The three-nucleon bound state with realistic soft- and hard-core potentials

NUCLEAR PHYSICS A

Nuclear Physics A551 ( 1993) 24 I-254 North-Holland

The three-nucleon bound state with realistic soft- and hard-core potentials A. Kievsky

and M. Viviani

Istituro Nazionale

di Fisica

Nucleare,

Sezione di F’isa, 56010 S. Piero a Grado,

lsriiuto Nnzionale

di Fisk

Nuclcare,

Sezione di &a,

S.

Piss, Italy

Rosati 560f0

S. Pier0 a Grudo,

i’isa.

Italy

and Dipararrimenro di Fisica,

L’niversirrj

di Piss, Piazza

Received

Abstract:

The wave function of the three-nucleon

and the radial amplitudes functions.

(CHH)

three pair-correlation hyperrddius

correlated

functions

is determined expansion

the Hamada-Johnston

expansions

per channel

expansion

in the CHH

in the LS coupling scheme

expansions.

technique. interactions

basis

namely the pair (PHH) The correlation

factor

case, whilst it is a product

case. The optimal

dependence

For the Argonne

are in very close agreement

noticeably

Pis~, ffaly

on a set of correlated

are investigated,

per channel in the PHH

allows for treating hard-core potential

are expanded

hyperspherical-harmonic function

by means of a variational

the results given by the PHH a!. The CHH

bound state is decomposed

types of correlated

includes only one pair-correlation

2, Ml00

17 June 1992

of ail the channels considered

Two different

and the Jastrow

Tarricelli

AV14

potential

with those of Kameyana

too, and the results obtained

improve those obtained

of

on the er for

with other methods.

1. Introduction

Recently ground-state

1,2) the variational method has been successfully applied wave function of a three-nucleon system with realistic

One of the results is that the convergence

in the number

to calculate the NN potentials.

of decomposition

channels

is faster than in the Faddeev 3*4) technique due to the absence of a partial-wave expansion of the potential. In ref. ‘) the radial amplitudes of the channels taken into account are independently expanded on a set of gaussian basis functions whose ranges are in geometrical

progression.

A sufficiently

large number

of basis functions

has been included in order to reproduce the radial dependence of the w.f. with great accuracy and a binding energy of 7.684 MeV (7.033 MeV) for 3H (‘He) has been obtained with the Argonne AV14 interaction ‘). The contributions from the disregarded channels and from a more detailed minimization with respect to the non-linear parameters have been estimated ‘) to give a contribution of about 1 keV. The accuracy of the method is therefore noticeable and it can be used for comparison with other different techniques for calculating the radial amplitudes. One of the Correspondence to: Professor S. Kosati, Grado, Piss, Italy. Elsevier Science Publishers

B.V.

Inst. NW. di Fisica Nucleaire,

Sea. di Piss, 56010 S. Piero a

A. Kircrky

242

important

reasons

to investigate

them to study also nuclear yet successful

constructing

other

systems

in the case of realistic

The choice of the expansion oscillator

et al. / Three-nuckon

(HO)

approaches

the A = 3 ground-state

.s,a,e

is the possibility

with A > 3, where the technique

of extending of ref. ‘) is not

NN interactions.

basis is a delicate

and the harmonic

hound

hyperspherical

point,

for example

(HH)

w.f. with accuracy,

bases

the harmonic

are inefficient

for

even when a large number

of basis functions were summed up ‘.’ j. One way to overcome such undesirable behavior is to multiply the expansion basis functions by appropriately chosen correlation factors. Along this line, the authors in refs. ‘.‘)) used the CHO “I) and CHH expansions, characterized by a correlation factor of a Jastrow form, namely a product of three pair-correlation functions, to calculate the ground-state w.f. of a three- and four-nucleon system interacting through semi-realistic purely central potentials. More recently “.12) the CHO and CHH expansions have been applied to the three-nucleon system with the RV8 version 13) of the Reid soft-core potential 14) and the Argonne (AV6 and AV14) model interactions; the results confirm the expectation that correlated bases can easily take into account also the minor details of the w.f. Motivated by these results, in the present paper the radial amplitude of every channel included in the A = 3 w.f. has been expanded on a set of pair-correlated hyperspherical-harmonic (PHH) functions; the flexibility of the expansion has been found to be very satisfactory ref. ‘) arc exactly reproduced. Another interesting aspect

and for the Argonne is the extensibility

AV14 interaction of the method

the results

of

to treat hard-core

interactions. In recent times, potentials containing a hard-core repulsion have not been used for the three-nucleon problem owing to the difficulty of performing accurate calculations. As an example, the Faddeev techniques are not directly applicable when an infinite repulsion is present. On the other hand a Jastrow correlation factor, as used in the CHH expansion, can easily guarantee that each channel in the w.f. will vanish when the distance of any pair of particles is less than the hard-core radius. The results reported in the present paper for the HamadaJohnston (HJ) potential “j represent a sizeable improvement with respect to those obtained by other variational approaches. The paper is organized in the following

way. In sect. 2 the expansion

of the w.f.

in terms of the PHH and CHH bases is outlined, in sect. 3 a procedure for obtaining in a simple manner quite efficient correlation factors is presented, and the numerical results are reported in sect. 4; the merit of the method based on correlated expansions and future possible applications to larger systems are discussed in the final section. 2. The PHH and CHH expansions The w.f. of a three-nucleon system with total angular momentum isospin T, T, can be written as a sum of three Faddeev amplitudes, 1Y=Ilr(x,,Y,)+~l(x,,Y,)+~CI(xk,Yk),

J, J, and total

(2.1)

A. Kievsky el al. / Three-nucleon hound state

where the Jacobi

coordinates Xi =d

In eq. (2.1) the spin-isospin The ith amplitude scheme

are (i,j, k = 1,2,3

cyclic)

(‘1 - Pk) 9

j$=&rr,+r,-2r,).

dependence

of the function

with quantum

(2.2) Y is implicitly

J, JL, T, T, is written

numbers

243

understood.

in the LS coupling

as +(xi,yi)

= ‘; Qn(xir yl)t?O(jk, ,, - I

oJlvW, 9 = I[ V,,(i,)

(2.3a)

i) ,

YL,.(~,)I,~,.[s~s~I.s,},,.C~~~~:~IT~ ,

(2.3b)

where xi, yi are the moduli of the Jacobi coordinates, NC is the number of channels taken into account and each channel is specified by the angular momenta I,, L, and .d, and the spin (isospin) .r!Zk(t!,k) and sf, (t:) of the pair j, k and the third particle i; moreover, 1, and L, are coupled to give il,, s’,” (tLk) and s:, (t:,) are coupled to give S,, (T). The antisymmetrization of the total w.f. requires 1,.+ ss + r!Yk to be odd; in addition l,, + L, must be even due to the even parity of the state we are considering. The channels satisfying such conditions are easily obtained and those selected in our calculation are specified in table 1 with the same ordering as in ref. ‘). In place of the coordinates x,, y, one can introduce the hyperspherical coordinates given by x, = p cos d,, and the radial

dependence

Y, = P sin 4,

of each a-amplitude

(2.4)

in the WI. (2.3a) can be expanded

Quantum numbers for channels (t = l-12 included in the partial-wave decomposition of the w.f. The ordering of the given channels is the same as in ref. ‘) IA .T,I

OI

1”

I 2”

4,

1 2 3

0 0 2 0 2 2 2 2 1 I

0 0 0 2 2 2 2 2 1

0 0 2 2 0

1 0 1

1 1 2 0 I

1

I I

4 5

6

7 8 9

IO II 12

I

1

I 2

s,,

P
t

0

f L

0

1 1

; ;

0 0

I 1 I I 1 I I

; 1 z 3 I

0 0 0 I I

I

4 3 2 3 ,

1

I 1

244

A. Kievsky

et al. / Three-nucleon

in terms of the PHH basis in the following

The hyperspherical

polynomial

‘2’Pk’.-(4i)

hound

stare

way:

is 16)

=e N>TLcF(sin &;)“*(cos ~,)!~pf;~~~‘~2.‘~~“‘~‘(~os24i),

(2.6)

where N$. ‘rr is a normalization factor and P:” is a Jacobi polynomial and the grand orbital quantum number is defined by K = ltr + L, + 2n, with n a non-negative integer. It should be noticed that the family of polynomials with fixed 1, and L_ values gives the hyperangular dependence of the solution of the Schrodinger equation in the case of a purely hypercentral potential. In eq. (2.5) K, = l,, + L, is the minimum grand orbital quantum number, I(,, is the maximum value selected so that the number of basis functions per channel is given by M<, =:(K-K,)+l,

(2.7)

namely the maximum value of the index n plus one. When the functions .L(x,) in eq. (2.5) are taken equal to one, the standard (unco~elated) HH expansion is recovered. Such an expansion is well suited to describe the structure of the system in the case of soft interparticle potentials, where a rather small number of basis functions is suihcient to reproduce the w.f. with reasonable accuracy I’). However, for realistic NN potentials containing a strong repulsion at small distances, the w.f. must be accurately determined for small interparticle separation values and co~espondingly the rate of convergence of the HH expansion turns out to be very slow ‘). The role of the correlation function f_(x) in eq. (2.5) is to fasten the convergence of the expansion by improving the description of the system when a pair of particles are close to each other. The PHH expansion has been successfully introduced by the authors in refs. ‘.I’) for studying the ground state of nuclei with A = 3,4 with central interparticle potentiats; the extension to treat the case A = 3 with realistic interactions is one of the purposes of the present paper. Of course, other forms can be chosen for the correlation factor, eventually also containing triplet correlations ‘). In the present paper the other form we are interested spherical-harmonic

in is of the Jastrow type and the corresponding (CHH) expansion is written as

correlated

hyper-

Such a product of correlation functions introduces an explicit dependence on the coordinate pj =$.F,, which produces a different channel mixing with respect to the one of the PHH expansion; however, as will be made evident by the numerical results in sect. 4, the consergence pattern is modified just slightly. On the other hand, it has to be noticed that the CHH expansion is well suited for treating also

245

A. Kieosky et al. / Three-nucleon hound State

hard-core

interactions,

Faddeev

techniques

satisfactory cedure,

where

choices

‘.4) are not

determined

the method

of ref. ‘) and the

For all the expansions

functions

can be obtained

considered,

by a simple

pro-

in the next section.

hyperradial

by means

expansion,

applicable.

of the correlation

as is discussed

The unknown

the PHH

functions

u:(p)

of the Rayleigh-Ritz

contained

principle

in eqs. (2.5) and (2.8) are

which can be expressed

by the

condition (&VlH-Ej1V)=O,

(2.9)

where 6,p is the change of the w.f. caused by an infinitesimal functions u’;,(p). From the latter equation, it follows that

variation

P “~‘L~,C(F,(2)p~‘~‘(~,)~yn(jk,i)p-EIv)~>=O,

of the

(2.10)

where F,, stands forfo(xi) orJ;,(x,)g~(xj)g,,.(X,) (PHH or CHH expansion, respectively), and the subscript 0 indicates that the integration over the hyperangles 4; and the angles 2,, 3, must be performed; in terms of the coordinates {&, fi, j,} one has d0 = sin2 (bi cos2 d, d&, d;, d$,. After the evaluation of the spin-isospin traces in eq. (2.10), it is convenient to introduce the variables zi = cos 24, and p, = zi.ji so that the integration

over R can be substituted IdR-nil_:ldp,

by

{:‘dzivG,

(2.11)

and standard numerical techniques can be used to calculate such integrals. In conclusion, by working out eq. (2.10) one easily derives a set of second-order differential equations which can be written in the form C [ W&(p)$+ u’.K’ with a’=l,...,

%$&4

$+

N, and K’= Klo,..

C’;,$(p)+;

EN&(~)]

., K&.. The total number

(2.12)

u$.(~) =o, of equations

is (2.13)

where M,, is given in eq. (2.7). The explicit expressions for the linear A <..<.’ Bz.‘$, ’ C’:TL, and N’& are given in the appendix. K.K’,

3. Correlation

coefficients

functions

As has been pointed out in the Introduction, when the interparticle interactions contain strong repulsions the standard expansion techniques reveal a slow pattern of convergence. The reason is that there are large cancellations between the contributions from kinetic- and potential-energy terms and therefore the w.f. must be very

246

A. KimskI

precisely

determined;

distances

where the potential

one can include system

the accuracy

to get convergence

w.f.

in the

configurations.

system,

rr 01. ! 77wec-~~ucleon bmnd

is required

can attain structure

should

become

particularly

for small

large values. Obviously, terms appropriate

As a consequence,

when all the remaining

.s,a,e

the necessary

smaller.

particles

to describe number

interparticle

just at the first step those delicate

of basis functions

We can notice that, in a generic

nuclear

arc far from a given pair of particles,

the

dependence of the total w.f. on the coordinates of those two particles is mainly determined by their mutual interaction. Therefore, the radial w.f. of the relative motion of the two particles in the state labclled by ,5, can be approximately described by the solution of an equation of the form

L [ T,dr) + Vp.dr)+ AasJr)l.fJr)

= 0:

(3.1)

B’ where

TL3,p,and

V,,,.

are the kinetic

and the potential ilj ( lfi + 1) r*

v,,.,,.=(l&t$l

V(jk)ll&y)

operators,

namely

7 1%K

.

(3.2)

The additional term A,,,,,(r) in eq. (3.1) has the role to simulate the effect on the pair from the other particles. Of course: the solution of eq. (3.1) can help to construct a good variational total w.f. It is clear that there is a large arbitrariness in choosing h,,,(r), since the only condition we have is ‘A,,,( r)l Q 1V,,,.( r)l when r is small. If the potential V,j,,. p rovides by itself a two-particle bound state, with the choice hp,p.(r) = 0 in eq. (3.1) the solution

But the presence of the expansion;

will have a radial dependence containing nodes. of nodes in the total w.f. could be negative for fast convergence therefore, we choose A,,,(r)

and

fix the depth

condition.

As

so

As an example,

= A’I,exp (- v)iip,,,

that the function for an uncoupled

&(r)

satisfies an appropriate state, WC can require that

.ffi(r) = 1,

when r)

R,

r.&(r) = 1,

when r> R,

(3.3) healing

(3.4a)

or (3.4b)

where R is large with respect to the range of the potential V,,,.(r). Both conditions (3.4a, b) can be easily satisfied when the function fP (r) is numerically calculated, but in practice they turn out to be equivalent in producing the convergence of the expansion for the A = 3 w.f. Also the precise value of y does not influence the final result ‘), and the value y = 0.5 fm -’ will be used here. The state fi in eq. (3.1) is determined by the quantum numbers for the pair j, k and it can be a single state or coupled to other ones. In order to give a quantitative

A. Kiecsky

er al. / three-nucleon

idea of the correlation

functions,

to the AV14 potential

and the healing

expansion, chosen

where the Jastrow

in a very simple

(3.1) is calculated the spin-singlet

their graphs

correlation

hound sfale

247

are shown in fig. 1 in correspondence

condition

(3.4a).

In the case of the CHH

factor is used, the function

ge (r) has been

way. For the states with 1 even or odd the solution

with the potential and the spin-triplet

V,,,+.(r) equal to the arithmetic

components

of the central

of eq.

average between

part of the potential.

Since the correlation functions correspond to states of pairs with definite value of the total angular momentum, the.jj coupling would be more convenient for the channel decomposition of the w.f.; however, in this paper we have chosen the LS coupling in order to do accurate comparison with the results of ref. ‘).

Fig. 1. (a) (Iorrelation 4,f,

functions j;,(r)

for the channels listed in table 1; ./, is used for channels

for channel 2, /; for channel 3 and channels 5 to X,j;, for channels 9 to 12. (b) Correlation g_(r).

g,, corresponds

to even channels (/,, even) and g,, to odd channels (I,, odd).

1 and

functions

A. Kieosky

248

et cd. 1 Three-nucleon

4. Numerical The problem

of the ground

of HH correlated

functions

the total

E and

energy

bound .staw

results

state of the three-nucleon

has been reduced the functions

system described

to the solution

u:(p).

by means

of eqs. (2.12) to obtain

The calculation

of the coefficients

X2,‘;,,‘( p), where X stands for A, B, C or IV, appearing in those equations, requires the integrations over the variables pi and zi which have been introduced in sect. 2; to this end, the Gauss-Legendre for integrating with respect to pi equations (2.12) must be solved technique adopted. The important

and the Chebyshev mesh points have been used and zi, respectively. Then, the set of differential and only a few details will be given here on the aim is to reduce as much as possible the number

N of grid points [p,, . . . , pryI in the hyperradius, without losing accuracy in the solution. Due to the nature of the problem, the coefficients X2:;. vary strongly only for smali values of p; therefore, the successive step lengths have been increased by a constant factor x, namely pk,, -pk = x(pk -pk-,). A new variable w is now introduced so that to the grid {,Q~}corresponds a grid {wli} of equally spaced values. The following relation between the variables p and w holds:

with h = P~--P~. For soft-core potentials pi =0 whilst for a hard-core potential p, =x/z r,, where r, is the radius of the hard-core; the last of the grid points is chosen such that pN = pmilx is about 20-25 fm. Eqs. (2.12) can be re-expressed in terms of the variable w and the differential operators replaced by finite differences, as given by the polynomial Lagrange interpolation formulas; values from 9 to 11 have been chosen for the degree of the Lagrange polynomials. At p = p, the solutions of eqs. (2.12) must be zero; for large p-values, the solutions have been required to possess a simple exponential behavior r9). However, it can be noticed that when the solution is taken to be zero at p = I)~ sufficiently large, ail the interesting results remain unchanged. The problem is then reduced to solving the generalized eigenvalue problem (%--E.JY-)G=O,

(4.2)

where G is a vector whose dimensionality is nCqx (N - 2). This generalized matrix eigenvalue problem has been solved in the way discussed in ref. “), by means of the Lanczos algorithm. The values N = 30-40 and x = 1.10-1.12 allow for a satisfactory accuracy. Ail the results given in this paper for the three-nucleon system correspond to the choice h2/ m = 41.47 MeV. fm’. of correlated expansions with the number of the In refs. x,y*‘s) the convergence HH states taken into account has been studied. In the case of semi-realistic, purely central NN interactions, as the ones given by Malfliet and Tjon 20) (MT-V) and Afnan and Tang ‘I) (S3), the conclusion was that few (5-7) hyperradial components were suficient to achieve convergence; now the situation is similar, but for each

A. Kieosky

er al. / Three-nucleon

249

bound slate

TARI.1. 2 Triton CHH

binding energy B and S’-, I)- and P-wave percentages expansions,

respectively,

for the AVl4

the number of equations

--.. B (MeV)

Py (%)

with the PHH

P,, (%)

P,, (%)

1

and M is

per channel CHH --

PHH M

potential,

and when only the first eight channels of table I are considered.

_______~__ R ( MeV)

Ps (“A)

5, (“/)

p, W)

1

3.722

3.616

7.33

0.059

7.425

1.216

8.559

0.060

2

6.810

I SO8

8.878

0.063

7.585

1.129

8.857

0.064

3

7.634

1.117

8.903

0.066

7.620

I.131

8.885

0.068

4

7.650

1.128

8.911

0.066

7.635

I.130

8.898

0.068

5

7.658

1.127

8.925

0.066

7.641

I.129

8.909

0.068

6

7.660

1.129

8.926

0.066

1.642

I.129

8.910

0.068

one of the channels included in the variational w.f. To give a quantitative idea of the pattern of convergence, we report in table 2 the results obtained for the AV14 potential when only the first eight channels listed in table 1 are considered and the same number of HH functions per channel are used to construct the radial dependence of the WI. In table 2, B is the calculated binding energy, Ps,, Pn and PIaare the probability of the mixed symmetry S-state component and the D- and P-state components of the w.f.; as can be seen by inspection of the results, few components (three for the PHH and CHH expansion) suffice for an appreciable accuracy. It has also to be noticed that the PHH converged results coincide with those of ref. ‘); the small differences with the converged results of the CHH expansion arise from the angular-momenta mixing caused by the Jastrow factor, which can be corrected by the inclusion of a larger number of channels. The higher-order channels (a > 8) give rather small contributions to the structure of the system; as a consequence, four hyperradial components for each of these channels are sufficient for the accuracy we are interested in. The results given at the convergence by the PHH approach, when the first 12 channels of table 1 are all considered, are listed in table 3 and compared with the evaluations by other accurate methods lv4). Since the results of ref. ‘) with N, = 12 are extremely close to those of the PHH, in table 3 we have reported

the results from ref. ‘) for N, = 26, to give an

Triton binding energy 0, average kineticcnergy for the AV14 Method

.B (MeV)

T (Me\‘)

potential

7 and S’-, D- and P-wave percentages with NC = 12 Py W)

p, (“A)

p, W)

PHH (12~)

7.678

45.644

1.127

8.962

0.076

ref. “) (34~)

7.678

45.670

1.12

8.96

0.08

’ ) (26~)

7.684

45.671

I.126

X.968

0.076

ref.

250

A. Kieusky Ed ul. / Three-nucleon bound s~nfe

idea of the convergence according to the channel number: it may be useful to recall that the extrapolated value of the binding energy is ‘) B = 7.685 MeV. We have also calculated

the asymptotic

normalization

w.f. and their values are in complete we report

in fig. 2 the ‘H point

constants

agreement proton

I.*‘) Cs, CL) with the PHH triton

with those of ref. ‘)_ For completeness,

and neutron

the PHH w.f. for IV,= 12 and the AV14 potential.

distributions

The distributions

calculated

with

are normalized

to one and the values for the r.m.s. radii are rp = 1.72 fm and r, = 1.88 fm, respectively. It can be noticed that in ref. “) the isoscalar and isovector density distributions have been given for the Reid soft-core potential; from those distributions one can obtain p,, and pn and the results are close to the ones displayed in fig. 2 corresponding to the AV14 potential. As has been stated in the Introduction, the Jastrow correlation factor allows for calculating the A = 3 ground state for the HJ potential, which has a hard-core radius rc = 0.485 fm. The convergence pattern for the CHH expansion turns out to be similar to the one displayed in table 2 for that expansion with the AV14 potential but a slightly larger number of grid points (N = 40-50) are necessary since the radial functions rapidly vary with the interparticle distances; as an example, the average kinetic energy in correspondence with the HJ potential is about 1.6 times larger than that for the AV14. The results presented in table 4 correspond to N,= 10; higher-order channels should increase the binding by about 0.03 MeV. Table 4 also gives the results obtained in correspondence with the choice @,Y(~,, L‘i9Pi) =Xj“Y,L“.f,(X,)g,(x,)R‘,(X~)

(4.3)

in eq. (2.8) and by determining the optimum form of the function fn, g, by means of a Euler procedure as the one described in ref. I’). The CHH results give a sensitive correction to those given by the Euler method, and also the improvement with respect to previous variational calculations 14.25) is noticeable, confirming, on one

Fig. 2. ‘H

point proton

and neutron

dktributions AV14

calculated potential.

with the PHH

w.f. for V,.=

12 and the

A. Kiec.sk_v er al. / Three-nucleon bound sfafe

251

TABLE 4 Triton binding

energy II, average kinetic cncrgy T and S’-, II- and P-wave percentages for the Hamada-Johnston potential with A’, = 10

Method

B (MeV)

T (MeV)

5, (Ya)

P” (%)

P,, (%)

CHH Euler 2’) ref. “I) ATMS ‘0

7.06 6.87 6.5 I 0.2 6.00

72.95 72.19

I .46

10.18 9.95 9.0 8.6

0.09 0.07

1.50

65.9

side, that the accurate description of the three-body system represents a difficult task and, on the other side, the power of the correlated expansions for treating hard-core

potentials

too.

5. Conclusions In most recent years the description of the ground state of the three-nucleon system, in the frame of the standard non-relativistic model with phenomenological NN potentials, has reached a very satisfactory degree of accuracy and it can be used as a useful test for comparison with new techniques. There are some motivations to devise novel methods for studying the structure of few-nucleon systems. First of all, it is necessary to look for the inclusion of non-nucleonic degrees of freedom which, at present, are only indirectly taken into account by the pair and three-body potentials constructed so as to reproduce a number of nuclear properties. The second important point is to attain the same accuracy in describing the structure of larger nuclei, beginning with the alpha particle. The use of a suitable expansion basis has a long history in the development of the theoretical analysis of nuclear structure. However, such an approach is successful only in the case of effective interactions which do not contain large repulsions. On the contrary, the so-called “realistic” NN potentials are characterized by a strong state dependence and a large repulsion at small distances; when these potentrals are used, it is necessary to set up new ad hoc constructed expansion basis. The method of correlated functions and correlated basis functions (CRF) has been proved to be very effective in the variational

approach

of inlinite

nuclear

matter

“*2h)

and light nuclei 27) too; for such a reason, an obvious generalization is to reconsider the usual expansion basis modified by the introduction of correlation factors. The main conclusion of the present paper is that for the A = 3 ground state, only very few correlated hyperspherical-harmonic functions per channel are sutlicient to obtain at least four correct figures for the quantities of interest. The correlation factors can be obtained in a quite simple way; however, if the correlation factors should be taken equal to one, with the corresponding number of HH functions considered in the paper no binding at all would be found.

A. Kiscsk~~ Ed al. / Three-nucleon

252

The PHH expansion

involves

hound sraw

only pair-correlation

functions,

therefore

there is

no mixing in the usual Faddeev amplitude decomposition in channels with definite values of the orbital angular momenta b, L, ; it is a merit of the Faddeev prescription to construct the w.f. and of the expansion basis adopted in the calculation that a detailed dence

description of the different

of the system is accomplished. amplitudes

interactions, containing also The advantage of the CHH with infinite NN repulsion; potentials have been used to

In conclusion,

are finely constructed,

the radial depen-

so that other types of NN

three-body forces, can be treated equally well. expansion lies in the possibility to treat also potentials in the literature there is plenty of works where such study the ground state of the A = 3 system: the results

here obtained for the HJ potential improve on the precedent ones by about OS1.0 MeV showing that the details of the structure must be constructed very carefully. The technique of correlated basis functions discussed in the present paper can be extended to the study of the A = 4 bound states in a rather straightforward way: the results will be the object of a forthcoming paper. The authors discussions.

would like to thank

Prof. L. Bracci and Dr. A. Bonaccorso

for useful

Appendix

The linear written

coefficients

of the second-order

differential

equations

(2.12) can be

in the form

XL;;UK’ p) = P(‘<.+L”+‘,,“.*J ‘( x x (F, qJy

-(&)G!/,,(jk,

i)(l‘,(F,.

“‘P’;T..:L”(~j.)~yn.(j’k’, i’)),?,

,,I’ (A.1) where

X stands

for the A, R, C and

f-(x,) (PHH) or fu(xi)gc,(xj)g,,(xk) expressing the laplacian operator

N terms of eq. (2.12) and F‘, can be either

(CHH). The operators f; are obtained after in terms of the hyperradial coordinates, namely

(A.21 where L2(0) verify that

is a purely

hyperangular

r, = r,

operator

16). It is then

straightforward

to

= 1,

(A-3)

A. Kievsky er I;.

is the sum of three

potential-energy

terms

operator.

al.

/

coming

The

Three-nucleon

hound

.sfu~e

from the kinetic-energy

first three

terms

are specified

2.53

and one from the by the following

relations: 2(1,,,+ L,,,) -5 P

1

a a~, F,,,

T,;F,,,;~[(I,.+L,,-2.5)(1,,+L,,,-1.5) P2 -(2n’+ I-,;F,,=

r,,,+ L,,)(2n’+

1,*+ L,,,-4)]F,;,

2Vf1F,,,‘Vn ;

%(I

-z~)~+V~,F~..V~,+V~,F~..V~,

I’

1 .

(A.4)

The matrix elements (A.l) are summed over the three permutations of the hyperangular variables {d, 9, a}. In order to calculate the spin-isospin traces and the integrals on the hyperangles, a set of reference is chosen, for instance the set i, and all the basis vectors C!/,, are expressed in terms of the vectors of that set, namely “‘Pb’-(~&)“3/,(nl,

m) = x ZI~K,I,IK,l “‘P~,‘l(~,)?!I~,(,jk, lK11

i),

(A.9

where [K,,] = [K, I,,, L,, _A,, sz’, tz’, S,] is the set of quantum numbers which specify the u-channel and [K,] = [K,, I,, L, , A,, SC,, t$, S,,,] is a set of quantum-number values required to perform the transformation. In the summation (A.5), the following conditions must be satisfied: K=K,. (A.61 A‘, = &, Se = S,“,1 The linear coefficients D in eq. (A.5) are a product ofthe spin-isospin transformation coetlicients and the Raynal-Revai I”) ones. Once the matrix elements (A.l) are referred to the prefixed set of coordinates, the spin-isopin traces can be performed and terms of the following general form are obtained:

=~(2n+1)C,(I,,12,11i,li,1’,n)Pn(CLi),

(A.7)

where Pn(pi) is a Legendre polynomial. The coefficients C, correspond to the different operators specified in eqs. (A.3) and (A.4) and they can be calculated by a straightforward algebraic procedure. Finally, the integrals in the variables q, /I, are calculated as specified in the text. References 1) H. Kamcyana, 2) H. Kamcyana,

M. Kamimura and Y. Fukushima, Phys. Rev. GUI (1989) 974 M. Kamimura and Y. Fukushima, Nucl. Phys. A508 (1990) 17c

254

A. KiecskJ et al. / Three-nucleon bound State

3) S. Ishikawa,

T. Sasakawa,

7‘. Sasakawa

4) C.R. Chcn. G.L. .I.].. l+ir,

Few-Body

and G.L.

1877;

Phys. Rev. C31 (1985)

266; C33 (1~86)

1740;

Payne, Phys. Rev. C36 (1’387) 1138

R.A. Smith and T.A. Ainsworth,

Strayer and P.U. Sauer, Nucl.

I’. Nunbcrg,

Phys. Rev. Lett. 53 (1984)

Syst. 1 (19X6) 3

Payne, J.I.. Friar and B.F. Gibson,

B.F. Gibson

5) R.B. Wiringa, 6) M.R.

T. Sawada and T. Leda,

and S. Ishikawa,

Phys. Rev. C29 (1984)

Phys. A231 (1974)

1). Prospcri and E. Pace, Nucl.

7) J. Bruinsma and R. van lVagcningen

1207

I;

Phys. A285 (1977)

Phys. Lett. B44 (1973)

58

221;

J.L. Ballot and $1. f:ahre dc la Ripelle, Ann. of Phys. 127 (IWO) 62 8) A. Kiessky, S. Rosati and hl. Viviani, Uuel. Phys. ASUSU1 (1989) 503; S. Rosati, M. Viviani 9) 5. Rosati, 41. Viviani

and A. Kiewkj,

i-ew-Body

and A. Kicvsky.

Nuo\o

10) C. Ciofi degli Atti and S. Simula, 11) A. Kicvsky,

Lett. Nuovo

S. Rosati and >I. \jiLiani,

12) S. Rosati, .4. Kievsky 13) R.R. Wiringa

and M. Viviani,

Syst. 9 (1YYU) 1

Cim. A, in prcsn Cim. 41 (19X4)

101; Phys. Rev. C32 (1985)

Syst. II

I’elv-Body

Syst. Suppl. 6 (lYY2)

and V.R. Pandharipande,

Rev. Mod.

(1991)

1090

I I1

I:cw-Body

Phys. 51 (1979)

563

821

$4) R.V. Reid, Ann. of Phys. 50 (196X) 411 15) T. Hamada

and I>. Johnston,

16) M. Fabre de la Ripclle,

Nucl.

Phys. 34 (1962)

i\nn. of Phys. 147 (1983)

3X2

281

17) G. Erens, J.L. Visschers and Ii. van M~ageningcn, Ann. of Phyc. 67 (1971) M. Beincr and 41. Fabrc de la Ripellc, 18) A. Kievsky,

M. Viviani

vol. 6 (Plenum, 19) A. Kievsky,

.M. Viviani

20)

R.A. Mallliet

21)

L.R. Afnan

IYYI)

and J.A. Tjon,

Nucl. Phys. A217 (1969)

and Y.C. Tang, Phys. Rev. 175 (1968) I1.R. Lehman E.L. Tomusiak

Delves and MA.

2s)

Y. Akaishi

26) S. Fantoni 27)

Hennell,

and S. Uagata,

and G.I..

Nuavo

( 1991 I

161

1337

Payne, I’hys. Rev. C25 (1982) Payne. Phys. Rec. C24 (IYSI)

Phys. A168 (1971)

Prog. Theor.

Phys. Rev. C43

28) J. Raynal and J. Revai,

and G.L.

Nucl.

and V.R. Pandharipande,

R.B. Wiringa,

nuclear physics, cd. L. Bracci

Piss 1989) p. 105

23) J.L. I-riar, B.F. Gibson, L.M.

matter theories, cd. S. Fantoni and S. Rosati,

and S. Rosati, in: Perspectives on rheorctical

Editrice,

461;

5X4

p. 391

22) J.1.. Friar. B.F. Gibson, 24)

Cim. I (1971)

and S. Rosati, in: Condensed

New York,

et uf., vol. 111 {ETS

Lett. Nuo\o

Phys. 48 (1972)

Cim. 68A (1970)

612

665

347 133

Phys. Rev. C37 (198X) 1585

1616

1697. and references therein