [~(~",~)sr + ~(~J~,O sr] ;

where M~,# and Nffi,# satisfying the Eqs. (4.la) and (4.1b), respectively, and other symbols have their usual meaning analogous to three-nucleon problem. The free terms ~(£ OST and ~(£ ate the appropriate initial asymptotic etates ~or two subsystems of four-nuc]eon problem.

Using the resonating group method of the three-nucleon case in Section I I I a s a guide, we write the completeIy antisymmetric wave function of four-nue]eon system as*

(ljk,1) Ht~:~~trifz

+~y _y (iJ, kl) Ht,lt~

(IJk,l) llt~f~~jkv~~~

+ _ y ..y (ti,M) llt~~t~

, : , , : . . ~..~j, Ptj,~, q~jk,t) I ~t,:.:,,, (ijk, t) > +

t[jn(l],kl)ST IL r8~s T~~ t~t~! [ls'ij, Igkl, $1J,kl) ] ~ tlttkl (iy, k l ) ~ ,

tO ~~'ij, PiLk) ~~T (quL/) } + ~t,e,,,,,,(zyk, 1) > (s.2)

~ , . ( % ) ~,.(k~,) ~~T,~ . , : ~ : , . ~~ (ij, kl) > ,

ffaTa where ~0~t~(k , p) and ~t~(k) ate the space parts of the normalized wave func- t ions for three-nuc]eon and two-nucleon (deuteron) bound states, respectively; ~~T(q) and gST(q) are the corresponding Spectator Functions [17] of two sub-

* Here d is tor t ion effect is n o t t aken in to account .

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120 v, K. SHARMA

systems; I t :S2sT2r\ and i:sasTar\ ~.ta~a%t / ~tat~ / are the spin-isospin states defined by*

I .t~. . . w ~ , / ) > I = (S;S:)~aS~~as~SXs > I (t, t j ) t a t ~ ' r a t t T 2 r > , ( 5 . 3 a )

~.S~sT2r I ~~a~~ (/y, ~t) > I = (s~s~)~a,(s~s~)s~S~s>l (t~tj)ta, (t~t~) t~T~~.>. (5.3b)

Substituting the form (5.2) for ~sr , the four-nueleon wave funetion into the FY equations (5.1a) and (5.1b) and eliminating the spin-isospin functions and further multiplying suecessively by ~rna~*, ~ta* ~0" and integrating over the internal variables using symmet¡ relations, we get a set of the integral equations with nonloeal separable potential as [9, 10]

~s~ (q; ~ ) ST = ~0n(q; ~) +

-q- . ~ dq ' < t a ~7 a T; a i .l'(t't'%%.~,x (q, q; g ) I tff ~b To> X t#abrb

. ~ dq'<taaa~a X • % s t (q; Ÿ 4 - ( - - 1)rt t,t~

~ST (q; ~ ) s t {V-~~)a I ---- ~0n (q; ~) 4- . ~ dq' • (5.4b)

fil crb" b .~

, _t~~t;~. , %sT %)

w h e r e

.-,ta,,.~,, [ f . " " Ut~t t , (P, Q t ; Q2 b) <t~aa~aIK~Ÿ q'; ~)]t#ab%> ----- Utp~b~b[Jap (p2 _q_ q2 _ Ÿ _ b) X

X F¡ (p; B) F~ "b (Q2; B) 4- (5.5a)

+ .~lldpdk,-- UT,~,~ (k', Qt; Q~ - b ) F#~. (p; B) FŸ162 ~~ (Q~; B) x t~ J~' (p2 + q 2 _ _ ~ __ b)

-1

X ~t~,(~b--q 2 k,2) X~.~.(p, k,; ~ _ q2)] - - t v t ~

l/tat$~a~a <taaa'~a [ -'~,~te (q, q'; ~) I t,t~ > = z~t*t~srn t~~..r. X

X dp UŸ (p, Rt; R~ -- b) ~.~. (p2 + q~ _ ~ _ b) X F~Ÿ (p; B) ~0,~ (R2) -4- (5.5b)

-4- dpdk' ut~tv t,~- , Rt; R12 -- b) F~,~,. (p2 4_ q] _ ~ _ b) ~ (p; B) ~pt~ (R2) X

- - Xt~.t, (p, k , Sb q2)], -1

X Tt',,, ( ~ q2 __ k.2) . . . . . . .

* Here s a + t a = 1 there fore s a is o m i t t e d in t he subsc r ip t o f ~'s.

Acta Phffdr Ar Sr ltungarieae 42, 1977


<t, t~ /~~x (q, q' ; ,%) I t~ ab %> = 2 ~t , o,,,

tTb~b f [ cpt~ (Sx) FiŸ (S2; B) ~t~(x) F # :r̀ (S2; B) X[(S~+q2_,%_b) q- dx (x 2 + q 2 _ , % _ b )

�9 tp(,% q2 S 2) (+) - x " ] - I

- - - - Yt=t~tpt~( ,S 1, ,% - - ~) �9

Here the following notat ions are used

(5.5e)

Q1 - 3q' + q Q2 = 3q -4- q' . 2 V 2 ' 2 V 2 '

R1 V3 q' - q R2 V3 q -- q' V~ V~

S~- - ~ q ' + q S 2 = -- 1/~ q - q' �9 ~;~ , ~;~ ,

(5.6)

with Bt=t~STtp"a'b, ~tpoa~b'g?t . . . . . and DŸ237 are the spin-isospin recoupling coefficients whose values can be obtained by using the explicit mat r ix representat ions of the symmetr ic group S 4 [8].

The scattering of a nucleon on a system of three nucleons in a bound state

For neutron -4- He3 (of p -F H 3) scat ter ing the various q u a n t u m numbers for spin-isospin states are S = 1, 0; T = 0, t = 0, ~ = 3/2, 1/2, T = 1/2. The inhomogeneous terms and energy parameters ,% of Eqs. (5.4) are given by

10 %0n = (2zr)3~(q -- % ) ,

~1o = 0 , ,% = q~ - B + i0 ; (5.7)

where q0 is the initial relative momen tum and B is the binding energy of the He 3 (or H3).

In analogy wi th the two- of three-nucleon problem (see Eq. (3.12)) one shou]d put the outgoing wave boundary conditions as

~sT (q; q~ ~ _ B) = (2=)~ ~(q - - qo) + 4= ~�91237 (q, %) (5.8a) [q~ - q ~ - i r ) '

~ ~ T (q, q0) [q~ - q~ - ir ] ; (5 .8b)

where the values of eg~Ÿ qo) for q2 = q2 ~ is the exact ampli tude for n + He s

Acta Physir Aeademiae Scientiarum Hungarieae 42, 1977

122 v . K . SHARMA

(or p + H a) scattering. Subst i tu t ion of Eqs. (5.8) with (5.7)in Eqs. (5.4) and part ia l-wave decomposition leads to a following set of integral equations

agur (q, 4o) 2n2 B) 4rt d4, q,2 K ~ (q, q'; qo ~ -- B) = Kur (4 ,qo; 4o ~ - - + (q,~ _ qg _ i~ ) •

(5.9~) f / rr~~f t t • ag~(q , ,qo) + ( _ 1)n4r t dq,4,~aur tq, q ;402-B) ~ s r , , ,

(4 'z -- 4~ - - le) ~ur t4 , qd,

lO = rrg~~ t 2 T~x + ~ (4,4~ = "++ a u r t+,~o; 40 + - B) + 4 + "+ + " ( - ~ _ - -~_ - -~ / ) in ~q, .o , ;

(5.9b) where

27~ __ r+l K'I~ (q, 4'; 4~ - B) = ~ ~" I dxP,(x)(q2 -- q~o -- le) X

8 V2t, o , , , J - t

• <t, aa~~ [ K~'~~ (q, q ' ; ~o - - B)Itp~£

(5.10a)

K ~ Ÿ (4, 4'; 4o ~ - - B) = - - 3 V ~ ~ ~+1

? J_ldxP: (x)(q 2 -- 4 8 - le)•

• <t=a~Tal KŸ237176 (q, q ' ; 4~ - - B)[tet;>, (5.10b)

a x P ~ ( x ) ( ~ - 4o ~ - ~~) x Kur(4,4 ;4~-- B) = ~ t,~~,,j_. (5.10c)

• q';4o~ - B ) I t~ ~b~b> ; q - q '

h e r e x = lql" Iq'l and the expressions for kernels ate given by Eq. (5.5). The

solution of Eqs. (5.9) for ampli tude wc�91252 ]eads direct]y to the phase shifts ~t ~ for n -1- He 3 (of p q- H s) scattering given by

' ~ ~ ( ~ , q o ) = o sin ~f(qo) (5.11) qo

The scattering of deuteron by deuteron

For d ~ - d seattering the various quan tum numbers for spin-isospin states ate S ~ 2, T = 0, t ----- 0, ~ ~--- 3/2, ~ = 1 /2 . The inhomogeneous te rms and energy pa ramete r ~b of Eqs. (5.4) are given by

~ o ~ = 0 , ~ = q o ~ - 2b + i 0 ,

= (2zO3tS(q -- qo); (5.12)

Acta Pb~ieo ~ , q e i ~ m Hungarir 42, 1977

RESONATI~G GROUP MODEL 123

where qo is the initial momen tum of relative motion of the centres of mass of deuteron and b is the binding energy of the deuteron. The outgoing wave boundary conditions in this case ate as follows (see Eq. (5.8))

~sT (q; q~ ~ _ 2b) = 4~t ags~ (q, qo)

[~f - 40' - i ~ ] ' (5.13a)

~ST (q; 40 ~ __ 2b) = (2=)s~(q -- q0) + 4~ o~F (q, qo) [ r - 40' - i ~ ]

; (5.13b)

where the value of • q0)for ~ = 4o ~ is the exact amplitude for d+ d sr ing. Substi tution of (5.13) with (5.12) in Eqs. (5.4) and partial-wave decomposition leads to a similar set of equations (see Eqs. (5.9)) but with different inhomogeneous terms as

ST ~,tt (4, 40) = 2~2( - 1)UKt~ffu (4, 4o; ~ -- 2b) +

lO ~ ~ * 2 + 4 ~ �91 K~ , (q , 4 ;4~ - ~b) ~ ~ (4', 40) + (4 ~ - ~ - i~)

+ (-- l ) 4 # f ; � 91 4" Kt~ff (q' q'; ~ -- 2b) ~Ÿ (4', qo), (~'= - ~o' - f*)

(5.14a)

f ; ,.,'g ~ .. t o*~(4 ,~0 ) = 4 ~ aq'~'= ~ ' ~ ~4,q ;qo' - 2b) _ o s T , ,

where

K~(q, q'; qo' 2b) 27 ~ 2 ['+' - = a x P t (x)(r - q~o - f~) x 8 l[~ t , o . , . a - 1

X <t= ffa *a I K t p # ab'~ (q, q t ; ~o - - 2b) ] tp ab % > ,

(5.a4b)

(5.15,,)

, ~t~t tft.T= x <t= a~ *a r , ,~ (q, q'; qo ~ - - 2b) I t~ t D ,

(5.15b)

K ~ (~,q' ;qo' 2b) 3 V3 +' - - = ~ 2 ~ d~ P , ( x ) ( ~ ' - - q~ - - i~) X V-2 ta a~%J - 1 (5.15c)

x <t,~ I K~,~'" (q, q'; ~ - 2b) I t~,~b ~ # .

Acta Physi~ Aea=&mfae Sd~r ltungari~e 42, J977

124 v.K. SHAR~

The solution of Eqs. (5.14) for amplitudes dSŸ leads to the phase shifts 5~ for d + d scattering given by

ei~ sT sin �91 (qo) ~ m (q, q0) =

q0 (5.16)

This solution only exists when I i s even.

The bound state of four-nucleon system

The energy of the system of four nueleons in a bound state is defined by a homogeneous set of integral equations corresponding to (5.4) with %0sn r = = ~0~~ = 0, ~ = -- $ + i0($ is the bindingenergy of the four-nucleon system). In this case the energy of the system is negative and does not exceed the total energy of possible subsystems in ground states.

VI. Conclusion and discussion

We ha-ce outlined a simple method to obtain one-dimensional integral equations for three- and four-nucleon problems where the main emphasis is on the resonating group structure of the wave function. It is clear that the integral equations (3.8) obtained b y t h e above method ate similar to that of MITRA [11]. Therefore, our method could be applied for four- of N-nucleon systems, since the resonating group method, which has been applied suecessfully in the calculation of scattering data of light nuclei, is closely related to the method of pole approximation by which the exact three-body integral equations can be reduced to effective two-body equations [18]. As regards to the four- nucleon equations obtained by other methods [7, 8, 17], our equations have the following advantages. Firstly, they are superior to the pole approximation because the effeet of other ehannels is also taken into aecount. Secondly, their inhomogeneous terms have a transparent physical meaning in terms of scattering meehanisms, i.e. inhomogeneous terms are plane waves. And, thirdly, the equations ate one-dimensional with the kernels containing the functions of two- and three-nucleon bound and scattering states.

For consideration in prineiple, therefore, our method can be extender to an N-nucleon system and the problem can be reduced to solve one-dimensional coupled integral equations. But in a more eomplex system, one eneounters prac- tical diffieulties sinee in resonating group calculations one employs a completely antisymmetrical wave funetion which, from a computational point of view, is feasible only for systems with a relatively small number of nucleons. There-

~4r Phy*ir Ar Sr Hun$arbr 42, 1977


f o r e , b y t h i s m e t h o d a p r a c t i c a l s o l u t i o n o f f e w - b o d y p r o b l e m s i n v o l v i n g a n y

n u m b e r o f b o u n d s u b s y s t e m s is o n l y f ea s ib l e w i t h i n t h e r a n g e o f m o d e r n

c o m p u t e r s .

Acknowledgement

The author is deeply grateful to Prof. A. N. MXTRA for bis stimulating interest in the subject of this work.

REFERENCES

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Compauy, Amsterdam--New York--0xford, p. 113, 1976.

Acta Physica Ar Scientiarum Hungoricae 42, 1977

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Resonating group model for few-nucleon problems