The bound states of 3H and 3He with the Reid potential

Nuclear Physics A246 (1975) 490--504; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

THE BOUND STATES OF 3H AND SHe WITH THE REID POTENTIAL M. A. H E N N E L L and L. M. DELVES

Department of Computational and Statistical Science, University of Liverpool, Liverpool, UK Received 8 March 1975 Abstract: A calculation o f the bound state parameters o f 3H and SHe with the soft-core Reid potential is described, and a comparison is given with the results o f other authors for this system.

1. Introduction

Two previous publications 1, 2) (hereafter called I and II) described the details of a series of variational calculations for the three-nucleon states below the breakup threshold for the Hamada-Johnston (HJ) and Gammel-Brueckner (GB) potentials. In this paper the results of similar calculations for the Reid 5) soft-core nucleonnucleon potential are reported. Preliminary accounts of this work can be found in refs. 3, 4). As in II only those features of the calculations which differ from those of I will be discussed, and the reader interested in the details of the calculation should read this paper in conjunction with I. The Reid soft-core potential is interesting not merely because it represents an alternative quality fit to the two-nucleon data but also because unlike the earlier HJ and GB potentials it does not incorporate a hard core. For this reason, it might be expected that its predictions for three-nucleon parameters such as the Coulomb energy and the electron scattering form factors of 3He could differ appreciably from those of the hard-core potentials. 2. The potential In ref. 6) a conversion is described whereby the Reid nucleon-nucleon potential is converted to a form similar to that used by Hamada and Johnston 7), namely

s(s+I)U~ ----s(s+1)U~(r)+ Sl2S(s+I)U;(r)_I_(L.S)S(S+1)U~s(r)+ gt2s(s+I)U~L(r)" This conversion is necessary for us since the suite of programs for the three-nucleon problem was written for potentials of the HJ type which have contributions in all two-body states. Reid potentials exist only for two-body states with J = 2. The converted potential used here reproduces exactly the Reid potentials for J _= 2 490

B O U N D S T A T E S O F 3H A N D 3He

491

(except in the 3P o state), but also includes contributions for the states with J > 2. The results quoted here will therefore differ from predictions made with the Reid potential by the contribution of these high-J states. It is expected however that this contribution will be small, differences arising from the different 3Po potential given by this conversion are expected to be extremely small, as it is known (see below) that the odd-parity potentials contribute very little to the three-body systems in the J~ = ½+ state. Throughout this paper it will be implicit that references to the Reid potential will imply the modified soft-core Reid potential for this calculation and the original potential when reference is made to work of other authors. 3. The trial function In I and II the three-body trial functions were constructed to be zero within and at the hard-core radius. Immediately outside the hard-core radius the trial functions rose rapidly with an initial slope of the order of 4.0 fro- 1. The Reid potential however has a large repulsion at small distances where the wave functions will be non-zero but small. The wave function will then rise slowly as the repulsive core gets softer until a knee is reached where the repulsion changes to attraction after which it will rise rapidly. In constructing a trial function with features of this type it was considered desirable to minimise coding alterations to the hard-core programme. As in I and II the trial function was split into three parts, • '~(trial) = P(channcl) + P(core) + PQ, where ~'e is a complete set of functions and Q is the expansion parameter. The construction of the systematic expansion set PQ proceeds as in I with the one-dimensional product wave functions given by ~b,(r) =

-),r

l--1

r ~

k=O

(1-e -6('-~,)~ - {I +

•

ak,e-2~('-~)}.

These product terms are identical to those used for the hard-core potentials with the exception that the non-linear parameters were crudely re-optimised by a parameter search, the choice being p = 1,

? = 0.1375,

6 = 1.75,

c = 0

for S-states,

p = 1,

? = 0.3,

6 = 1.5,

c = 0

for D-states.

The value o f p was set at unity so that the product terms were non-zero at the origin. The parameter c is the hard-core radius and this was set at zero. The channel function ~(channel) used for N-d scattering and core wave function ~P(core) were constructed as in I with the exception that the functions g(r) [see eq. (12a) of I] were replaced by if(r) = ( 1 - e - 6 ( ' - c ) ) ' ( 1 - e - " ) .

(1)

492

M.A. HENNELL AND L. M. DELVES

It is these functions which were used to reproduce the expected behaviour of the solution for small interparticle distances. Again the values of these parameters were obtained by parameter searching. In the event the overall convergence rate of the product wave function ~'a was found to be poor, and a considerable amount of effort went into obtaining good core functions. For instance, in a series o f pilot calculations the g-function was fitted (for small r) to the shape of the deuteron wave function but this did not give any improvement. The values chosen for these parameters are as follows: Core 1

c

Core 2

3.0

6.5

3.0

5.0

-0.I

-0.01

n

3.0

10.0

fl

0.25

0.20

7

0.50

0.65

For details of the parameters fl,Y see eqs. (12c) and (13) of I. In table I we compare the performance of the dements of the Reid trialfunctions with those of HJ and GB. F r o m this table it can be seen that the first term o f the TAnLB1 Binding energy estimates (in MeV) for various components of the trial functions

Core 1 Core 2 Q = 3, no core

HJ

GB

--4.35 --4.73 16.82

--5.54 --5.76 12.46

Reid 0.206 0.121 581.7

systematic expansion set (Q = 3, no core) gives a very poor result for Reid compared with H J and GB. Similarly the Reid core wave functions are inferior to those of H J and GB but nevertheless they give a reasonably good starting point for the systematic set. With this particular set of trial functions, the only changes to the coding used for HJ arose in the coding of a routine to supply the new g(r) of eq. (1). 4. Numerical accuracy

Two sources of inaccuracy contribute to the error in a variational calculation: (a) Inaccuracies in the quadrature rule used (numerical errors), and (b) the truncation of the systematic expansion set ~Q (truncation errors). We discuss these briefly here.

BOUND STATES OF 3H AND aHe

493

(a) Numerical errors. The quadrature rule used for soft-core potentials, described in ref. : 1), is a special case of the rule used for hard-core potentials. For a given number of points per dimension n, the total number of mesh points is ~n(n + 1)(n + 2) which is less by a factor of about 3 than for the hard-core problem. The integration rule incorporates a scaling parameter ~ and the value taken for this parameter was initially that used for the GB potential. Later investigations showed that this parameter was poorly chosen since the bound-state integrals were evaluated with greater accuracy than necessary whereas the accuracy of the longer -70 &

-~0' -~r0

rn

&

I

]

5

6

-I,.0 E) -:~0"

-2-0-

-'r00~.~

I 3

I,

a

7

Fig. 1. The aH binding energy as a function of the expansion parameter Q. ranged scattering integrals was inadequate. Rather than increase the number of points per dimension it was decided to repeat the calculations with a value of biased towards greater accuracy for the scattering integrals. It was found that the soft-core results were much more sensitive to variations o f x than was the case for the hard-core potentials. This behaviour was not expected from experience with simple three-body calculations with and without hard cores 14). The final accuracy attained for the bound states is of the order of 1 ~o for the binding energy and bound state expectation values using n = 14 points per dimension, these results being independent of values o f ,t in the range 0.1 to 0.2. However for the scattering calculations a value of ct = 0.3 was chosen after a careful investigation. Nevertheless satisfactory results for the scattering problem have not been obtained with n = 14 and current results indicate that n = 16 is still not enough. Fur-

494

M.A. HENNELL AND L. M. DELVES

ther investigations have had to be curtailed at the present time due to the transfer of the programs to a new computer. In fig. 1 the aH binding energy is shown as a function of the expansion parameter Q for the various quadrature meshes used. The curves for n = 10, ct = 0.1 and n = 14, = 0.2 coincide to better than 1 ~ for all values of Q, the error being smaller at small Q. The justification for increasing u with n comes from ref. 14) where the behaviour of the quadrature rule was investigated. Two other curves are also presented; these arise from the attempt to obtain better scattering results. Both curves have = 0.3 and n = 10 and 14, respectively. They lie essentially parallel to the main curve, indicating that the choice 0t = 0.3 is adequate for kVQbut yields poor accuracy for the core wave function ~(core). The variation of EB with ct for the core terms has been investigated for both n = 10 and n = 14 and found to be sensibly independent of ,t over the range 0.1 < ct < 0.2. (b) Truncation errors. The convergence curves of fig. 1 for E(aH) show that, firstly the core wave functions for the Reid potential do not contribute as much to the final result as they do for both HJ and GB. Secondly, the convergence for the product terms is worse for Reid than for either of the hard-core potentials. Both observations indicate that the Reid potential is "more dimcult" than the hard-core potentials; we attribute this to the strongly repulsive core, which our wave functions have difficulty in fitting. For the hard-core potentials we had of course the exact solution (zero) for these regions. The best bound obtained is Eu = - 7 . 0 0 _ 0.1 MeV. Using the extrapolation procedures discussed in subsect. 6.1 of I gives the following estimate of the binding energy for Q = oo ET = --7.75 _ 0.5 MeV. This extrapolation procedure is discussed further below. The error quoted is almost wholly due to poor convergence, the error due to the numerical quadrature being negligible in comparison. An attempt was made to extend the calculation to the case of Q = 8 but owing to severe computing problems with a machine in the last hours of its life, this attempt was abandoned t. Once again the computed Temple lower bounds have proved to be of little direct value, the best bound being - 4 5 7 MeV.

5. Properties of the 3H and SHe bound states In table 2 are presented the computed values of various bound-state parameters for both 3H and aHe, compared with the predictions of H J and GB. t Both authors greatly lament the retirement of the Chilton Atlas computer, a truly remarkable and friendly machine.

B O U N D STATES OF aH A N D 3He

495

TABLE 2 Expectation values of various parameters of 3H and 2He for the Reid HJ and GB potentials, compared with experiment

3H parameters Binding energy (MeV) Ec (point) (MeV) E~ (finite) (MeV) 23o (n-d) (fro) (MeV) <* Vc+> (MeV) ½ (fln) ~" (fm) P(S) (70) p ( s ' ) (70)

HJ

GB

Reid

--6.54-0.2 0.56 0.55 2.54-0.3 --49.44-1.0 --33.04-1.0 1.85 1.90 89.2

--7.754-0.2 0.62 0.60 2.0±0.3 --43.14-1.0 --23.64-1.0 1.65 1.74 91.5

--7.754-0.5 0.62 0.59

1.8

1.8

P(D) (70) q2 (diff. min) (fro -2) q2 (sec. max.) (fm -2) IFch[ (see. max) (× 10 -2)

9.0 13.44-0.3 20.0 0.16

6.7 16.04.0.3 22.0 0.15

aHe parameters Binding energy (MeV) ~" (fro) /r (fro) P(S) (70) P(S') (70)

--5.954-0.2 1.90 2.11 89.4 1.93

--6.734-0.2 1.74 1.90 90.43 1.98

P(D) (70) q2 (diff. rain) (fm -2) q2 (see. max.) (fm -2) [F©h[ (seg. max.) (× 10 -2)

8.63 12.54-0.3 21.0 0.1

1.78 1.86 89.54-0.5

Expt.

--8.47 0.764 0.1-0.7 1.704-0.05 ~ 90

~ 1

9.54-0.5 13.84-0.5 ~ 20 0.23

--7.164-0.5

4-10

--7.728 1.87-4-0.005 i

7.56 14.84.0.3 18.0 0.11

12.8-4-0.3 ,,~ 18.0 0.2

11.8 18.0

It may be seen from table 2 that contrary to expectations the Coulomb energy for the soft-core Reid potential is comparable with the results for the hard-core potentials. Clearly the discrepancy with experiment for this parameter still exists. The binding energy prediction for the Reid potential of - 7 . 7 5 _ 0.5 MeV is comparable with the result obtained for the GB potential and substantially less than the result of - 6 . 5 MeV for HJ. We note however that quite widely differing results for this parameter have been reported in the literature, and we compare these results in sect. 6 below. The charge radius is consistent with the experimental value which is perhaps to be expected since the binding energy is also close to the experimental value. Numerically we find that the rate of convergence of these expectation values is faster than that of the binding energy. The S- and D-state probabilities of Reid are much the same as those of HJ which is again to be expected as the two-body D-state probabilities are close (6.47 ~o and 6.96 ~ respectively).

496

M.A. HENNELL AND L. M. DELVES 1.0 3He

0"1 /EXPI'. levi

0.01 IF, hi

/

/

ii •

,,,%

,~.,.,,~, /

oo0% ~ 4 ~ 6 ~oj2,~ ~Ik ~o ~,

0"001

,IJIl

0-000'

¢7 in~2 Fig. 3. The aH charge form factors. a

Fig. 2. The 3He charge form factors.

The electron scattering charge from factors are shown in figs. 2 and 3. In fig. 2 a comparison is made with HJ and GB whilst in fig. 3 the experimental results are also included. The Reid predictions for the 3He charge form factors give a minimum close to that of H3, although the agreement with experiment for low qZ is much better (reflectting the closeness of the charge radius to the experimental results). The secondary maximum of Reid is in roughly the right place and is higher than that of HJ by a factor of about two; however it is still lower than experiment by a factor of about three. 6. Comparison with other calculations 6.1. THE BINDING ENERGY The predictions for the Reid potential reported here appear to differ appreciably from calculations performed with this potential by other authors. In table 3 the results

B O U N D STATES O F aH A N D 3He

497

of the currently available calculations are surveyed. The calculations are grouped so that a comparison between those which have used the same number of components of the Reid potential can be made. The column containing the binding energy results contains multiple entries for those variational calculations for which a rigorous bound exists, and where also an extrapolation to an infinite number of trial functions TABLE 3 Reid potential predictions for the triton binding energy Potential components

Ref.

1So aSx

11) io) ix) xo) a) 11)

ISo ' aSz.SDz

Method

ET a)

lo) 9) 2o)

~ Faddeev t (2 components) i ] Fadeev (3 components) Faddeev (4 components) | Faddeev j (5 components) perturbative

6.8=[=0.5 6.71 6.4 6.39 6.39 6.9 6.7 7.0 7.02

iv)

variational

6.05

iso ' sSz aD2 1D2 ' aDz

lo)

22)

Faddeev variational

6.8 6.72k0. i

I P t , aPo

1~)

variational

6.26

3p3.aFz

9) s) 19)

variational Faddeev variational variational

6.3 7.27 6.65 7.0

Reid local

b)

6.9 I 0 . 5 6.6 ± 0 . 2 6.8 ±0.1 7.1 ±0.5 6.8 ±0.2 6.5

7.3 ~0.3 7.75±0.5

,. b) Column ") gives the upper bounds where applicable and column b) the extrapolated prediction. In the latter case two entries occur where a reassessment has been made of the extrapolated value by the authors of this paper. TABLE 4 The results of an analysis of a 50 × 50 matrix constructed with no odd-parity states, compared with the results for all states present

Core 2 Q = 3 Q = 4 Q = 5 Q --- 6

Full calculation

No odd-parity state calculation

0.152 -0.065 - 1.62 -4.40 -6.22

0.172 -0,057 - 1.66 -4.31 -6.18

498

M . A . HENNELL AND L. M. DELVES

can be made: The bounds are tabulated on the left of this column and on the right is the extrapolated value. In some cases two entries occur for the extrapolated values; the lower of these are the values quoted by the original authors, the upper results are the result of a reanalysis carried out by the authors of this paper, the details of which can be found below. In ref. 4) the authors suggested that the major discrepancy between the present work and the others listed in table 3 could be due to neglect of the odd-parity twobody states in the calculations of the other authors, these states contributing possibly 1.0 MeV to the 3H binding energy for HJ. This point has now been investigated for the Reid potential by explicitly omitting all odd-parity states. The results of these calculations are shown in table 4 where the analysis of a 50 x 50 matrix is shown. From this table it may be seen that these states appear to contribute less than 0.1 MeV. Gignoux and Laverne s) have also estimated the contribution of the oddparity potentials with J < 2 and their estimate (obtained by perturbation theory) is 0.27 MeV. Clearly whilst the inclusion of these states is such as to increase the binding energy they do not account for the current discrepancy. Referring to table 3 it can be seen that for the IS o and 3S1 potentials there is excellent agreement between the two available calculations. The same is true for the tS o, 3SI-3D 1 potentials when three Faddeev components are included but for the case of five Faddeev components it is noticable that the four calculations (when the original results are taken) separate into a pair at 6.7 MeV and a pair at around 7.0 MeV. With the re-evaluation of the results of Bruinsma et al. given in appendix A this calculation comes into agreement with calculation s, 9) (see table 3). This suggests that there is a source of error in the five component calculation of Harper et aL When the group of calculations t~-t6) is considered it must first be noted that there is an error in ref. 9) which is likely to affect the accuracy of this calculation and a re-calculation has been carried out by Strayer t 9). There appears to be a duster around 6.7 to 6.8 MeV and another around 7.1 to 7.27 MeV. However the revaluation of ref. 17) again leads to an extrapolated value in agreement with the second duster. The calculation of Demin et aL 22) was also re-evaluated in ref. 2t). Despite the apparently good convergence achieved, reasons were suggested there why the result might be erroneous. The final (present) calculation has a slightly different potential in that high-J states are included, but nevertheless this calculation has produced a rigorous bound at - 7.0 MeV. The fact that the extrapolated value of this calculation is significantly greater than the other binding energy estimates may reflect an unexpectedly large contribution from the high-J states, or may merely reflect the difficulty of extrapolating with only a few terms (see appendix). The results of this analysis is that a value of the binding energy in the range 7.0 to 7.5 MeV is consistent with the results of most of the current calculations, and we suggest that the best available value is E(3H) = 7.3 +_ 0.2 MeV (Reid soft core).

BOUND STATES OF aH AND aHe

499

6.2. THE F O R M FACTORS

Another point of contention lies in the aHe electromagnetic charge form factors. The results shown in fig. 3 agree to within a few percent with the results of Jackson et aL 9) and of Strayer et al. x9) but differ considerably from those of Harper, Kim and Tubis I o) and Malttiet and Tjon t 1). The form factors of Gignoux and Laverne a) are closer to those presented here in that their secondary maximum although displaced towards higher qZ is roughly of the same height. Their minimum however is still located at a higher value of q2, i.e. q2 __. 15.0 fm -2. TABLE 5 Analysis of a 50 × 50 matrix constructed with only the Reid 1St-3Dt states in the modified potential, compared with the corresponding results for the full potential Wave function components

Full calculation

~So and aS~-SDt states only

Core 2 Q = 3 Q = 4 Q = 5 Q = 6

0.152 -0.065 -1.62 -4.40 -6.22

0.267 0.033 -1.58 -4.18 -6.00

TABLE 6 Contribution to the aH binding energy o f the Reid states other than xSo and aSt-aDt Bruinsma et al GignotLx et aL Kim et al. Hennell et al.

0.2i 0.27 0.1

0.22

The fact that Jackson's form factors agree with those presented here, but his bindding energy does not, can be understood by the following remarks. Using a systematic expansion of the trial function we have studied the covergence of the form factors with respect to the parameter Q which determines the quality of the wave function. We find that when the wave function has improved to a point where the binding energy prediction is of the order of 5.0 MeV the form factors have essentially converged. This suggests that any variational calculation with the Reid potential which predicts a binding energy greater than 5.5 MeVwill also predict the converged form factors. It also suggests that the minor error in Jackson's work is unlikely to affect the form factors; this is borne out also by the quite good agreement between the results of Jackson and of Strayer and Sauer. We have no direct explanation for the difference between our form factors and those derived from finite difference solutions of the Faddeev equations. However, the fact that the variational calculations use analytic wave functions in form factors calculations whereas the other methods use discretisations of the wave function,

500

M.A. HENNELL AND L. M. DELVES

suggests that possibly there may be instabilities in this latter approach at high values of q2. 6.3. EFFECTS OF HIGHER TWO-BODY STATES In order to estimate the contribution of states other than the 1So and aSI-aD t states, a recalculation was made including only these states in the basic Reid potential. It must be noted however that this calculation is not directly comparable with the calculations 7, s) of table 3 since the way in which the Reid potential enters our calculation ensures that contributions from two-body states with J > 2 are still present. The results of this calculation are given in table 5 from which it may be seen that our estimate of the contribution of the states other than 1S o and aSo-aD 1 is of the order of 0.22 MeV. This estimate is in exellent agreement with the estimates made by other authors. This agreement is illustrated in table 6. In a similar calculation 12) with a super soft-core potential it was found that the difference between a calculation with all two-body states and one with J =< 2, was negligible [note that the binding energy quoted in ref. 12) should be - 8 . 3 MeV and not - 7.8 MeV]. 7. Conclusions

We attempt here to summarise the results of this paper and the best available data for the Reid potential. Our conclusions are: (i) The best available calculations strongly indicate [E(aH)I > 7.0 MeV. (ii) Our recommended "best value" for the full Reid potential with J _<_2 is E(3H) = ---.~-o.2 ~ a+°'3 MeV which suggests a value E(aH) ~ - 7 . 5 MeV

(Y < 2), (all d-values).

(iii) There is less agreement on details of the 3He or 3H charge form factors. However, three variational calculations give results in excellent agreement, predicting a minimum for ZHe at q2 ~ 12.5 fm -2 and a secondary maximum at between 18.0-20.0 fm -2, although the height of this secondary maximum varies between calculations by a factor of 2-5 below the "experimental" data. Appendix EXTRAPOLATION ERRORS All calculations in this field involve in principle one or more truncations or extrapolations: of a sequence of quadrature rules; of an expansion of a trial function; of a truncated two-body angular momentum decomposition; of a step length in a

B O U N D STATES OF aH A N D aHe

501

space discretisation. Too often these truncations pass without comment; the error involved is, presumably, estimated by the authors using experience built up from more accurate calculations on similar systems. Such experience is fallible, and the absence of published information on how the errors have been assessed, makes a comparison of results difficult where they disagree, and inconclusive when they agree. We would make a strong plea for details of the error analysis to be included in future published calculations; and we here discuss briefly the relevent analysis for a variational calculation such as our own. In this calculation two extrapolations are used. The first involving the quadrature rule has already been discussed. The second involves the extrapolation of the binding energy to the limit of an infinite number of terms in the expansion set. The basis of this extrapolation procedure is the theoretical prediction 15) that in the "asymptotic" region the rate of convergence ? in the binding energy is given by EQ = E . -

a__ + O ( Q _ ~ _ : ) ' Q~

(A.1)

where Q is the expansion parameter (not the number of terms, except for one-dimensional problems) and u is some constant. The extrapolation problem is to estimate the parameters u, ? and especially E~o from the available computed values. Such problems are notoriously ill-conditioned, and we would like to stress the dangers inherent in a blind extrapolation. If the data do not well fit (A.1), it is important that this be noticed. This may happen, for example, if the range of Q covered is too small ("not in the asymptotic region") or if the numericals errors associated with each point are significant. The extrapolation process was illustrated in ref. 21), for several recent calculations. There, the purpose was to emphasise the effect of these numerical errors and one way in which they can be detected, rather than to give an efficient way of estimating u, ~. The procedure which we use in practice is graphical in basis, and is also designed to ensure that these points will not be missed. We difference (A.1) to yield

A'~ =

EQ+m--EQ - - (Q+m) ~

=~_~m "" At2 Q,+I +O(Q-~-2)'

Qy'

(A.2)

where fl = ~ty, and hence retaining only the leading terms, E® = EQ+

~ tim e=Q,m p~,+l "

(A.3)

Formulae (A.1) and (A.3) only give the same extrapolation for large Q. A graph of log A Q against log Q will thus have a slope of ~ + 1, enabling ~ to be estimated.

502

M . A . H E N N E L L A N D L. M. DELVES

TABLE 7 The binding energy E(3H) with the Reid potential, using a hyperspherical basis: taken from Bruinsma et al. XT) Q EQ LIE

10

12

14

16

18

20

2.325 1.982

4.307 0.798

5.105 0.593

5.698 0.421

6.119 0.139

6.258

0.3 0.2 0-1 00

i

-0.1

I~AQ -0"2 -0-3l -04.

-05 -0"6 -0-7 -06 o

-o.81.o-

1t

1-2

\ 1-3

1"/,

t0g Q Fig. 4. Log A o plotted versus log Q.

The quality/~ (and hence ~) can be found either from the intercept of the graph or by use of (A.2). Finally using either (A.1) or (A.3) and estimate can be obtained for Eco.

Two points immediately arise from this analysis. Firstly the error in using (A.2) rather than (A. 1) will be smallest ff n is large. Secondly the difference between E. and E~o will be small ff T is large and ~ is small. When these three conditions appertain extrapolations can be made with a small error.

503

B O U N D S T A T E S O F 3H A N D 3He

In the following we analyse, where possible, the current variational calculations.

(i) The calculations o f Jackson et al. 9) and Strayer 19). These are variational calculations in a hyperspherical coordinate system using harmonic oscillator trial functions. Theoretical considerations t 6) show that the asymptotic convergence rate for these trial functions is 7 = 2.0. Ref. 19) includes an extrapolation by the authors, based on (A.2). The measured value of T is somewhat greater that 2.0, but appears to be dropping with increasing Q; that is, the asymptotic re#on is reached only slowly in this calculation. (ii) Bruinsma et al. xT). In table 7 we give the quoted results from this reference and in fig. 4 we show a plot of log AQ versus log Q (note that in this case m = 2). We see that three of the five points lie on a straight line, whilst the second and fifth are significantly displaced. If we for the moment ignore the points at n = 12 and n = 20, the value of ~ from the graph is ? = 2.45 + 0.2. Putting this value in eq. (A.2) for n = 16 gives tim = 5800 -22oo-+42°°Then from (A.I) using n = 18 we get Eoo = 7.1 + 0.2 MeV. However this error estimate takes no account of any inaccuracy in the points used. In view of the range of the extrapolation a more cautious error of +0.5 MeV would seem advisable. TABLE 8 T h e b i n d i n g energy E ( 3 H ) with t h e Reid potential; calculations o f this p a p e r Q

3

4

5

6

7

w

-0.103

- 1.67

-4.47

- 5.98

-7.00

An explanation for the behaviour of the points at n = 12 and n = 20 has been given by Van Wageningen who points out that the parameter n [this parameter is the K of ref. tT)] may not be the appropriate parameter for use in extrapolation since for large values which are multiples of three an extra completely symmetric state appears. The effect of this is that these 'triples' cause small perturbations to the smoothness of the curve of EB versus K. Unfortunately there are insufficient of these triples available to use them alone for extrapolation. Nevertheless it is extremely disturbing that in fig. 4 the point for n = 18 lies so far off the smooth line through the other points. It is to be expected that in the case where n is not an appropriate expansion parameter but a multiple of n is, a curve drawn through the special points and one drawn through the other points should be parallel. A typical case is given by examples showing an "odd-even" effect - a difference between odd and even values of n. The full-range Fourier expansion of a "nearly even" function, for example, shows such an effect; for an example of its occurrence in a variational calculation, see, e.g. ref. 23). Clearly from fig. 4 this is not happening and in the authors' opinion the point at n = 20 which is the point most susceptible to numerical error must be suspect. Even if this point is accurate, it clearly is useless for extrapolation purposes and a line through the non-special points must suffice.

504

M. A. HENNELL AND L. M. DELVES

(iii) Hennell and Delves. I n table 8 are presented the calculated values. Here the picture is far f r o m satisfactory: It is clear that the convergence m a y n o t yet be asymptotic. This is a consequence of h a v i n g a core wave f u n c t i o n ; for H J a n d G B the prod u c t set alone gave very s m o o t h convergence e n a b l i n g extrapolation to be c a r d e d o u t with confidence whereas convergence with the core terms was less smooth. W i t h the Reid potential the p r o d u c t terms by themselves give results which are too far f r o m convergence to extrapolate, so we are forced to rely o n the results with a core wave function. W e have t a k e n a weighted least-squares fit to the points a n d have tried t o c o m p e n s a t e for the u n c e r t a i n t y by a d d i n g a substantial error estimate. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23)

L. M. Delves and M. A. Hermell, Nucl. Phys. A168 (1971) 347 M. A. Hennell and L. M. Delves, Nucl. Phys. A204 (1973) 552 M. A. Hennell and L. M. Delves, Phys. Lett. ,10B (1972) 20 M. A. Hennell and L. M. Delves, Proe. Conf. on few-particle problems, Los Angeles, 1972 (North-Holland, Amsterdam) p. 419 R. Reid, Ann. of Phys. 50 (1968) 411 M. A. Hennell and L M. Delves, Acta Phys. Acad. Sci. Hung. 33 (1973) 103 T. Hamada and I. D. Johnston, Nucl. Phys. 34 (1962) 382; T. Hamada, Prog. Theor. Phys. 33 (1965) 769 A. Laverne and C. Gignoux, Phys. Rev. Lett. 29 (1972) 436 A. D. Jackson, A. Land6 and P. U. Saner, Phys. Lett. 36B (1971) 1 E. O. Harper, Y. E. Kim and A. Tubis, Phys. Rev. Lett. 28B (1972) 1533 R. A. Malfliet and J. A. Tjon, Proc. Conf. on few-particle problems, Los Angeles, 1972, ed. I. Slaus (North-Holland, Amsterdam) p. 441 M. A. Hermell, Proc. Conf. on few-particle problems, Quebec, 1974, to be published J. W. Humerston and M. A. Hennell, Phys. Lett. 31B (1970) 423 D. H. Bell and L. M. Delves, Nucl. Phys. A146 (1970) 497 L. M. Delves and K. O. Mead, Math. Comp. 25 (1971) 699 L. M. Delves, Adv. in Nucl. Phys. 5 (1973) 1 J. Bruinsma, R. van Wageningen and J. L. Visschers, Proc. Conf. on few-particle problems, Los Angeles, 1972 (North-Holland, Amsterdam) p. 368 D. H. Bell and L. M. Delves, J. Comp. Phys. 3 (1969) 453 M. Strayer and P. Sauer, Preprint IPNO/TH 74-16, Manchester (1974) I. R. Afnan and J. M. Read, Proc. Conf. on few-body problems, Quebec, 1974, to be published L. M. Delves, ibid. V. F. Denin and Yu. E. Prokrovsky, Phys. Lett. 47B (1973) 394 T. L. Freeman, M.Se. Thesis, Liverpool, 1971