On the symmetric S- and D-state components of the triton wave function

I I

1.B

I I

Nuclear Physics AI05 (1967) 649--664; (~) North-Holland eublishinq Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

O N T H E S Y M M E T R I C S- A N D D - S T A T E C O M P O N E N T S OF THE TRITON WAVE FUNCTION M A L C O L M M c M I L L A N a n d M E L V Y N BEST t

Department of Physics, University of British Columbia, Vancouver 8, B.C. Received 13 June 1967

Abstract: Approximate forms for the symmetric S- and D-state components of the triton wave rune-

t/on are found using the Feshbach-Rubinow equivalent two-body method. Two coupled ordinary differential equations for thc components are obtained, and are solved numerically with the Feshbach-Pease potentials. A further approximation involving one variational parameter is shown to yield good results. Appendices contain the Feshbach-Pcasc variational wave function written in Derrick-Blatt and Sachs notations, and the detailed relationships between the DerrickBlatt and Sachs basis functions.

I. Introduction

This p a p e r is the sequel to one dealing with the symmetric S-state c o m p o n e n t o f the triton wave function 1), a n d represents the second step in a calculation o f a simple a p p r o x i m a t e triton wave function which m a y be used to analyse triton form factor a n d p h o t o d i s i n t e g r a t i o n data, a n d which does not wholly rely u p o n variational p a r a meters. T o w a r d s this end, we e m p l o y the F e s h b a c h - R u b i n o w 2) ( F R ) equivalent two-body method. The F R a p p r o a c h has the a d v a n t a g e over the variational p a r a m e t e r a p p r o a c h o f allowing a calculation o f the functional form o f the a p p r o x i m a t e wave function. Indeed, a c o n t i n u a t i o n o f the F R p r o c e d u r e w o u l d lead to a set o f coupled secondo r d e r o r d i n a r y differential equations for the wave function c o m p o n e n t s . Clearly, however, when one considers the n u m b e r o f c o m p o n e n t s o f the triton wave function, this a l l o w a n c e is in practice t o o generous, and one is led to seek further meaningful a p p r o x i m a t i o n s . F o r example, one m a y o m i t certain c o m p o n e n t s , or m o d i f y the F R p r o c e d u r e to allow the i n t r o d u c t i o n o f variational parameters. Here we shall do both. It is well k n o w n t h a t the symmetric S-state is the d o m i n a n t c o m p o n e n t o f the triton wave function, a n d that the D-states r a n k next*t. R e g a r d i n g the D-states themselves, one expects that the D e r r i c k - B l a t t 5) ( D B ) kinetic energy a r g u m e n t m a y be used to conjecture t h a t the symmetric D - s t a t e c o m p o n e n t be the most i m p o r t a n t , followed b y t Present address: D e p a r t m e n t o f Physics, M a s s a c h u s e t t s Institute o f Technology, Cambridge,

Mass. tt Calculations by Blatt et al. ~) indicate that the combined probabilities of the remaining states is less than one percent, and that good results for the triton energy can be obtained when they are neglected. The Feshbach-Pcase 4) (FP) calculation is an outstanding example of this. 649

650

M. MCMII_I_ANAND M. BESI

the mixed symmetric components, then the antisymmetric component*. As shown in appendix l, the FP results also indicate this ordering. In this paper, as a first step towards the treatment of the D-states by the FR procedure, we retain only the symmetric D-state along with the symmetric S-state. Sect. 2 contains a deriwttion and discussion of the two coupled ordinary differential equations for the symmetric S- and D-state components which result when the FR approximation is made to both components. Expressions for the D-state probability and for the Coulomb energy of 3He are also given. Sect. 3 contains an equation which results when the FR procedure is modified to include a single variational parameter; sect. 4, the results of the numerical solution for the FP potentials; and sect. 5, our conclusions. Appendices contain the FP variational wave function written in DB s, 6, 7) and Sachs 8) notations, and the detailed relationships between the DB and Sachs basis functions.

2. The coupled Feshbach-Rubinow equations Derrick 6, 7) has given the set of coupled partial differential equations for the internal functions in the DB expansion of the J = T = J three-nucleon wave function. In the FR approach, however, the dynamical statement is a variational principle from which differential equations are obtained using the Euler-Lagrange equations. In the problem at hand where only the symmetric S- and D-state components (i.e., components 1 and 13 in DB notation) of the triton wave function are retained, the relevant variational principle ** is

6fdzL d

= 0,

(2.1)

where

dr23

dz = ~

L = ~

{

h2

----

VI-~fl ]2+ c o s -

drlzr23 r31 rlz ,

dr3a

~0

(2.2)

~ Jr23-rzll

L~fl {~fl + N 5 i{Gfl3~2+ c o s

03 . . . . . . . . . .

(~f13Cifl31]

03-

-- ½[ V~( r, 2) + I/'C'(rt 2)]f~ -[N5 ~ 2~F~

V:t(riz)+(FZ+G2)(2 • 3-½a cosec ;.+-~-- cosec 2 2 - - c o t 2 2)V'(ri2)]f:3 COSgC

2-3-÷)+Gb cot 2]Vt(rl2)flfl3 } +E(f(+

N5/123),

(2.3)

where throughout we have adhered to the DB notation. We have in addition written + We follow the Derrick e) notation for the D-states. • * That this is a suitable variational principle may be checked by noting that the corresponding Euler-Lagrange equations are the dynamical equations given by Derrick ~, 7).

651

T R I T O N ~X/AVE F U N C T I O N

the two-nucleon potential as t

V,2 = ¼ ( 1 - a , • ~2)V"(r,.,)+1(3+a , • az)V':t(r,2)+S,z Vt(r,z ).

(2.4)

The functions f~ and f~ 3 are both completely symmetric functions of the three interparticle distances *t and are normalized according to

f

dr(f~ z + N5/2a) = 1.

(2.5)

We now assume thatJ'~ andf~3 depend only on the single symmetric variable

s=

½(r23 + r3, + r, 2),

(2.6)

i.e., we assume that ,,t f l = f~ (s),

(2.7a)

f~ 3 = ,1~3 (s).

(2.7b)

Eqs. (2.7) are generalizations of the dominant parts of the FP trial function ~ which has been shown 3) to give quite accurate results for the triton energy. When approximations (2.7) are made, some of the integrals in eq. (2.1) may be easily evaluated to yield

d I=ds { h= [(du] 2- 5u (du) .o

-

Z L\dsl

-s

ds

25 u2 + 4

(dwl2 - 9w (dw I

-~ + \ d s l

81 w2l

S- \~tsl + -4 s2_]

- w+(s)u 2 - [w°'(s)- -r'(s)]w 2 - 2,,r'(s),,w + ~ E ( , , ~ + ~2)} = 0,

(2.8)

where u and w are defined by

fl(s) = V 7~-- u(s);_,

(2.9a)

S-_~.

9 f13(s) -- -7_ 5 W(s)

(2.9b)

s~-,

so that normalization integral (2.5) becomes

() ()] fo•[

u 2 s + w 2 s ds = 1,

(2.1o)

The FP potentials are o f this form. t÷ The functions f l and f13 are proportional to the Sachs functions f~ and f7 respectively, as may be seen using the results in appendix 2. **÷ Eq. (2.7a) is the original F R approximation. ~t See appendix 1.

Xt. McMILLAN AND M. BEST

652

and where the potential terms are ~/'+(s) = 2 4 f ] d z z Z ( l - z + ~ z 2 ) { } [ V ~ ( s z ) +

V~t(sz)]},

~v'¢t(s) = -2-016f ldzz2(l - 3z + a~ z z _ ~O_z3+ 4z 4 _ 6zS + ~_3ffz6) VCt(sz) ' 55 Jo

~K"(s) = 6 7 2 f ' d z z 4 ( 1 - 2 z 53o

+xs~-597"2--]-8"5 Z212

3--..i_ l 5 4 z 4 ) V t ( s z ) '

3 6V ' F

~/~"(s) = - ~ - , ~ 3 0dzz'(l - z + lzZ) V'(sz).

(2.11) (2.12)

(2.13)

(2.14)

The Euler-Lagrange equations which follow from variational principle (2.8) are h 2 {dZu m \d~

15u~ 4s2] +"Y+(s)u+'~t(s)w

14,= TsCU,

h 2 {d2w 63w] - m-- ~dj2 - ~ - / + [V°'( s ) - ¢"(s)]w + ¢'(~)u = 154-e'~'

(2.15a) (2.15b)

Thus, use of assumption (2.7) reduces the problem of finding the symmetric S- and D-state components of the triton wave function to the solving of two coupled ordinary differential equations. We call eqs. (2.15) the "coupled FR equations". It is worth stressing that the employment of variational parameters has not been resorted to, rather the triton energy appears as the eigenvalue of the coupled FR equations. The correspondence between the coupled FR equations and the well-known coupled pair of equations for the deuteron as 1 and 3D~ radial functions is apparent. In particular, one sees that the triton S and D components are also coupled by the tensor force (as of course is wellknown), and that the first potential term in eq. (2.15b) involves the difference between central-triplet and tensor potentials. On the other hand, the first potential term in eq. (2.15a) contains a singlet spin potential (which of course does not appear in the corresponding deuteron problem), and also the "centrifugal barrier terms" involve (~-+ l)(} + l + 1) rather than l(l+ 1). We close this section by giving expressions for the D-state probability (I'D), and for the Coulomb energy (Eco,0 of 3He. It follows from normalization integral (2.10) that

eo =

;7

w2(s) ds,

(2.16)

and from (6) and (7) that

.

LY12

FI2A

TRITON

WAVE

653

FU'NCTION

which, when approximation (2.7) and definitions (2.9) are used, becomes 25e 2

oo _

__

243 (,oo wZ(s)dsl

"

(2.17)

3. The modified Feshbach-Rubinow equation The eigenvalue of the coupled F R equations (2.15) will, because the equations follow from variational principle (2.8) without approximation, be better (i.e. lower) than a triton energy computed in any other way with f l and.f~3 of the formgiven by eqs. (2.7). Nevertheless there is some value in considering further approximations at this point. Indeed, if one were to take into account the remaining components of the triton wave function and further employ the FR single symmetric * variable approximation, one would arrive at a set of sixteen coupled ordinary differential eigenvalue equations, the solution of which would be difficult. We now give a modification of the procedure used in the preceding section which leads to a considerable simplification and which yields good results as we show in the next section. We assume here that the functions u and w defined by eqs. (2.9) are related as follows

(3. I)

w(s) = ~.u(s),

where :~ is a parameter. When approximation (3.1) is used, variational principle (2.8) contains a single function to be determined, and the Euler-Lagrange equation reads --

h2[ --

m

(]..1_

2" d2u 15+63~ z u :()ds z 4 "~

1

+ [Y/'+(s) + 2=Y;'t(s) + ct2(Y/'~'(s)- Y/'t(s))]u = I--}E(1 +

otZ)u.

(3.2)

Thus, use of assumption (3.1) reduces the problem of finding the symmetric S- and D-state components to the solving of a single ordinary differential equation. We call eq. (3.2) the "modified FR equation" *t. The eigenfunction u and eigenvalue E depend upon the parameter ~; the best triton energy possible with approximation (3.1) is, of course, the minimum E with respect to :c The normalization integral, D-state probability and the Coulomb energy of aIle in this case follow immediately from eqs. (2.10), (2.16), (2.17) and approximation * The sixtecn DB internal functions may be expressed in terms of sixteen symmetric functions and, as ill eq. (A.7), explicit functions of the internal variables which determine the required transformation properties under the operations of the symmetric group of degree three. In this connection, as is wellknown, the Sachs 8) wave function follows from an incomplete treatment of mixed symmetric and antisymmctric internal functions and as a result contains only eight symmetric internal functions. tt Setting ~ = 0 yields the original FR equation 2).

654

M, McMILLAN AND M. BEST

(3.1). O n e has

fo

u2(s)ds

(3.3)

1 +12 '

-

52

PD -= Ec°ul

25e2 ( 1 + 14-

(3.4)

I ..}_~ 2 '

243cz2~ fo~ u2(s)ds 275-/Jo s

4. N u m e r i c a l

(3.5)

results

In o r d e r to p r o v i d e a c o m p a r i s o n with existing results i n v o l v i n g t e n s o r forces, t h e e i g e n f u n c t i o n s a n d e i g e n v a l u e s o f the c o u p l e d a n d m o d i f i e d F R eqs. (2.15) a n d (3.2)

']'ABLE l

Comparison of results Potential

E(MeV) symmetric Sstate only

FP FP FP FP

No. No. g ~ No.

1 2 0 3

PD(~) S-- D FP

exponential function

FR

FP

-5.45 -3.27 --2.01

--5.89 --3.77 -2.47 --0.92

--10.05 --9.06 --8.40 .-7.50

Ecoul(MeV)

coupled modified FR FR

FP

coupled modifie FR FR

coupled modified FR FR

--8.10 --6.29 -5.37 --3.26

--8.03 --6.24 --5.32 --3.22

2.2 2.8 3.1 3.6

1.9 2.1 2.5 2.3

1.6 1.9 2.4 2.4

1.088 1.059 1.038 1.008

1.061 0.997 0.961 0.861

1.059 0.995 0.960 0.861

The first column designates the FP potential; the columns headed " F P " contain the final FP results; those heade "coupled FR", the results of solving (2.15); those headed "modified FR", the results of solving (3.2) with that valt of the parameter :~ giving the lowest value of E. The second column contains the results of using the best trial e: ponential function; and the third, the results of solving the FR equation. (The experimental results for the trite energy and the Coulomb energy are E = --8.492 MeV and Ecoul -- 0.764 MeV, respectively.)

h a v e b e e n f o u n d n u m e r i c a l l y for the F P p o t e n t i a l s t. T h e results are s u m m a r i z e d in t a b l e 1 a n d figs. 1 to 4. T a b l e 1 c o n t a i n s the e i g e n v a l u e s t* o f t h e c o u p l e d a n d m o d i f i e d F R e q u a t i o n s , a n d the D - s t a t e p r o b a b i l i t y a n d t h e C o u l o m b e n e r g y o f 3He as g i v e n by eqs. (2.16), * Details of the numerical work, which was done on the U.B.C. IBM 7040, are contained in ref. 9). tt Note that in all cases the eigenvalue ofthecoupled FR equations is lower than that of the modified FR equation, which is consistent with the first sentence of section 3.

TRITON WAVE FUNCTION

655

(2.17), (3.4)and (3.5). The final FP results * and the triton energy calculated with the best trial exponential function *t are also included. The eigenfunctions of the coupled FR eqs. (2.15) are plotted in figs. 1 to 4. The similarity between the eigenvalue of the FR equation and the energy calculated with the best exponential function t**, and between the eigenvalues of the coupled and modified FR equations has led us to calculate various overlap integrals. We define I, =

\/7J

d z e - ~ y l ( s ) = ~/Tsx 3

s~e-~Uo(S)ds,

(4.1)

lz=jo~(s)u(s)ds/~:uZ(s)ds,

(4.2)

['3 ~ f:W(S)W(s)ds// f:W2(S) dS,

(4.3)

t

TABLE 2 Overlap integrals Potential FP FP FP FP

No. No. g ~ No.

1 2 0 3

I~

Is

13

0.81 0.83 0.85 0.86

0.97 0.96 0.95 0.93

0.89 0.91 0.93 0.95

See eqs. (4.1), (4.2) and (4.3).

where u and w are the solutions of the coupled FR eqs. (2.15), ~ is the solution of the modified F R e q . (3.2) with that value of ~ giving the lowest value of E, ~ is the corresponding function given by eq. (3.1) and u o is the solution of the modified FR equation when 2 = 0. The solutions of the coupled and modified FR equations are normalized according to eqs. (2.10) and (3.3). Finally, the best trial exponential function ~ appears in eq. (4.1). The numerical values of these overlap integrals are contained in table 2; their nearness to unity provides a further measure of the relative merits of the approximate functions ~. Thus one sees that the difference is rather large between the trial exponential function and the solution of the ~ = 0 FR eq. (3.2), See table 2 o f ref..I). The FP No. 1, FP No. 2 and FP No. 3 potential parameters a n d results are given in the first, second and fifth lines, respectively. T h e FP g ~ 0 potential parameters and results have been obtained by interpolation. This latter potential contains only four parameters and gives a reasonable fit to the low-energy experimental data. t ' See fig. 2.1, chap. V o f ref. ,o). *.+* See also rcf. z). T h e c o r r e s p o n d i n g value o f • m a y be obtained from chap. V o f ref. 1o). *%* We have not attempted to c o m p e n s a t e for the fact that the c o m p a r e d functions correspond to slightly different triton energies.

o.7

I

I

I

I

1

I

I

FP

0.6

I

I

I

POTENTIAL

E = -8.10

MeV

0.5~

k.--. 0"4u IF.

"-'o3o

0.2{

0.1(

0

I

2

3

4

5

6

7

8

9

IO

s (fm) Fig. 1. T h e eigenfunctions u(s) a n d 0.70

l

I

I

:

w(s) versus s for the FP N o . 1 potential. 1

I

I

FP 2

0.6(

I

I

POTENTIAL.

E = --6-29

MeV

0.5(

0.4( _l,..,j I E 0.3I

V/ 0

w(s)

I

2

I

I

I

I

I

I

3

4

5

6

7

8

s (fm)

Fig. 2. T h e eigenfunctions

u(s) and w(s) versus s for the FP N o . 2 potential.

9

I0

0.70[

I

I

I

I

I

I

I

i

I

FP g =0 POTENTIAL E = -- 5..57 Me"/

0601--

0.50

/

O.4O -IN

J~ 0.50

0.20

0.10

0

I

2

3

4

5

6

7

8

9

I0

(fro) Fig. 3. The eigenfunctions u(s) and w(s) versus s for the FP g ~ 0 potential.

°-7°I

I

I

I

I

I

I

I

I FP

i 3

I

POTErlTIAL

E =-3.26 Mev

0.601 -0.50~--

°4°I-

/

:21

\

/

o.2ol-

0

\

t

-I

~

I

2

3

4

5

6

7

0

9

s (frn) Fig. 4. The eigenfunctions

u(s) and w(s) versus s for the FP No. 3 potential.

tO

658

M. McM1LLAN

0.70

I

I

i

AND

I

M, BEST

~

[

i

i

1

FP I POTENTIAL 0"60L

I1= 0.81

5/~ - K ~

/

_

\

°~°F #

X~

;o~oL //

\~

o.,o

0.20~--

'

I

0

I

I

I

2

I

3

!

4

5

6

7

8

9

I0

s (fro) Fig. 5. T h e e i g e n f t m c t i o n

0.08

i

uo(s)

I

and the best t r i a l e x p o n e n t i a l f u n c t i o n versus s f o r the FP N o . ! potential.

I

1

1

I

FP

0.07

i

I

I

POTENTIAL

0.06

Q05

0.04 E 0.03

O.02

O.01

d/

0

I I

I 2

I 3

] 4

I 5

[ 6

I 7

I 8

I 9

I0

s (fro)

Fig. 6. The eigenfunctions

~(s) and w(s) vcrsus s for the FP No. 1 potential.

659

TRITON "~VAVE FUNCTION

somewhat less between the D-state functions resulting from the modified and coupled FR equations, and still less between the S-state functions resulting from these two equations. Figs. 5 and 6 show the functions whose overlaps 11 and ! 3 appear in the first line of table 2. The functions u and fi whose overlap 12 appears in table 2 are, on the other hand, sufficiently similar to eliminate the need for a separate figure. Another comparison of approximate functions is provided by the test suggested by T a n g et al. 11), the calculation of Hsfl/fl where 7) Hs = ( ~ i , H.~¢t) • If one set I

I

L

1

I

6.0

I

I

FP g =0

1

i

I

POTENTIAL

2.0

-2.0~ HS fl ft _ 6.0V {MeV),

\ FESHBACH-RUBINOW

L

-.0.0~ EXPONENTIAL

/

-i4.01~ -18.0

o

I 3

I 4

I 5

I 6

I 7

I 8

1 9

I I0

I II

I 12

s (fml Fig. 7. The function Hsf~/f ~versus s for the FP .q = 0 potential with the FR and best trial exponential functions.

equal to zero the tensor part of the FP 9 = 0 potential and had the resulting exact /'1, then Hsf~/fl would, for all values of the interparticle distances, be equal to a constant *. Thus deviations of Hsft/ft from a constant value for an approximate Ji should indicate the fidelity of the approximation. Fig. 7 shows llsf~/fl for the FP 9 = 0 potential with f l determined from eq. (2.9a) and u 0, and also equal to the best trial function ,t. The meaning of this figure is as follows: Hsfl/fl may be regarded as a function of s, z = r23/s and z' = rails, for any s the allowed region in the zz' t This would not be true for the other FP potentials because of the presence of the (small) mixed symmetric S-state component. *+ The function Hsfa/]~ was calculated for these cases by David Maroun, whose help we gratefully acknowledge.

660

M. M c M I L L A N A N D M. BEST

plane being a fixed isosceles triangle. The shaded bands in fig. 7 contain for each s the values of Hsfl/j] for points in this triangle where r12r23r31/s 3 = (2-z-z')zz' > 70 % of its maximum value (which occurs at the centroid). This includes about 80 % of the area of the triangle, regions near the vertices and corresponding to the smallest interparticle distances being omitted. (The selection of this bound is o f course quite arbitrary, with larger areas yielding wider bands.) The horizontal lines in the figure denote the values of the triton energy found using the corresponding approximate functions. One sees that the solution of the FR equation seems to be the more accurate, and that the trial exponential function seems, in particular, because of the large negative values of H s f l / f l , to fall off too rapidly for large s. 5. Conclusions

We have in this paper extended the FR equivalent two-body method by including the symmetric D-state component of the triton wave function along with the dominant symmetric S-state component. The triton energy and wave function have been determined from the solution of eigenvaIue equations. Table 1 shows, however, that for the FP potentials our calculated triton binding energies are smaller than those obtained by FP using the Ritz procedure. We attribute this mainly to our inclusion of only the symmetric D-state component along with the symmetric S-state component. (We consider assumptions (2.7) to be of lesser importance in this regard since the dominant parts of the corresponding FR functions are of this lbrm.) Our calculated D-state probabilities, when compared with the corresponding FP quantities, further bear out the assumption that the symmetric D-state be the dominant D-state component, but it is clearly necessary to include the remaining D-states for very accurate results. The general similarity between the results obtained from the coupled and modified FR equations indicates that approximation (3.1) is very accurate. Indeed, it leads to far less error than the replacement of the solution of the FR equation with the best trial exponential function, since both the computed energies and functions are similar. One is thus encouraged to extend the present work by taking the remaining D-state components into account using approximations analogous to eqs. (2.7) and (3.1). Work along these lines, and on the calculation of triton form factors and photodisintegration cross sections, is in progress. We wish to thank Alvin Fowler of the UBC Computing Center and Raymond Vickson for help during the initial stages of the numerical work, and Dr. G. Derrick for a letter regarding the contents of appendix 2. One of us (M.B.) also thanks the National Research Council of Canada for a Bursary.

TRITON WAVE FUNCTION

661

Appendix 1 T H E FESHBACH-PEASE T R I T O N WAVE F U N C T I O N

In view of its success 3), the FP triton wave function is here written in the DB and Sachs notations, which are now in c o m m o n usage. The FP function is an expansion including a 2S~ function and three #D÷ functions denoted by 7.s, ZD, XD', ZD" and related to the Sachs basis functions 9 m, 4Ds, 4 D , 4 D as follows * Zs = (P, ZD

=

Zo' = ½[4O~-'N,"3(4b)l,

ZD. . . .

.~[~ RZ(4Ds) + F('*~) + G(4D)],

(A. 1)

1 as in appendix 2, (.o = (p'=5 and 4/~ = __6.\/3 (4D); and where R 2, F, G are defined by Derrick 6,7). The final triton wave function given by FP eq. (8) may be written

where,

~1 = Ao(s)q) + [A l ( S ) - 2fA3(s) + 29GAs(s)](4D) - 2 O[A z(s)- 2 f A g ( s ) - FAs(s)](4D) +

}g,]3Ea2(s)- 2fA 4(s) +-~x,:SR 2A 5(s)](4Ds),

(A.2)

where

Ao(s ) = As, e - 2,s + Ax(s ) =

3

As 2 e -

Z2s,

2

•2-[-AD + 3(AD+ + 2ADo)S]e -"s,

A : ( s ) = - ¼\/3lAD, + ~-(AD,+ + 2AD,o)s]e- v,, 1 /A3(s ) = zV 3(AD+ -- ADo)e T M ,

=

As(s ) =

-~(Ao,o- AD,+ )C ~" , 3 A

g,,D,,e

-o~s

(A.3)

in which the FP variational parameters appear, and where S = -~-(r23 + I"31 +

rl 2),

f = ½~3(r23+r31--2r12), g = rz3--r31.

(A.4)

Using the values of the variational parameters given in table 3 of ref. 4), one sees that for all FP potentials the six functions Ai(s) are non-negative (with the unim* In arriving at eqs. (A.1) we have used the equations R ~- = ½(p2--3r~-), F =- ~'v"3 (p'-'--3r2), G ~: - - r - p, where r and 0 are defined by Sachs, and the identity p2A(r, r)+r'-'A(p, p ) - - 2 ( r , p)A(r, p)'-A ( r × p, r × p) "-- 0, where A(a, b) ~ (~lz ' a)(tr3 " b ) I - ( g ~ "~a)(a~2" b)----](a • b)(e~2 • ~3), which identity incidentally also explicitly demonstrates the well-known interdependence of the four GerjuoySchwinger ~-) D-states.

662

M. Me,MILLAN AND M. BESI"

portant exception of Aj and A 2 for large s), and that the dominant functions (in the sense of attaining the largest values and having the largest areas when 1 fm is set equal to unity) are A 0 and A 1 with A o > A~. The main part of the FP Step 1 trial function, which is a good first approximation, is of form (A.2) with A 3 = A 4 = A s = 0 and with the second terms of A o, A~, A z omitted. The function ~ is antisymmetric under the interchange of particles 1 and 2 (the two neutrons); it has been written in the above form to make more transparent the transformation properties of its constituents under the operations of the symmetric group of degree three* In order to write the FP function in DB notation, expression (A.2) has been augmented with the isospin formalism and the relationships of appendix 2 used. Finally, the FP approximation corresponds to writing the triton wave function as

7s =f~ @'a-f, ,,2ffJl ~,, + f , , , , 7Y,,, z +f,2 c~,2 +f~ 3 ~,3 __f14,2@'14,1_bf14,10¢14,2__flS. 27~lS, l q_f15, 1 O?/ ~15.2,

(A.5)

where ~ t , . . . , ~ 4 , 2 are the DB 7" = ½ S- and D-state basis functions, where the mixed pair of T = ~ D-state basis functions (¢Yls, 1, ~?¢/a5,2) are here defined by **

\.%~, ~i

= \'D]

(A.6)

v3,

where v3 is a T = ~ isospin function 5), and where the internal functions are fl

~\/3

'

= Ao(s ) = -

(fo) [Az(s)+ ~']3R=As(s)]+ \

2f9

] A4(s)'

I 2

f~ = (fG-gF)As(s), f13 = Ai(s)+(fF+gG)As(s),

(A.7)

f l 4-, 2

- ½,,/~ i f , ~, + T h e quantities R 'z, s, 4D, are each completely symmetric; the pairs (F, G), (f, 9), @, q), (4/3, ~D) are each mixed symmetric. t~ O n e m a y show that ?//'15, where ,;. is given by Derrick ~), a n d where thc o r t h o n o r m a l T = ~ D-state basis functions ~gs, ~9, '~10 are obtained from the original DB T ~ ~ D-state basis functions ~s,~, ~g~,2,~u,1 by replacing in the latter v.~ by r':~.

TRIION WAVE FUNCTION

663

The function 7j given by eq. (A.5) is completely antisymmetric; that it is the correct extension of ~k given by eq. (A.2) may be checked by noting that is, as required, proportional to ip. One sees from eqs. (A.5) and (A.7) that the FP approximation involves keeping the DB symmetric S-state component and all of their D-states. In addition to these T = I components, a T = ~ D-state component is included. In all cases, the variational parameters are contained in the completely symmetric functions A i(s); the remaining factors in the internal functions (e.g., the completely antisymmetric factorfG-gF in f~2) are simple functions of the interparticle distances which determine the required transformation properties of the internal functions under the operations of the symmetric group of degree three. A comparison of the functions Ai(s) should indicate the relative importance of the various components. The statement following cq. (A.4) implies that the symmetric S- and D-state components (states 1 and 13) are the dominant ones; further comparison of the functions A~(s) implies that the antisymmetric D-state component (state number 12) and the T = ~ mixed symmetric D-state component (state number 15) are the least important. This ordering of the T = ½ D-states is consistent with the DB kinetic energ2~ argument 5).

Appendix 2 T H E R E L A T I O N S H I P S B E T W E E N T H E D E R R I C K - B L A T T A N D SACHS BASIS F U N C T I O N S

The equations relating the basis functions given by DB and Sachs are (pF/- ~ t / = (pr/+ q30 = ~0~ "}- 0 ~ ]

--

4n~J,, 4nJW2, \@/3, 2

'

(2p)~l+(2p)# = 16nA~.¢,,,

(2p)O_(2p),~ = _

{(2p)q_(2p)q})= (4P) ( q O ) =

16,~Aq¢5,

16hA \~¢6'.2]' --16,f3nA{~¢7,'~ \ ~--/7,2] ' V-"-G i, 2! '

( 4 D ) r / + (4/~)q =

q_ 4n,7/, 2,

(4D)O-('tD)q = -4n@'~3 , (aD)O +(,,D)rl] = - 4 n \ ~14, 2] ' where in the Sachs functions* we have taken m = - t = ', then omitted these

664

M. McMILLANAND M. BEST

superscripts and used the notation .4 = - ~ x / 3 , 4 . The symbol A denotes the area of the triangle formed by the three particles. The main features of the S- and P-state correspondence equations follow directly from simple symmetry arguments. For example, the Sachs basis function c¢~-q3r/ is an S-state which is antisymmetric under joint permutation of spin and isospin variables, and ~1 is the only DB basis function with these properties. Thus, the two basis functions are proportional, the proportionality factor being a symmetric scalar function of the spatial variables. Neither basis function depends upon spatial variables, however, so the proportionality factor must be a number, the magnitude of which can be determined by comparing the normalizations of the states. Similar arguments may be made for the remaining S-states and the P-states ,t The main features of the D-state correspondence equations, on the contrary, cannot be obtained by the above symmetry arguments since the above D-states do not have definite permutation symmetry under joint permutations of spin and isospin variables. These arguments imply only that each Sachs basis D-state is a linear combination of all DB D-states, the expansion coefficients being functions of the internal variables and having the appropriate symmetry. On the other hand, the spatial dyadics which are relevant to each of the Derrick 6) functions are essentially identical to the spatial dyadics out of which each of the Sachs D-states may be formed. This accounts for the simple relationship between the two sets of D-states. A detailed examination of the basis functions has also been made: the operations implied in all Sachs functions have been carried out, the space-fixed components of r and p which then arise have been expressed in the DB body-fixed coordinate system (which step introduces Euler angles and rotation matrices), and the results have been compared with the DB basis functions. The procedure is quite straightforward and need not be repeated here. * In addition to the basis functions involved in Sachs' eqs. (8.17) to (8.27), the above equations contain the three basis functions whose (antisymmetric) internal function coefficients Sachs took to be zero. See also table 1 of ref. ~a). *+ The factor A in all P-state correspondence equations arises because the Sachs functions contain internal variables only through the quantity r × p.

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10)

M. McMillan, Can. J. Phys. 43 (1965) 463 H. Feshbach and S. 1. Rubinow, Phys. Rev. 98 (1955) 188 J. M. Blatt, G. H. Derrick and J. N. Lyness, Phys. Rex'. Lctt. 8 (1962) 323 H. Feshbach and R. L. Pease, Phys. Rev. 88 (1952) 945 G. H. Derrick and J. M. Blatt, Nuclear Physics 8 (1958) 310 G. H. Derrick, Nuclcar Physics 18 (1960) 303 G. H. Derrick, Nuclear Physics 16 (1960) 405 R. G. Sachs, Nuclear thcory (Addison-\Vesley Publishers Co., Inc. Cambridge, Mass. 1953) M. Best, M. Sc. Thesis (1966), University of British Columbia J. M. Blatt and V. F. Weisskopf, Theoretical nuclear physics (John Wiley and Sons, Inc., New York, 1952) 11) Y. C. Tang, E. W. Sciamid and R. C. ttcrndon, Nuclear Physics 65 (1965) 203 12) E. Gerjnoy and J. Schwinger, Phys. Rev. 61 (1942) 138 13) L. Cohen and J. B. Willis, Nuclear Physics 32 (1962) 114