A simple and realistic triton wave function

Nuclear Physics A342 (1980) 404-420;

@J North-Holland Publishing Co., Amsterdam

Not to be reproduced by photoprint or microfilm without written permission from the publisher

A SIMPLE

AND REALISTIC

J. LOMNITZ-ADLER

TKITON

WAVE FUNCTION’

and V. R. PANDHARIPANDE

Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

Received 19 November 1979 (Revised 11 February 1980) Abstract: We propose a simple triton wave function that consists of a product of three correlation operators

operating on a three-body spin-isospin state. This wave function is formally similar to that used in the recent variational theories of nuclear matter, the main difference being in the long-range behavior of the correlation operators. Variational calculations are carried out with the Reid potential, using this wave function in the so-called “symmetrized product” and “independent pair” forms. The triton energy and density distributions obtained with the symmetrized product wave function agree with those obtained in Faddeev and other variational calculations using harmonic oscillator states. The proposed wave function and calculational methods can be easily generalized to treat the four-nucleon a-particle.

1. Introduction

Variational theories of nuclear matter often use the wave function: Y = p3,

(1.1)

where 8: is a suitable many-body correlation operator, and @ the Fermi gas wave function. Detailed calculations ‘92, of the E(p) of nuclear matter have been carried out with a rather simple fsp, BSP =

n Fij,

y

i
(1.2)

that consists of a symmetrized product of two-body correlations operators FiY The Fij contain central, spin, isospin, tensor and spin-orbit correlations, and are generally written in the forms: p=l,n

= fix1 +

c

u$OE).

(1.3)

p=2,n

The f;.; are functions of Iri - rjl, and OG=‘*’ are defined as operators: O$=‘** = 1,

bi

’

aj, Zi’

Tj, (ai * aj)(ri ’ ~j), Sij, Si,Jzi’ zj), tL *s)ij9 (L *s)ijt’i ’ zj).

’ Supported by NSF PHY 78-26582. 404

(1.4)

J, Lmmitz-Adler, V. R. Pandharipande / Triton

wave

fwtction

405

The superscripts c, tr, r, err, t, tz, b and bz are often used to denote functions associated with these operators, i.e. f”, uoZetc. The 9 is often expanded in powers of non-central correlations u~‘~,* = fP/fc. An alternative form gIP used by Owen 3, allows spin-isospin correlations between independent pairs only. Symbolically gIP may be written as: R BlP

=

lJ

Fij,

(1.5)

i
where the restricted product flR omits all terms of type ~5’ ‘u$> l 05’ ’ 0;;'i, describing spin-isospin correlations between connected pairs, in the expansion of fli
406

J. Lomnitz-Adler,

V. R. Pandharipande

2. The variational

In the independent pair approximation

/ Triton wave fiction

wave function

a general three-body Y,, can be written as:

YY,,= U-I _@{I +

c

c

$>@Y

(2.1)

cycp=2,8

CYC

respectively represent the product and sum of the three terms ncyc and ccyc obtained with ijk = 123,231 and 312. We first examine the properties of @ and use them to limit the sum over p to p = 6, t. The non-interacting triton @ is given by: where

@ = &&1(t)z1(p)X2(t)z2(n)X30X3(n) E JZs

n

Xi(a)Xi(z),

(2.2)

CYC

where d is the antisymmetrizing operator, and xi(f, 1) and Xi(n, p) are spin and isospin wave functions for the particle i. The @ has spin and isospin 4 and the spinand isospin-exchange operators satisfy pTjpij @ = - @. The following properties of @ may be easily verified: Zi’ Tj@ = -(2+oi.

Oj)@,

(2.3)

(ai. Oj)(Zi . Zj)@ = - 3@,

(2.4)

Sij(Ti. Tj)@ = - 3Sij@,

(2.5)

(L. S),,@ = 0.

(2.6)

These relations imply that the wave function Y*p = ( C

XT){1 + C (ii;Oi *

CYC

Oj+

U:jSij)}@,

7’ =f”(l

-2u’-3C),

ij” = Ub-UT, ii’ = u’- 3UZT, is equivalent to (2.1) in all terms linear in up, and the variational IP approximation are carried out with it. From the results obtained in nuclear matter it appears that the are much more important than the spin-isospin correlations, dominated by ufr. So we simplify the Y,, tremendously by taking term in u’~: Yy,p =

(2.7)

CYC

{nJjs}{l+

C(ii~ai.aj+u:jSij+ilsii:jii5kOij,)}~,

CYC

(2.8) (2.9) (2.10) calculations in the tensor correlations and that they are only the quadratic

(2.11)

CYC

where Oijk

=

{Sijtij,

SjkZjk},

(2.12)

J. Lmnnitz-Adler, V. R. Pandharipande / Triton wave fiction

407

Using relations (2.3H2.5) we obtain: wijk@

=

3[SijSj,(2+Oi. Oj)+Sj,SiJ2+Cj

’

O&.)1@*

(2.13)

We find the energy does not change too much on going from Y,, to this approximate Ysp, and thus further refinements in it may have small effects. We also use a more general variational wave function YYSPT = f :“,3‘ys,,

(2.14)

that contains a spin-isospin independent three-body correlation: f:“23 = 1 + C U:pU:.f: COS8,

(2.15)

CYC

where ei is the internal angle of the triangle ijk. It is similar to the one used in liquid 4He [ref. ‘)I to describe Feynman-Cohen backflow. In principle we should determine the functions 7” which define the Y by minimizing the triton energy. In practice this variational problem is much simplified by describing these functions with a few variational parameters which are determined by minimizing the energy. The long-range behavior of the 7” can be obtained from the following arguments. If one particle (say particle 2) is taken very far away from the other two then Y must decay as: Y(r*+m)a~

x

(l+aY” (1+:+$ z

x = ~lr2-~311,

(2.16) (2.17)

where R, 3 is the c.m. of 1 and 3, and rc is related to the one-particle separation energy J% x=

-2@, 2 J-- ii

p

’

=

(2.18)

{m,

while CIgives the D-state amplitude. This asymptotic behavior is imposed on the Y,, by requiring that J‘j(rij > d) = exp (-grcrij)/~, ${rtj

> d) = & CXp (-$Crtj)

1+ $

( p(rij

>

d) = 0,

i,

(2.19) +

3 (Klij)Z>I Jlii,

(2.20) (2.21)

and the E, and d are taken to be variational parameters. The correct short-range behavior of fP, which is dominated by the strong potential, is obtained as follows. Let fi and f3 be radial correlations in singlet and triplet states: fi = J;” - 37@,

(2.22)

f3 =f”+p.

(2.23)

J. Lomnitz-Adler,

408

V. R. Pandharipande

/ Triton waue function

We assume that at Y -C d the fr, f3 and 7’ are solutions of the two-body “Schrodinger” equations in singlet and triplet even states: (2.24)

- ~V’+~:,(‘S,)+I,T,(~)-E,

-~V2+l:.(3S,

-3Dl)+&r3(r)-~3

f3(r < d)+8u,(3S,

-3D,)f’(r

< d) = 0, (2.25)

{_g( v

2

6)

- 7

xJ“(r < d)+ v,(‘Sr - 3DI)f3(r < d) = 0.

(2.26)

The sl, s3 and E,are chosen to make the second derivatives of fr, f3 and 7’ continuous at r = d, while A,, A, and the D-state amplitude CY are chosen so as to make rf(r) = 0 at Y = 0. Thus only the functions T(r) are left undetermined. They represent the radial dependence of the modification of the two-body interaction due to the third particle, and we can always choose them so that T(d) = 0. The T(r -+ 0) is unimportant because the f are dominated by the strong u at small r. An adequate variation of the T(r) is probably provided by taking: Ti(r) = (d-r)+Pi(d-r)2,

i=

1,3.

(2.27)

The /?r and /I3 are taken to be the variational parameters. By means of T(r) we essentially vary the fp at intermediate range without influencing their small and large-r behavior. Solution of eqs. (2.18)-(2.27) provides us with yP(/?r, fi3, E,, d). As per definitions the U” in wave functions (2.7) and (2.11) is taken to be f”/T’. However we provide for a little extra variation by taking: ii’ = /?J’/f”.

(2.28)

The parameter fi, simply varies the overall magnitude of the tensor correlation, and we should obtain a minimum in energy at /I, z 1 if the above calculation of fp is adequate. At first glance there there may appear to be too many variational parameters in this wave function. However, PI and fi3 have little effect on the energy; the short/longrange behaviors of f” are sufficiently restrictive. The equilibrium value of 8, is generally close to unity, and the calculated energy has a very shallow minimum with respect to variations in E, at E, = -E the triton binding energy. As a matter of fact once d is large enough so that v(r > d) is small the E(/3, = /I3 = 0, /3, = 1, d N 3-4 fm, E, = -E) is quite close to the minimum energy. Thus most of these parameters are primarily intended to verify the arguments underlying eqs. (2.19H2.26) used to calculate the f-functions. The ufb(r) are described with two parameters Btb and y that vary its range and

J. Lomnitz-Adler, V. R. Pandharipande / Triton wave function

strength: ufb(r) = Jj& I exp ( - yr’).

(2.29)

Thefb was introduced in ref. ‘) primarily to minimize the contribution of terms like -(h2/m)VJ, V,TF2 to the energy. These terms are not too large in Reid models of either nuclear matter or the triton, and so the j”Ib has small effect on the energy. 3. The calculation of the energy The variational parameters the energy expectation value

in the wave function are determined by minimizing

(3.1) and the minimum value of(E) gives an upper bound to E,, the ground-state energy for the Reid model of the triton. If the minimum E,,,, E,, and ESPTobtained respectively with the Iyip, Ys, and Ys,, are close to each other one may hope that the chosen variational wave functions are general enough, and ESPTx E,. The numerator and denominator of (E) are calculated separately. Let us first consider the calculation of ( Y 1Y) : Cyly)

=

{n X!(“)Xt(T)~d~+~{~ s

where we have antisymmetrized is given by:

(3.2)

Xi(o)Xi(r)l7

CYC

CYC

only the left hand Y*. The d (operating on its left)

d = (l+e,,+e,,+e,,+e,,e,,+e,,e,,),

(3.3)

where eij are spin-isospin exchange operators: eij =

--$[l+bi~~j+ri*rj+(Oi~crj)(ri*rj)].

(3.4)

Inserting (3.4) in (3.3) we get: d = $+&(l-+

1

(Oi.

&7,*oj)(l-

cq.zj)

CYC

CYC

oj)(zi

* Zj)

-+(ol

* CT2

x

cT3)(t1

. z2

x

Q.

(3.5)

CYC

The ~~9 does not contain any isospin operators so we can take the matrix elements of the q * ?j and r1 . z2 x z3 between the ncyc x!(r) and neyc xi(r). Those of zi * zj for ij = 12, 23 and 31 are respectively - 1, 1 and - 1, while that of 7(. z2 x T~ is zero. This gives (3.6)

410

J. Lomnitz-Adler,

V. R. Pandharipande

/ Triton wave function

where d,

= +--&

* 63.

(3.7)

The uij has ri * zj operators which need to be eliminated before the isospin part of (Y/HI Y) can be calculated as above. We use the Reid potential in its so-called us form 1,2), obtained by expressing the potentials in the ‘S,,, 3S1-3D1, ‘Pi and 3P2-3F2 states as a sum of eight terms associated with ejC1**. It is convenient to define spin and isospin space operators O:, ij, OF,ii: O~,~f’” = 1, pi ’ ~‘j, S,j, (L ’ S)ij,

(3.8)

OY.r.2 t r.,’ 7,IJ = I Y 7.’

(3.9)

The equation Vij =

C

C

(3.10)

V~O~,ijO::ij,

p=1,4q=1,2

defines the functions a+“; for example u$’ is the central potential v’(rij), uz” is the potential afT(rij)associated with operator Sij(zi. rj) etc. The relation: zi * Tj#

= $Tq. Tj@ = - 9(2 + cri. ajp,

(3.11)

obtained with eq. (2.3) enables one to eliminate the T operators from f+Hg, and we get :

+{C

1

cycp=1,4

C vr”f,

ij8tdql - 6q2(2 +

q=1,2

Oi ’gj)> Xi(a)J* 1
CYC

The spin overlaps in eqs. (3.6) and (3.12) can be easily calculated with the following expectation values:


(3.14)

CYC

(Cai’

Oj)

=

-3,

(3.15)

CYC

(~Uix4crjd3)

= -A-B+i(AxB),.

(3.16)

CYC

It is also covenient to introduce the general tensor operators ‘) c&4, B): a&4,B) = ~ai.Aoj.B+aj.Aai.B)-A.Bai.oj, sij = c$,(Pij,P,), where Pij are unit vectors.

(3.17) (3.18)

J. Lmmitz-Adler, V. R. Pandharipande / Triton wave function

411

To calculate the ( YlY) we expand the ~~9 in powers of iit. In the IP approximation: dPBIP=

{n.Ey2){1+

C CYC

CYC +

C

p’=a,t

{

C

UG[20:,ij

p=a,t

(ii$O$ ijO~‘ij+U~~(Of,ij, 0:: j,})]}}. ’ ’

(3.19)

The product O:, ij Opb’, ij can be trivially expressed as a sum of terms with operators 1, gi * oj and S,,, while the anticommutators are given by: {Gi* aj, oj. Ok} = 2oi. Ok,

(3.20)

{Sij, aj ’ bk} = 2Crik(P,i, 3ij),

(3.21)

(Sij, Sj/c} = 18pij’ Pjk0 i . PijcTk. bjk- 2crik(Pij,Pij)- 2cli#j,, Pjk)- 20,. Ok.

(3.22)

Let us consider the expectation value of the term 2ii:jSij in ~~9. It is zero, but it illustrates the method adequately. This expectation value is given by:

{fl Xt(“)>do{rI Jj~‘>{C

=

ulZaij(31*~

J

CYC

CYC

CYC

do)1

p12)>{fl CYC

cn ~~‘}((3 C bi’ P,,Oj. P,,)-( C pi’ bj))U:2 = 0. (3.23) s CYC CYC CYC Eq. (3.23) is obtained by relabeling the ri in the three terms of xEyCU:jSij such that each depends only on r12. The {nCyCxG2} is invariant under relabelings of ri. The other terms in (YIY) can be similarly calculated and we obtain: =

(YIY),,

= z[1-3u’;,(z~,+1)+3u:,z:,],

Z[x] E

d3r,,d3r,,{n s

z;,

z:,

=

$‘}x,

(3.24) (3.25)

CYC

1-5ii;z+u;3+ii631,

= 4u:,--u;,(3+1)-u~,(3+1)

(3.26) (3.27)

and ci, i = 1,2 and 3 are the cosines of the internal angles of the triangle rl, rz and r3. The kinetic energy expectation value (T),, can be similarly calculated. We define (3.28) fi

= (vy/p)ij,

;fi = (v2fiqij,

Y = C (Jj+_$jj;.$j). CYC

(3.29) (3.30)

J. Lomnitz-Adler, V. R. Pandharipande / Triton wave function

412

The ( 7’)rp is conveniently

divided into three terms: (T)IP = TI,,P+ Tz,rp+ T3,IPY (3.31)

T3,rp = -2

T1,IP

=

T2,IP

=

~I[y{l-31?;,(2~~+1)+3ir:,z;,j],

-

-

~~[-3C;,Z;,+3(;;,-61i’/r:,)Z;,1,

(3.32)

~l[(j12+~,CIH-3ri;,Z;2+jt;lZ;Z) + lg~:,~:3U&c1

These terms respectively

(3.33)

+c,c%“LcA1-4/~,,1.

contain contributions

from V2{nCYCXT}, V2{CEycahOt>

and V{flCYCA;> 9V<~,, $Pf.$ To calculate (u),~ we replace the ~~0~. ijg(dq, -6,2(2

+ Qi ’ aj)

(3.34)

in eq. (3.12) with: UWO$,ij(6,r -6,2(2 + Oi * Oj))f + Lf;O:, ijsq2[ai ’ Oj, 81. The contribution

(3.35)

of the first term above may be calculated with a GPO: defined as:

C

1

C iFOf: = p=1.4

PO~(dql -dq2(2+Oi

’

(3.36)

Oj))*

q=1.2

p=1,4

We have v” = v=--a=-

The expectation

3v*r,

(3.37)

ijo = vo-v=,

(3.38)

v’ = v’-3lP,

(3.39)

fib = ab_3$’

(3.40)

values (f+‘),, are given by: (P)rp = I[{1

(3.41)

~j}{l-3U~~(Z~~+l)+3U~~Z~~}],

CYC (Ua)lp

=

r[~,{-3(22~2-1)+3u:,(2z:2-4u:,)-39(u~2)2+30u”,,ii~, - 12u;,u;,

(J),,

-6(U;,)2 +24(1&&~ -6ii:,U:,(3c:

- 1,}],

(3.42)

+ I,}],

(3.43)

= Z[~l2{6(1+ti~2)z’l2-6U:2(22:2-4ii:2+16~J+U:3(3c~-l) x(-12u;,+6oG;,

-24ii;,)-

1&;,&(k,C,c,

J. Lmnnitz-Adler, @),p

=

I[$‘,(-9i&

V. R. Pandharipande / Triton wave

function

413

(2zi, -4ii:,)+u:,r,,c,(-36u:,+9ii:,(3c~-l))/r,,

-9u:lr12(~2~:+c1cJXii:3/r23+I^i:3)-

18ti:,ii:,r,,c,[(l

-c’,)f&

-(crcz + c3) zm-

(3.44)

The expectation values (6uPq),, denote contributions of the commutator eq. (3.35). These are non-zero only for 4 = 2, and are given by: (6v’),, = 1[36u;,ii;,(2ii:,

-i&,(3c;

(8uoT)rp = I[o~~{36U:,(2ii:,-U:,(3~3(Bu”)rr=

- l))],

(3.45)

l))+ 144ii;,(ii;,-U”,,)}],

Z[u:f,U~,{144(3c;- l)(ii& -U~3-~~3)-72ii:1(3clczc,+

(6ub’),, = I[~b,;{54r~?ii:~[u:~c~(3~3-

term in

(3.46) l)}], (3.47)

l)/r,,-(ii:3/r23+~:3Xc2c23+c1c1)]

(3.48)

- 108(u:,)2r,2c2/r2,)l.

The total (u)rr is given by:
(3.49)

~,,m-%p),

(3.50)

+ ,~),
Our approximate Ysp is correct only up to second order in up, and so we calculate (YlYu>s,, (7% and (u)sr only up to second order in up. At this level those terms that contain iiijUfkgive different contributions to the expectation values with Y,, and Y,,. We obtain:
(3.51)

l)],

TLIP -61[yU:2U;3(3c; - l,],

=

(3.52)

G;)ii:2U\,(3c$-l)],

(3.53)

(~“),p+61[ii:2ii:3(3c3-1W2~~2+~,)],

(3.54)

($)sp = (vT)rp_61[{C CYC W)sp

=

(d)sP = (v’),p+ C&P

=

12I[&$:,{ti:,(3C;-

1)+ii;3(3C,C2C3+

I,}],

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

(3.55) (3.56)

The T2,3 and (80Pq) are identical in the IP and the approximate SP calculations. In order to calculate expectation values with the Y,,, we must ammend the definitions of Z(x) [eq. (3.25)] and y [eq. (3.30)] as follows: I(X) = Y = E

s

d3ri2d3ri3{n

(.Ej+_$j_&j)-f

~f2}(f:23)2%

(3.57)

(vi_fZ3/_f?23)2*

(3.58)

CYC

1 CYC

414

J. Lomnitz-Adler, V. R. Pmdharipande / Triton wave function

The gradients of j&

defined as: +

(V&J

(3.59)

= $Vij+J$Pik

123

give additional contributions interactions. We obtain :

denoted by A(Cb) to the expectation

value of L..s

(3.60)

A@‘b),pr = SPT

The coefficient is 18 for the IPT wave function and 9 for the SPT. The other expectation values have the same expressions in SP and SPT or IP and IPT, only the definitions of Z(x) and y change.

4. Calculation

of density distribution

The proton and neutron density distributions are spherically symmetric functions of I in the c.m. frame. The distribution of particle i denoted by pi(r) is given by:

s

Pi(r)= d3rjkY*Y(rjk,Rj, = -+, and is normalized so that

ri

=

(4.1)

r),

s= s pi(r)d3r

The proton and neutron distributions

(4.2)

are given by:

p,(r) = 1 ”

1.

d3rjk’Y*$f1 Tt,,i)Y,

(4.3)

CYC

=

d

d3rjky*$l TtZ,Jg{n ~t(a)~t(r)I.

CYC

(4.4)

CYC

The z,,~ commute with 8, and so we obtain:

s

p,(r) = d3r23{nx/(a)Id,gtg{flXi(a)), P,(T) =

d3r,3{n s

X!Co)>d,g’g{n CYC

Xi(O)} +d3r12{n CYC

(4.5)

CYC

CYC

X!(a)Idgtg{II CYC

Xi(O))- (4.6) CYC

three integrals in eqs. (4.5) and (4.6) are generally not equal because the expectation values of ci * oj and CJ~ * Aaj * B depend upon i and j. Consequently p,(r) # $p,(r) and the two have to be calculated separately. We define a function

The

J. Lomnitz-Adler,

w,[X(rij,

rjk,

rki)] as:

s

W[x] = d3r2,{n f;:f}2x(r23,R23 = -ir, r1 and obtain :

415

V. R. Pandharipande / Triton wave fiction

=

(4.7)

4,

CYC

p,(r) = W,[1-6ii”,3+6(ii~,)z+9(ii”,3)2--6ii~~ii~~

+ 12(ii;,)2- 12i&,i?&; p,(r) = 2~,[1 -~II:~+

- l,],

(4.8)

12(ii~,)‘+3(U”,3)‘-6ii~2U”,3 +6(ti\,)* +6(ii\3)2-

12ii\,ii\3(3Ci- l)]*

(4.9)

The charge form factors given in the next section are calculated from the proton density distribution : (4.10) F(q) = fJm&)~

s

sin

F,(q)= p,(r)7

(qr)

d3r.

(4.11)

We assume that the proton form factor is given by: F,(q) = l/(1 +aq2)2,

a = 0.055 fm2.

(4.12)

5. Results We minimized the (E) with the three wave functions IP, SP and SPT separately and obtained the following optimum energies (in MeV): (T),,

= 43.96,

(V),,

= -49.97,

(E)

(T&

= 47.81,

(V),,

= -54.93,

(E&a = -7.14,


= 49.00,

(I’),,,

= - 56.20,

(E&*

= -6.01,

= - 7.20.

Fig. 1. The optimum correlation functions 3, 9 and I? in 3H.

J. Lomnitz-Adler, V. R. Pandharipande / Triton wave fwtction

416

The optimum 7, U”and U’in SP calculations are shown in fig. 1. These functions are not very different from the optimum functions in IP calculations. If we calculate (E) using an IP wave function, but with the y”, U” and U’ which optimize the SP energy, we obtain (T&

= 47.92,

(V);,

(E&

= -53.85,

= 5.94.

In SPT calculations we used the optimum SP 7, ii” and U’. In all of these calculations a step length of h = 0.03 fm was used to solve the differential equations, while the integrations were done with a step length of 6h. As a test of convergence we performed integrations with a step length of 4h, and obtained a difference of only 0.006 MeV in the energy. Considering that the errors in our integration go like h’, we can estimate that our energy has a numerical error of about +O.Ol MeV. We note that the errors in the kinetic and potential energies separately are larger, because they are both integrals of rapidly varying functions which add up to a smoothly varying function. This cancelation is ensured by the construction of the f’s from equations (2.24H2.26). The integrations were carried out to 10 fm. On extending them to 12.0 fm we pick up an extra 0.001 MeV. TABLE1 Variation of (I?‘& with “healing distance” d, the values of other parameters are as given in the appendix d

( n,,

( us,

(E)SP

3.0 3.25 3.50

48.65 47.81 48.02

- 55.11 -54.95 - 55.05

-7.12 -7.14 - 1.03

TABLE2 Variation of(E),,

with separation energy Es, the values of other parameters are as in the appendix

ES

( %P

1.25 7.15 8.0 8.25

47.813 49.485 50.299 51.099

( V)s, -

54.932 56.618 57.432 58.228

(E)SP -7.136 -7.152 -7.152 -7.148

TABLE3 Variation of (E&

with y (= width off,,), Y

0.0 0.1 0.2 0.3

the values of other parameters are as given in the appendix

(E),,, -7.12 - 7.20 -7.19 -7.15

J. Lomnitz-Adler, V. R. Pandharipande / Triton wave fiction

417

The variation of the energy with changes in the variational parameters is very smooth, as can be seen from tables 1-3. The separation energy E, for which (E)SP is minimum is not 7.25 MeV as used in most of our calculations, but around 8.0. It was thought that, since the dependence of(E) on E, is very weak, one may take E, - -(E).> The two parameters & and & which determine the “effective potential” r have their optimum values at /I1 = -0.5 fm-‘, and the corresponding

& = -3.5 fm-‘,

values for the di are

1, = -7.21 MeV.fm-‘,

1, = -0.72 MeV+ fm-‘.

The (E) is quite insensitive to these parameters; if they are both taken to be zero, a choice which corresponds to a linear r, we get I, = - 1.68 MeV . fm- ‘, (T&a

= 52.54,

(V),,

1, = +3.26 MeV. fm- ‘, (E)SP = - 7.00.

= - 59.54,

The (E& is calculated exactly, however there remains the question of how accurate are the results obtained with the SP and SPT wave functions. As stated in sect. 3, I

I

I

I

f (fm)

Fig. 2. The. SET and IPT p. and p, in “H are compared with (i) the results of Faddeev calculations *) and (ii) the p, of “He deduced from electron scattering 9).

418

J. Lomnitz-Adler, V. R. Pmdharipande / Triton wave function

we have only computed terms which are quadratic or less in ii’. Both ( Yy(Y),, and ( YlH( Y),, have cubic and quartic terms in U’which we have neglected. Their magnitude is difficult to estimate, but it should be less than half of the energy gained in going from the IP.to the SP. There appears to be a concensus among workers in the field 5, that the triton energy calculated with the Reid potential is 7.3 f 0.2 MeV. Our results with the SP and SPT wave functions fall into this range. In fig. 2 we plot our neutron and proton densities for 3H in the IPT and SPT cases, along with those obtained in the Faddeev calculations performed by Gignoux and Laverne 8), and the point neutron density deduced by Sick ‘) from electron scattering off 3He [refs. lo* “)I. (One assumes that point neutron density in 3H equals the point proton density in 3He). We can see that our SPT densities have higher shoulders than those obtained with the Faddeev calculations, and our proton density in SPT approximation has a dip of about 10 % magnitude at the center of mass. The IPT central density is much too large. We have not plotted the density corresponding to the SP wave function because the three-body correlations are so weak that the differences between psp and pspT are negligible. None of the calculations can explain the large anomalous dip in the neutron density deduced from experiments. This dip is related to the observed enhancement of the charge form factor of 3He after the first diffraction minimum. Assuming that the p, in 3H equals pp in 3He we have calculated the charge form factor squared of 3He with the SPT wave function. It is compared with the results of a variational calculation by Hennell and Delves I’), a Feddeev calculation by Gignoux and Laverne *), and

Fig. 3. The 3He form factor obtained from the SPT wave function is compared with those obtained with Faddeev *) and variational calculations “), and that deduced from electron scattering I”).

J. Lomnitz-Adler,

V. R. Panabipande

1 Triton wave function

419

experiment lo) in fig. 3. All the calculations are in essential agreement, and none can reproduce the observed enhancement of the form factor after the first diffraction minimum. It has been proposed recently by Giannini et al. 13),that the observed enhancement of F(q) can be attributed to the polarization of the nucleons by the electromagnetic fields. The mechanism that they propose requires the photon to cause a transition of the nucleon to the d-isobar state. However, such a process cannot occur in the case of the 4He nucleus, due to the isovector character of the transition operator. It thus cannot explain a similar enhancement of F(q) observed by electron scattering from 4He. One of the advantages of our method is that the three correlation functions p, TABLE 4

The correlation functions J”, tie, U’ r

7

P

U’

r

0.06 0.12 0.18 0.24

0.00899 0.00899 0.02346 0.05411

-0.15009 -0.15009 -0.11039 - 0.08296

0.02397 0.09578 0.11008 0.11679

1.74 1.80 1.86 1.92

0.59439 0.57782 0.56167 0.54598

0.01442 0.01313 0.01186 0.01064

0.09937 0.09684 0.09437 0.09195

0.30 0.36 0.42 0.48

0.10226 0.16657 0.24219 0.32508

-0.06163 - 0.04425 -0.02992 -0.01810

0.12211 0.12654 0.13019 0.13308

1.98 2.04 2.10 2.16

0.53078 0.51608 0.50188 0.48819

0.00947 0.00837 0.00732 0.00635

0.08960 0.08732 0.08510 0.08294

0.54 0.60 0.66 0.72

0.40748 0.48498 0.55397 0.61235

-0.00838 - 0.00043 0.00601 0.01117

0.13522 0.13664 0.13737 0.13746

2.22 2.28 2.34 2.40

0.47498 0.46225 0.44997 0.43814

0.00545 0.00463 0.00388 0.00321

0.08086 0.07883 0.07687 0.07498

0.78 0.84 0.90 0.96

0.65933 0.69508 0.72040 0.73649

0.01523 0.01836 0.02068 0.02232

0.13698 0.13601 0.13461 0.13286

2.46 2.52 2.58 2.64

0.42612 0.41570 0.40507 0.39479

0.00261 0.00209 0.00164 0.00125

0.07315 0.07138 0.06967 0.06802

1.02 1.08 1.14 1.20

0.74469 0.74635 0.14273 0.73496

0.02338 0.02394 0.02408 0.02388

0.13082 0.12856 0.12612 0.12356

2.70 2.16 2.82 2.88

0.38486 0.31526 0.36597 0.35698

0.00093 0.00067 0.00045 0.00029

0.06644 0.06491 0.06344 0.06203

1.26 1.32 1.38 1.44

0.72402 0.71072 0.69575 0.67966

0.02339 0.02266 0.02176 0.02070

0.12091 0.11821 0.11547 0.11273

2.94 3.00 3.06 3.12

0.34828 0.33985 0.33168 0.32376

0.00017 0.00009 0.00004 0.00001

0.06067 0.05936 0.05811 0.05692

1.50 1.56 1.62 1.68

0.66286 0.64571 0.62846 0.61131

0.01955 0.01831 0.01703 0.01513

0.10999 0.10728 0.10460 0.10196

3.18 3.24

0.31608 0.30864

O.OOOOO O.OOOOO

0.05577 0.05468

f

lP

lit

420

J. Lomnitz-Adler, V. R. Pandharipande / Triton wave function

ii” and U’,and the variational parameters fit,, and y completely specify the wave function. In the appendix we list the optimum values for the variational parameters which minimize (E&r, and tabulate the correlation functions: f’, ii” and U’. It appears from our calculation that Y,, is a good deal better than Y,,. To verify this, however, we should perform an exact calculation with an SP wave function which has been modified by an additional parameter lo

Y

SPQ

=

~

C

{Jj}(l+

(5.1)

(u~aij+i;fjtij+.ifijPoii:jii:.kOijk))~.

CYC

If SP is a good approximation value of /?o N 1.

the lowest energy (E) should be obtained at a

The authors wish to thank Dr. I. Sick for sending them the proton and neutron densities he obtained from Faddeev calculations done by Gignoux and Laverne. Appendix

In this appendix we present our optimal correlation functions and the corresponding variational parameters. The values for these parameters are d = 3.24 fm, PI = -0.5

E, = 7.25 MeV,

fm-‘,

/I& = 0.94,

& = -3.5 fm-‘, y = 0.1 fm-*.

& = 0.04,

The functions J‘“, U” and ii’ are tabulated in table 4 for I 5 d. For r > d one uses the asymptotic forms (2.18H2.20). The normalization is given by (Y(Y)

= 275.419(1+0.082 a;). References

1) V. R. Pandharipande

and R. B. Wiringa, Rev. Mod. Phys. 51 (1979) 821 R. B. Wiringa and V. R. Pandharipande, Nucl. Phys. A317 (1979) 1 J. C. Owen, Phys. Lett. 77B (1978) 9 L. D. Faddeev, JETP (Sov. Phys.) 39 (1960) 1459; 12 (1961) 1014 For a review, see A. C. Phillips, Rep. Prog. Phys. 40 (1977) 905 R. V. Reid, Ann. of Phys. 50 (1968) 411 V. R. Pandharipande, Phys. Rev. B18 (1978) 218 C. Gignoux and A. Laveme, Phys. Rev. Lett. 29 (1972) 436 1. Sick, Int. Conf. few body problems and nuclear forces, Graz 1978 J. S. McCarthy, 1. Sick and R. R. Whitney, Phys. Rev. Cl5 (1977) 1396 R. G. Arnold, B. T. Chertok, S. Rock, W. P. Schiitz, Z. M. Szalata, D. Day, J. S. McCarthy, F. Martin, B. A. Mecking, I. Sick and G. ‘ramas, Phys. Rev. Lett. 40 (1978) 1429 12) M. A. Hennell and L. M. Delves, Nucl. Phys. A246 (1975) 490 13) M. M. Giannini, D. Drecbel, H. ArenhBvel and V. Tomow, Preprint (1979) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11)