Lippmann-Schwinger equation in the harmonic-oscillator basis for the trinucleon bound-state problem

Nuclear Physics A296 (1978) 134-140; © North-Holland Pubfiahlng Co., Amaterdarn Not to be reproduced by photoprlnt or microfilm without wrlttea pamiaion tkom the publirhar

LIPPMANNSCHWINGER EQUATION IN THE IiARMONIC-0SCILLATOR BASIS FOR THE TRINUCLEON BOUND~STATE PROBLEM E. TRUHLIK T

Laboratory oj7neoreticaf Physics, Joint lnstftutejor Nuclear Research, Dubna, USSR Received 22 June 1977 A6atract : The matrix element of the n-body free Green function between harmonic~oscillator states has been brought into the form of a one~imensional integral which is useful for practical calculations. Using the explicit expansion for the three-body wave function, its asymptotic form is derived by a new method . Starting with the Lippmann-Schwinger equation better convergence for the binding energy calculations is obtained as compared with the method of the diagonali7ation of the energy matrix.

I.

IOtIOd0Cb00

Several years ago the problem of the trinucleon ground state was extensively studied ') by the method of the diagonalization of the energy matrix (DEM) built up from the Reid'soft-core interaction s) in a harmonio-oscillator basis. Strayer and Sauer') improved these calculations by enlarging the basis considerably . Their results are compatible with those derived by other methods 3). The convergence of some of the basic three-nucleon bound-state quantities like the binding energy and S' state probability was lacking, however. It is known, that this trouble lies in the form of the spatial dependence of the oscillator wave functions. A great number of them are needed to build up a realistic three-nucleon wave function which (a) is strongly suppressed at small relative distances and (b) decreases exponentially at large relative distances 4). The complications arising from (a) can be partly avoided by using the super-soft-core two-nucleon potentials s). Nunberg, Pace and Prosperi e) attempted to solve the problem by introducing two oscillator radii instead of the usual one. The employment of another nonlinear parameter gave them a tool to change once more the radial form of the oscillator wave functions so that the convergence would be better . They estimated that they improved it by x 0.2 MeV in thebindingenergy, at the maximum number ofoscillator quanta Qm.~ = 36. In this paper we attempt to make the convergence of the three-nucleon binding energy calculations better using the Lippmann-Schwinger (LS) equation. This On leave from Institute of Nuclear Physics, fEeï near Prague, Czechoslovakia .

134

TRINUCLEON BOUND STATE

135

approach is based on the observation'~ e) that the incorrect asymptotic behaviour of the wave function from the DEM method is due to the truncation of the basis in which the kinetic energy operator is acting . Quite recently, this method has been used 9) to solve the two-centre problem with realistic potentials . A potential V was expanded in terms of harmonic oscillator functions V -" ~~ I i~(iI V~~(jI. Thus, the resulting approximation to Vis a rank-N separable potential, and the well-known methods can be applied to solve the problem. Here, we consider the problem more formally and use rather the completeness of the basis to simplify the treatment of the kernel of the LS equation. The formalism needed for practical calculations is described in sect . 2. The numerical results for the triton binding energy are given in subsect. 3.1 . Our main results concerning the three-body wave function properties are presented and discussed in subsect. 3.2. 2. Formalism Let us write down the LS equation for the trinucleon bound-state problem

I~) _ -Go(~VI~)~

(1)

In eq. (1) the Green function Go(E) is

where E is the energy eigenvalue and Ho is the kinetic energy operator of the system . Further, V stands for the operator of the potential energy and is equal to the sum of the two-nucleon potentials. Using the completeness of the basis constructed io,s) from harmonio-oscillator functions (rlnl) = b-~R,(rlb) [b is the oscillator length parameter, b = (blu~M)~, where m is the oscillator frequency and M is the nucleon mass], we write eq. (1) as (3) I~Y) _ - ~ Go(~Il)~ t, ~

We define the basis states I k) (k = i, j) according to eq . (2) of ref. 3). In principle, the sums in eq . (3) contain an infinite number of terms. We take the first Ni (N~) of them for the sum over i (j). We do not demand that Nt = N~ . The reason is that the sum over i may turn out to be saturated easier than the sum overj. The equality N, = N1 wouldjust mean a waste of the .computer time when performing the sum over i numerically. Now wé transform eq. (3) into the system of homogeneous equations for the coefficients c,~ _
=

-

N~

N~

~

~ <

1=i

r=i

klGo(~li)
k = 1, . . . Np

It is immediately seen that the problem of the solution of eq. (1) is simplified considerably . In eq . (4) the potential energy m _ .riac
13 6

E . TRUHLIK

DEM method. Here we use the Eikemeier-Hackenbroich potential l '). This potential has a core of ~ 800 MeV and reproduces satisfactorily the nucleon-nucleon phase shifts in the singlet-even and triplet-even states up to 300 MeV and the binding energy of the deuteron . We restrict ourselves to the trinucleon symmetric S-state. In this case the matrix element of the Green function G o(E) between the oscillator states ~i~ and ~k~ is

t~~~~

x
(5)

The sum in eq. (5) is restricted by the conditions nl +n 2 +l = dl +riZ +! _ ~ Q, ni+n2+P = rti+nZ+~ _ ~Q' .

(6)

The numbers of the oscillator quanta Q and Q' are related to the states ~k) and ~t~, respectively . The sum over i in eq. (4) is different from zero, only if the selection rule ~(Q+Q')+t+f = even is satisfied. Due to this restriction about one half of the terms is ruled out. The symbols of the type
fo

(

~

()

k=1

In eq. (8) ~k (k = 1, 2, . . ., r) are the absolute values of the Jacobi moments of the (r+ 1 }body problem '~ and a2 = -2E/u~i~. For the practical calculations the following integral representation of the function F is useful : t,t= . . .t~

X

J

~~

a=r

II r(~k+>~t+Ik+~)

_2

~

Z E±. " t(Ar+~iJ

k=~

k=1 ~

k

F(-1'l , -Itk ; -j~k-~lk-~ ;

1-1,/ZZ).

(l,~)

13 7

TRINUCLEON BOUND STATE

Here I'(a) and F(a, ß ; y ; x) are the gamma and hypergeometric functions' 2), respectively . In order that the system of equations (4) be solvable, the condition N,

det ~5~+ ~ < k~G o (É)Iii
(11)

3. Resolts and disca~oo 3.1 . THE THREE-NUCLEON BINDING ENERGY

We found numerically the lowest energy eigenvalue to be a function of three nonlinear parameters b, Ni and N~ . The results are presented in tables 1 and 2. The convergence for Q~ = 20 (Nj = 6~ is similar to the case when Q~ = 16 (table 1). It is seen from table 1 that the sum over i in eq. (4) is saturated for N, = S3 (18 oscillator quanta). T~e~e 1 The triton ground-state energy -E (in MeV) from eq . (4) as a function of the number N, of the intermediate states lei and of the oscillator length b (in fm) ; the dimtnsion of the determinant is 41 N,

b

6 .38 6 .42 6 .33

0 .7 0 .8 0 .9

6.31 6.47 6.38

6.37 6.50 6.37

6 .40 6 .51 6 .35

6 .51

T~ 2 The triton ground-state energy Es,, (in MeV) from eq. (4) as a function of the number N~ of the basis states corresponding to the total number of oscillator quanta Q.a ; b = 0.8 fm and the number of intermediate states In, taken into account, N, ~ 53 (18 oscillator quanta) N~(Q~ - Ee,,

41 (16)

67 (2l1)

6 .51

6.65

In table 3 we present the results derived within the DEM method for the same potential ") [see also ref 's)] . The comparison of tables 2 and 3 shows that the convergence in the binding enèrgy for the LS equation is better. The other investigations of the triton ground-state energy E~ with the potential used here give 6.98 MeV [ref. t 4)], 7 .00 MeV [ref. l')] and 7 .03 MeV [ref. '6 )]. Strayer and Sauer a) estimated that due to the lack of convergence their computed

13 8

E. TRUHLIK

The triton ground-state energy

Ee ,, (in

Teet.~ 3

MeV) from the DEM method as a function of the total number of oscillator quanta Qm ; b = 0.9 fm

Q~ - Em,e

5.93

6.40

6.64

6.76

binding energy was about 0.5 MeV smaller than the extrapolated ground-state energy (this extrapolated value is in agreement with the value from other calculations). Here we see, that using the super-soft-core potential, the rate of convergence in the DEM method is better. 3.2. THE THREE-NUCLEON WAVE FUNCTION

Solving eq. (4) for the coefficients ck , we obtain the wave function of the problem x, (12) I~) _ - ~ Go(~li)dr, ~=t

with

The coefficients c~ are also restricted by the normalization of the function I>/r). In the coordinate representation, the wave function from both the LS equation and the DEM method is of the form i%(Xt,xz)=~Dk(1+8 ~ t=t .,A= ) ,; leven

(21+1)~

4

Yu+~(xt)Yi~(xz)

X (~ti, >~tzhUln t l, nzl)IÂ,~,(xt, xz~(~, ~)"

(14)

In eq. (14) Nk = N, for the wave function from the LS equation and N,, = N~ . for the DEM method . The sums over k - (ntnz l) and (ritrizl) are restricted by the conditions of eq. (6) ; the coefficients Dk = -(2/cufikik with d~ from eq. (13) for the LS equation. For the DEM method Dk result from the diagonalization procedure. Further, 9(S, 7~ is the spin-isospin wave function . We choose the dimensionless internal coordinates z t, iz according to refs . to .3) Finally, for the wave function from the LS equation, (16)

TRINUCLEON BOUND STATE

where

139

(1 _ 2z)~~+a~

(1 +ZZ)~~+~= +2I+3 I+

Xt

l l+ ( Xz

l

xL~,~Cl-4zz1L"~#ll-4zzl' N __

tt .zl

T 2Ch(nt+~+~)l'(nz +'! +~)~(xiXZ)~

~(xi),

~

17)

(18)

The function L~(t) is the Laguerre polynomial 'z). For the DEM method I~ (z t+ xz) = G!~~,(z v zz~ 0) = Ne-~ .t{ + .t~>L^~ ~(xi)L^i e,Az

(

(19)

with N from eq. (18) . It is the factor exp [ - ~( z; + zZ)] in eq. (19) which dictates the asymptotic behaviour of the wave function from the DEM method and forces it to go down too rapidly. In eq. (17) the effect of this factor is reduced by the denominator (1 +2z). Consider now the asymptotic form of Io°o(xt ; xz) from eqs. (16}{18), °° -a~s-p2 /2(t +zz) dz e ~ (1 +2z)s o After some simple transformations we obtain Iôo(xt~ xz)

x

pz =

zi +zZ.

(20)

Generally it can be shown that the function from the LS equation [eqs. (14) and (16}{18)] has the asymptotic form, eq. (21) . The proof will be given elsewhere. The asymptotic shape of the type exp(-ap)/p# for the trinucleon bound-state function was derived from the Faddeev equations by Merkuriev a) (true three-body asymptotic form). It follows that the wave function built up here from the LS equation is more correct than that obtained by thé DEM method [eqs . (14) and (19)]. Consequently, the convergence in the binding energy calculations is also better (see tables 2 and 3). My thanks are to Dr. M. Gmitro, Dr. L. Majling and Dr. J. Révai for discussions . The help in programming by E. D. Fediunkin is also acknowledged . References 1) A. D. Jackson, A. Lande and P. U. Sauer, Nucl . Phys . A156 (1970) 1 ; Phys. Lett . 35B (1971) 365 ; S. N. Yang and A . D. Iackson, P6ys . Lett . 36~B (1971) 1 2) R. V. Reid, Ann. of Phys . SO (1968) 411 3) M. R. Strayer and P. U. Sauer, Nucl . Phys. A231 (1974) 1 4) S. P. Merkuriev, Yad. Fiz. 19 (1974) 447

140

E. TRUHLIK

5) D. Gogny, P. Pires and R: de Tourreil, Phys . Lett . 32B (1970) 591 ; H. Eikemeier and H. H. Hackenbroich, Nucl . Phys . A169 (1971) 407 ; D. W. Sprung and R. de Tournil, Nil. Phys. 201A (1973) 193 6) P. Nunberg, E. Pace and D. Prosperi, preprint LNF-76/4 (P).1976; P. Nunberg and E. Pace, Nuovo Cim. Lett . 7 (1973) 785; P. Nunberg, D. Prosperi and E. Pace, Phys. Lett. 40B (1972) 529 ; Nuovo Cim. Lett . 15 (172) 327 7) L. M. Delves, in Advances in nuclear physics, ed. M. Haranger and E. Vogt, vol. 5 (Plenum Press, New York, 1972) p. 168 8) J. Révai, preprint JINR, E49429, 1975 ; A. L. Zubarev, Teoreticheskaya i Matematicheskaya Fizika 30 (1977) 73 9) F. A. G .reev, M. Kh . Gizzatkulov and J. REvai, Nucl . Phys . A~6 (1977) 512 10) M. Moshinsky, The harmonic oscillator in modem physics (cordon and Breach, NY, 1969) 11) H. Eikemeier and H. H. Hackcnbroich, Z. Phys. 19S (1966) 412 12) I. S. Gradstein and I. M. Ryzhik, Tablitsy integralov (Nauka, Moscow, 1971); H. Bateman and A. Erdelyi, Tables of integral transforms, vola . 1 and 2 (McGraw-Hill, NY, 1954) 13) L. Majling, J. i~fzek and Z. Pluhai, Czech. J. Phya. B24 (1974) 1306 14) M. Fabre de la Ripelle, preprint IPNO/I'H 71-13, 1971 15) I. R. Afnan and Y. C. Tang, Phys . Rev. 17S (1968) A1337 16) S. Fantoni, L. Panattoni and S. Rosati, Nuovo Cim. 69A (1970) 80