M1 transitions in 208Bi

Nuclear Physics A197 (1972) 583--592; (~) North-Holland Publishin9 Co., Amsterdam

Not to be reproduced by photoprint or microfilmwithout written permissionfrom the publisher

M 1 T R A N S I T I O N S IN 2°SBi

A. OLIN, O. HAUSSER, D. WARD and D. L. D1SDIER * Atomic Eaeryy of Canada Limited, Chalk River Nuclear Laboratories, Chalk River, Ontario, Canada Received 7 August 1972 Abstract: Electromagnetic matrix elements of some low-lying states in 2°SBi have been determined. Lifetimes of the following levels have been measured using the reaction 2°Tpb(TLi, a2ny)2°SBi and the Doppler-shift attenuation method: 1.095 MeV (r ~ 0 •19-o.os +0.07 ps), 0.886 MeV ( r = 026 +°"12 ps), 1.069 MeV (r - - 0 6• 4 -O.lS +°'3x ps), 1.033 MeV (~ . . . .1. (~d-+°'38 • -o.os o.2,, ps), 0.936 MeV (3 :> 2.5 ps) and 1.539 MeV (~ > 1.7 ps). The lifetimes of these levels are very sensitive to configuration mixing through the residual nucleon-nucleon interaction. We compare our results to predictions from nuclear structure calculations by Kuo. NUCLEAR REACTIONS 2°7pb(TLi, 0t2n~')2°SBi, E = 31.5 MeV; measured ~y-coin, Doppler-shift attenuation. 2°SBi levels deduced B(MI) and lifetimes• Enriched target.

I. Introduction

Lifetimes o f low-lying states in 2°SBi have been m e a s u r e d using the D o p p l e r - s h i f t a t t e n u a t i o n m e t h o d ( D S A M ) . These states have been studied previously in light-ion r e a c t i o n s t - 3 ) , where low recoil velocities m a k e m e a s u r e m e n t s o f lifetimes from D S A M o r recoil-distance m e a s u r e m e n t s impossible. W e have excited 2°8Bi states in the reaction 2°Tpb(TLi, ~2ny)2°SBi with recoil velocities o f 0.005c. This allows us to m e a s u r e lifetimes in the range 0.02-2 ps, a p p r o p r i a t e to M1 decays. T h e o r e t i c a l studies 4, 5) have successfully described the low-lying 2°SBi levels as r e a s o n a b l y p u r e shell-model states. Predicted lifetimes from these studies will sometimes be very sensitive to c o n f i g u r a t i o n admixtures, a n d thus are a sensitive test o f these calculations. Extensive particle s p e c t r o s c o p y on 2°8Bi has been d o n e recently by A l f o r d et aL ~) using the 2°Tpb(3He, d), / ° 7 P b ( ~ , t), 2°9Bi(d, t) and 2°9Bi(3He, ~) reactions. F r o m the o b s e r v e d spectroscopic factors a n d a n g u l a r d i s t r i b u t i o n s they were able to assign spins a n d parities to m o s t levels, a n d to d e t e r m i n e d o m i n a n t configurations. Precision ? - r a y spectroscopy, including 7-7 coincidence work, has been r e p o r t e d recently b y Proetel et aL 3) using the reactions 2°8Pb(p, n7) a n d 2 ° s p b ( d , 2n),). This w o r k was used to confirm o u r identification o f 2°8Bi "f-rays. It c o n t a i n e d careful measurements o f the b r a n c h i n g ratios needed to extract matrix elements from o u r lifetime mcasurements. T h e decay scheme from this w o r k is given in tig. 1. * Permanent address: Groupe de Laboratoires de Strasbourg-Cronenburg, 23 rue du Loess, 67 Strasbourg 3, France. 583

584

A. OLIN

CONFIGURATION ~-0

et al.

SPiN RANGE

~--

,.~ 2,}

,+-8÷--

2401.3 2308.3

2 0 7 7 . 4 42+)

2,} , , }

2,-5,

1920. t ( 4 - )

.../

1836,5 4 5 - ) 1801.9 45"+)

g< // //

1703.4 i .157H q~, .~,~.5

,¢. |i~

3p /

6---7-

-

-

=,~ ¢n

43-) ,o-

44+, 5 +) --4539.3 4 2 + ) * " t 5 2 9 . 5 (3+, 4 ÷ ) 1 4 6 9 . 3 45-1-)

¢o

-:"2. • .-~

,',4

-*00

~0

=,.. ~ .=. ~_ o,,,o o_ o_,,,oo • r ~r

J J i

" " ~ ~ m

i

~ , , ,

,094.9

I I

--1069.2 ~035,5

I

,~,~=_-~;,°'® QOOJ~N • , 1 , 1 , I, I, I, I, Il Il Ii Il {•

2,} 3,~- 3,-%

m =

1

i

i

i

958.9

~936.2 _"924.8

"886.4

/

,,~ 2,~

/

6+ 5+ 4+ ,÷ 3+ 2+ 5+

/

. . . . . . . . 1

2t_.

_7---

~633, I -628,2 "601,5

3+ 5+

4+

510.2

6.1-

JI

62.9

4÷

|

0

5+

Fig. 1. Partial decay scheme of 2°SBi [rcf. a)]. Only y-rays of interest to our lifetime measurements arc shown.

2. Experimental method A 30 nA, 31.5 MeV beam o f 7Li 3÷ ions was incident on a 10 m g / c m 2 foil o f 2°Ypb, and subsequently passed into a shielded beam dump. Back-scattered particles were observed in a 300 # m thick annular surface-barrier counter. A 45 cm 3 Ge(Li) detector was used to observe "/-rays at 40 °, 75 °, and 135 °, and the time relationships and pulse heights for particles in coincidence with ?-rays were recorded on magnetic tape. Precision mercury relay pulser peaks were inserted in the Ge(Li) preamplifier at the upper and lower ends o f the ?-spectrum to stabilize the gain and zero o f the ~-detcctor system using a method described by Broude6).

2°aBi MI TRANSITIONS

585

Fig. 2 shows a typical spectrum for particles in coincidence with y-rays. From previous experiments 7) we identify the particles from 18.6 to 30 MeV as mostly 7Li and 6Li, those from 10.6 to 18.6 MeV as 4He, and the 3.5 to 10.6 MeV particles as tritons. This previous experiment allowed us to choose our beam energy to obtain grazing incidence for back-scattered particles, thus maximizing our transfer cross section. Gamma rays observed in coincidence with particles of energy 10.6 to 16.5 MeV

31.5 MeV "tLi a°TPb TARGET

-'OSBli

/'

dz

(Li, aZn7) 2°IIBi__~

z I (,.) r,

( 71.-i,a n X)

,ooc

z 0 0

5

I0

] (rLi,;'7) Li .

m

Z°rpb

( rLi, eLi 7") 2oapb

I

15

20

25

30

ENERGY (MeV)

Fig. 2. Spectrum of back-scatteredparticles in coincidence with reactionT-rays.

are mainly from the decay of levels in 2°SBi, although the strong 2°9Bi v-rays at 1.608 MeV and 0.8966 MeV are also seen; the latter obscures the 0.859 MeV 2°SBi line. We see all the strong 2°8Bi transitions reported in ref. 3), our energy determinations agree very well, and, as expected, we observe that higher-spin levels are preferentially excited in the 7Li reaction compared with the (p, n) reaction. 3. Data analysis The peak centroid positions were determined by fitting the data with a non-linear least-squares code 19). Gaussian peak shapes of variable width and a quadratically varying background were assumed. A typical fit is shown in fig. 3. Skewness o f the Doppler-shifted peaks at these recoil velocities was washed out by the counter resolution, justifying the use of Gaussian peak shapes. The recoil velocity of the 2°8Bi ions was calculated by considerhag the reaction 2o7pb(7Li ' a)alOBi ' and adjusting the excitation energy of the 2t°Bi to give the observed ~-encrgy; this corresponds to levels around 16 MeV in 21°Bi and v / c = 0.0050 for the Bi ions. The decay 21°Bi ~ 2°SBi+2n will leave the 2°8Bi with 3 MeV of excitation if the neutrons each carry off 2 MeV in kinetic energy; this is sufficient to populate the observed levels.

A. OLIN et al.

586

2°'Pb ('tLit a2n 7)z°eBi

z z

oc 5 0 0 03 FZ

2

0

.',.t

I

o.sz

I

I

o.86

I

I

o.9o

I

o.94

I

I

0.96

I 1.00

!

I 1.04

I

I

".''

I

1.08

ENERGY (MeV)

Fig. 3. G a m m a - r a y spectrum at 40 ° m coincidence with back-scattered particles o f energy t0.6-I 6.5 MeV.

The observed energy shifts, expressed as a fraction F of the calculated full shift, is related to the nuclear lifetime z by F =

_.1

(cos ¢(t))v(t)e-ti~dt,

(1)

• V 0 ~. 0

where ¢(t) is the scattering anglc of the ion as defined by Blaugrund8), and v(t) is the recoil vclocity of the ion calculated using the Lindhardt theory of stopping powers 9). The computer code for this calculation was written by Broude x0). Table 1 contains our lifetime measurements. Partial mean lives are calculated from thc total mean life of the level, as determined by (1), and the experimental branching ratios of rcf. 3). The quality of our data for measuring intensities and energies was considerably poorer than those of ref. 3), who were able to observe the y-rays in singles and without Doppler broadening. Furthermore our lowcr discriminator was set at 500 keV. However those branching ratios that we were able to extract (1033/970 and 886/823) were in good agreement with Proetel's values 3). In these cases the weighted mean of the F-factors from both branches was used to determine the total mean life. Quoted errors includc an aliowancc for l0 ~ deviation from the Lindhardt theory added directly to the statistical crrors. No experimental information is available for the stopping of Bi in Pb, especially at these low vclocities where nuclear scattering is dominant, so this assumption of a l0 ~o error is arbitrary and possibly low. However, it is clear from cq. (1) that relative lifctimes with comparable z-values are not very sensitive to thc choice of stopping powers. The 0.8965 and 1.608 MeV ,?-rays from Z°9Bi are unshifted. Since the lifetimes of

2fk(2q) -~

°) Ref. a).

1.5393

(2 + )

34.

4+

1.0330

0.9363

3+

1.0694

2f.~(3p~)- '

5+

h}(2q)-'

hl.(3p~)-

h i (3p:~.)-

h~(3p~) - l

6+

final state

initial state

Main configuration

0.8862

1.0950

E~ (MeV)

1.0330 0.9703

5+ 44.

0.9061

1.0067

44.

3*

0.8231

4+

0.8731

0.8862

5+

4+

1.0950

5+

(MeV)

TABLE 1

/-forbidden /-forbidden /-forbidden

0+0.62 --0.39 4.2 + 1.5 4.0 +co

3.0 +co

0N~fl + 0.006 .... -0.005 NIN+O.O03 .... -0.003 Ngl + 0 . 0 1 3 .... --0.021 017+0.013 -0.017

0.033(0.015) 0.047(0.039)

0.17 (0.06)

0.121 (0.025)

. . . .

1.16

'77+0.37 "----0.21

1-179+0.026 .... -0.024-

0.19 (0.04)

-0.06

--1.3

--1.5

-1.0

0.464

0.20 +o.o8

0.30 (0.11)

0 ,~N+0.21 '~v--0.14

/-forbidden

0.048

0.219

0.117 0.697

0 ~;d + 0 . 2 3 '~---0.16

0.15 +0.06 --0.05

0.037

0.43 (0.08)

-0.06

1.16

B ( M I ) partial mean (/tNz) life (ps)

10+0.07 "'---0.05

partial mean life (ps)

Pure configuration

0.23 +o.oa

0e~~)

B(MI)

Exp.

0.47 (0.05)

2°aBi lifetimes

~0.74

1

0.249

0.614

0.83

0.52

0.48

1

ratio ~)

Branching

..o

Z

Z

.q

588

A. OLIN

et al.

these levels are known to be long, this is a good test of the stability of our electronics. No fully shifted lines were available for an independent determination of v/c. In a previous experiment ,1), using the Z°Spb(TLi, e2n)2°gBi reaction, we were able to compare v/c as determined from fully shifted lines with our value calculated from kinematics, and the agreement was satisfactory. No allowance was made for feeding of the levels of present interest from higher levels. We have examined our spectra for evidence of such feedings and find they are very weak. The only significant feeding comes from the decay of the 2.308 MeV level through the 1.069 MeV level; the intensity of the 1.239 MeV ~-ray is about 25 ~ of the intensity of the 1.0063 MeV line (see fig. 1). The properties of this 2.308 MeV level are not known. Other known feedings are less than 10 ~ of the intensity of the level of interest.

4. Electromagnetic matrix elements Let us consider 2°8Bi within the framework of the shell model, as a l p - l h system with an inert 2 ospb core. Using the matrix elements and phases of Rose and Brink 12) we may write the electromagnetic matrix elements for this nucleus as

<.i~pjihd~llG[(rp)+ G[(rh)ilJfpJ'r,Jr> = ~/2Jf + 1 ( - )Jrp-j"'-L X {~(Jih, Jfh)(-- )s'~/2J, p + ] W(jJvJrpJiJf; Ljih) + 6(j~p, jr,),/2,),~ + i (-)s,. W(j,,je,J, Jf, Lj~p)}

(2)

Here , the reduced matrix element for the electro-magnetic operator of multipolarity L and parity 7r, may be calculated within the simple shell model using eqs. (4.5) and (4.6) of Rosc and Brink 12). The reduced transition probabilities, widths, and mean lives obtained from this calculation are called 'pure configuration' values in tables 1 and 2. introducing a residual interaction between the core and the valence particle or hole effectively renormalizes the single-particle matrix elements - In the case of M1 transitions, the effective mechanism seems to be the excitation of lh~(lh~) - t protons and l i÷ (1 i~)-a neutrons from the core t 7, xs). Two-body effects and mesonic exchange currents are also cxpected to contribute 2~). In E2 transitions a similar renormalization occurs through the excitation of vibrational states in the core ix). Since calculations of these renormalization factors are not particularly reliable for M 1 transition rates, we use experinaental values taken from measurements of magnetic moments, reduced transition probabilities, and mixing ratios in 2°9Bi and 2°Tpb. The matrix elements used in our calculations (colmnns 6 and 7 of table 2), together with references, are collccted in table 3. Harmonic oscillator wave functions were used in the evaluation of the effcctive charges; the oscillator parameter used was mw/h = 5.324 fm 2. In order to justify the use of these renormalized matrix elements in eq. (2) we must

(2 + )

1.5393

0.4058

0.4002

0.1467

5~

3+

5÷

3+

0.9061

0.8731

0.9703

4"4"

4~

1.0330

0.4361

3+

5+

0.4677

4+

4.77

13.7

0.0044 0.050

0.032

0.00016

0.00015

0.0000002

0.000001

0.00003 0.000004

0.0006

0.027

0.059

Fl~_, (meV)

0.00002

0.084

0.050

0.263

2.74

1.77

2.16

5.64

l~tl ~) (meV)

0.035

0.0011

0.056

1.05

1.95

6.15

i ~ 1 bj (meV)

Calculated widths

3.00

5.58

17.8

(meV)

lMl )

a

configuration and bare nucleon g-factors. configuration experimental single-particle matrix elements. wave fi,nction experimental single-particle matrix elements. ~).

3+

0.9363

Pure Pure Kuo Ref.

4+

1.0330

a) b) ~) d)

3+

1.0694

0.8231

1.0694

4+

5+

0.8862

5+

5+

5+

0.8862

1.0950

1.0067

5~

6+

1.0950

E^/

(MeV)

4+

Final spin

Initial spin

E~

(MeV)

T A B L I ~, 2

2°8Bi widths and. brartching ratios

-0.008

. . . .

0.1 ~R+0.047 ~ - 0.042 0 f~9/~+0.006 -0.005 0 (~/)~1 +0.0015 . . . . . --0.0014 0 f~9+0.018 . . . . --0.016 0.17+o.1 o -o.17 99+0.17 "----0.22

0 ~0.,I+ 0.034 . . . . -0.031 0 t3~A+ 0.019 . . . . --0.017 +01 0.39_0:12

. . . .

99+o.49 "---0.36 1 -~7+o.53 '---0.39 0 R~+0.31 "~-0.28 0 ~gd.+ 0.009

• -0.9

3 5 +1"2

(meV)

l'¢xp

0.0014

0.00003

0.0003

0.006

->- 0.74

1

0.097(0.012)

0.008(0.002)

0.032(0.002)

0.249(0.011 )

0.614(0.023)

0.052(0.005)

0.016 0.99

0.091 (0.006)

0.023 (0.005) 0.085

0.019

0.830(0.031)

0.52 (0.027)

0.45 0.88

0.48 (0.031)

1

exp a)

0.55

calc

Branching ratios

,.q Z ,.q

70

A. O L I N e t al.

590

TABLE 3 Electromagnetic matrix elements in the Pb region Transition

B(MI) (p.N2)

Renormal- B(E2) Effective ization (e 2 • fm 4) charge factor fl~

Source

gs/g . . . . ~1 h.~. -> lh.,3_ zt2f.1. -> lh.~ zt2I~. --~ I h ~ v(3p½) - 1 __> (3p~)- 1 v(2f~) -1 ~ (3pt.) -~ v(2f].) -1 -~ (2f~.)-' v(3p~)- 1 ~ (3p.~) - ~ v(3p,t.)- 1 ~ (3p,l.) - ~ v(2f,1.)- ~ ~ (2f~)- ~ ~2f.~. ~ 2f I.

4.86 0.0045 a)

0.36

0.25

0.93

0.14 0.41 0.091 0.49 7.03

0.48 0.59 0.25 0.55 0.64

280 30 480

1.55 2.36 0.89

71

0.94

61

0.80

16

1.08

~) The reduced matrix element is chosen negative. b) Ref. ~3). ~) Ref. ~'). a) Ref. z4).

e) Ref. ~5).

#, Q in 2°9Bi b) B(E2)~ ¢), 6 z d) B (E2)~" ¢) /z in z°TPb b) B(E2)~" ¢) /z in 2°7pb e) z°7pb ~' lineshape, B (E2)~ ¢) calculated p ¢) B(E2) calculated ~), 62 ') calculated !z ¢)

f) Ref. 1~).

examine how the presence of the extra valence particle in/°SBi affects the renormalization of the transition matrix elements. From fig. 4 we see that the extra terms introduced in 208Bi are two-body terms of at least second order in the residual interaction. There will also be a slight change in the 2°Tpb matrix elements because of blocking

-X

a)

I--x I-i b)

~ c)

--X

d)

Fig. 4. Diagrams for calculations o f M1 rates in 2°SBi: (a) Direct single-particle contributions from z°7pb and z°9Bi. (b) Some diagrams for the first-order core polarization correction. This term appears also in the 2°TPb effective M1 operator. (c) A second-order diagram appearing also in the 2°Tpb effective M1 operator. (d) A second-order diagram corresponding to the two-body part o f the effective MI operator.

2°SBi MI T R A N S I T I O N S

591

effects on the l h ~ ( l h ~ ) -x excitation. The magnitude of these terms is estimated to be about I0 % of the total renormalization 2o). The residual interaction acting between the two valence particles will mix configurations in the two-particle wave functions, and we must replace the states ]jpjhJ) in (2) by a sum of configurations

~(J) = Z Ci,iJJp]hJ)" JpJh

These wave functions have been computed by K u o 5) using an H a m a d a - J o h n s o n potential for the residual interaction. Widths calculated using K u o ' s wave functions and experimental matrix elements are also given in table 2 together with calculated branching ratios. For some of the weakly admixed states experimental matrix elements are not known and we have used calculated values in table 3. The configuration mixing calculations will be very uncertain for the small widths, since they involve cancellations among a large number of matrix elements. For the best cases, (3p_~)-~ ~ (3p~.) - I transitions, there is a 15 % error arising from uncertainties in the experimental matrix elements. 4.1. D E C A Y O F T H E lh~_(3p~_)- t Q U A R T E T

The measured widths for these/-allowed transitions are systematically smaller than our calculated values by a factor of almost two. The measured branching ratios for the 5 + and 3 + states are a sensitive test of the Kuo wave functions and are found to be in excellent agreement with our calculations. The observed F-factors for the two decay modes of the 5 + level agree, but in each case the statistics are rather poor. The observed lifetimes of the 6 + and 5 + states are in satisfactory agreement with the calculations if we consider the 15 % error in the (P~I[MIIIP~r) matrix element. Varying this matrix element within its range would not seriously change the branching ratios. The small Doppler shift of the 3 + state may be the result of feeding from the 2.308 MeV level. Since the predicted lifetime for the 3 + is of the same order as the 5 + and 6 + lifetimes, this discrepancy is independent of the stopping power assumed in our F(z)calculation. 4.2. D E C A Y O F T H E 2fzt_(3p.~)-x D O U B L E T

The decay of the 3 + state of this configuration is observed to be highly retarded, in agreement with the calculations. Both the width and branching ratio of the 4 + state disagree violently with the calculations. Afford et al. x) have suggested a 10 % admixture of I h~(3p.~)- 1 in this state to explain its spectroscopic strengthin 2°9Bi(d, t) reactions. Using this admixture we calculate M1 widths of 0.35 meV for the 1.033 MeV 4 + ~ 5 + transition, and 0.19 meV for the 0.9704 MeV, 4 + ~ 4 + transition. This agrees very well with the observed width and branching ratio. 4.3. O T H E R D E C A Y S

The transition originating from the 1.539 MeV level is observed to be hindered, as predicted by the AI = 0 selection rule. The 0.862 MeV y-ray from the lh~(2f~) -1

592

A. OLIN et al.

2 + state was observed, b u t poor c o u n t i n g statistics did n o t allow us to extract a D o p p l e r shift. F o r this level the calculated a n d measured b r a n c h i n g ratios disagree by a factor of two.

5. Conclusion We have measured the lifetimes o f a n u m b e r of levels in Z°SBi to a n a b o u t 31)-40 %. These lifetimes d e p e n d sensitively o n small admixtures functions of these states, a n d thus are a good test of nuclear structure for 2°SBi. The Z°SBi wave functions calculated by K u o give generally m e n t with o u r observed lifetimes a n d b r a n c h i n g ratios. However, o u r

accuracy of in the wave calculations good agreeresults also

confirm the I0 % admixture of lh~(3p~) -1 configuration in the 1.033 MeV level, as reported by Alford ~). Because of the large uncertainties of o u r lifetime m e a s u r e m e n t s we c a n n o t draw conclusions a b o u t the existence of additional r e n o r m a l i z a t i o n s in the particle-hole system. The data for this experiment was collected in 32 h of r u n n i n g . However, even if considerably longer r u n n i n g times a n d more counters are used to increase the statistics, large errors will still r e m a i n in the F-factors because of the complexity of the spectra. We wish to t h a n k Dr. F. C. K h a n n a for his elucidation o f the properties of effective M1 operators a n d his suggestions regarding the i n t e r p r e t a t i o n o f o u r m e a s u r e m e n t s .

References t) 2) 3) 4) 5) 6) 7) 8) 9) 10) I1) 12) I3) 14) 15) 16) !7) 18) 19) 20) 21)

W. P. Alford., J. P. Schiffer and J. J. Schwartz, Phys. Rev. C3 (1971) 860 J. R. Erskine, Phys. Rev. 135 (1964) BII0 D. Proetel, M. Dost, H. J. Korncr and P. yon Brentamo, Nud. Phys. AI61 (1971) 565 Y. E. Kim and J. O. Rasmussen, Phys. Rev. 135 (1964) B44 T. T. S. Kuo, Nucl. Phys. A122 (1968) 325 C. Broude, Chalk River report AECL-3512 O. H~iusser, A. B. McDonald, T. K. Alexander, A. J. Ferguson and R. E. Warner, Phys. Lett. 38B (1972) 75 A. E. Blaugrund, Nucl. Phys. 88 (1966) 501 J. Lindhardt, M. Scharff and H. E. Schiott, Mat. Fys. Medd. Dan. Vid. Selsk. 33, no. 14 (t963) C. Broude, Can. J. Phys. 45 (1967) 3415 O. Hhusser, F. C. Khanna and D. Ward, Nucl. Phys. A194 (1972) 113 H. J. Rose and D. M. Brink, Rev. Mod. Phys. 39 (1967) 306 G. H. Fuller and V. W. Cohen, Nucl. Data Tables A5 (1969) 433 G. R. Hagie, R. C. Lange, eald J. T. McCarthy, Nucl. Phys. 84 (1966) 62 H. J. Korner, K. Auerbach, J. Braunsforth and. E. Gerdau, Nucl. Phys. 86 (1966) 395 P. H. Stelson, W. G. Smith and F. K. McGowan, Phys. Rev. 116 (1959) 167 H. A. Mavromatis, L. Zamick and G. E. Brown, Nucl. Phys. 80 (1966) 545, and refs. quoted therein A. Arima and H. Horie, Prog. Theor. Phys. 12 (1954) 623 W. G. Davies, private communication F. C. Khanna, private communication M. Harvey and F. C. Khanna, Nucl. Phys. A152 (1970) 588