Deuteron emission following 6He beta decay

Physics Letters B 322 (1994) 17-21 North-Holland

PHYSICS LETTERS B

Deuteron emission following 6He beta decay F.C. B a r k e r Department of Theoretical Physics, Research School of Physical Sciences and Engineering, Australian National University, Canberra ACT 0200, Australia

Received 10 November 1993; revised manuscript received 9 December 1993 Editor: C. Mahaux

The measured spectrum shape and branching ratio for deuteron emission following 6He beta decay are reproduced in a onelevel R-matrix approximation, by including an external contribution to the beta decay matrix element that is calculated using wave functions with the correct asymptotic forms.

1. Introduction Riisager et al. [ 1 ] and Borge et al. [ 2 ] have measured the energy spectrum and branching ratio of deuterons observed following the beta decay o f 6He. The branching ratio for deuterons with energies greater than about 360 keV is given as ( 7 . 6 + 0 . 6 ) X 1 0 -6 [2]. Various models have been used to calculate the deuteron emission and, although most give approximate consistency with the observed shape o f the spectrum, the calculated branching ratios vary widely and generally are much too large. A standard one-level R-matrix calculation [ 1 ] gave a branching ratio exceeding the above measured value by well over an order of magnitude. A two-level model was constructed [ 1 ] that could give agreement, but only by using parameter values that were unacceptable for other reasons. Models that assumed beta decay directly into continuum states [ 1,3,4 ] also gave branching ratios that were too large. Two calculations have obtained agreement with experiment. In the standard R-matrix calculation, the 6He beta decay produces an unstable state of 6Li which subsequently breaks up into an alpha plus deuteron; Borge et al. [ 2 ] extended the R-matrix calculation by including a contribution in which the initial breakup o f the 6He into an alpha plus dineutron is followed by beta decay o f the dineutron to a deuteron. The two contributions tend to cancel and, by adjusting the strength o f the additional contribution, they were able

tO fit the observed branching ratio and also obtain a reasonable spectrum shape. In a semi-microscopic model using three-centre harmonic oscillator basis functions, Baye et al. [ 5 ] found cancellation between the internal and halo contributions to the G a m o w Teller ( G T ) matrix element. This cancellation is very sensitive to the description o f the halo and to the effective interaction, and Baye et al. found a particular choice that gave agreement for both the branching ratio and spectrum shape. This letter points out that agreement can also be obtained in a one-level R-matrix approximation; one would expect this to be possible because o f the known structure o f the 6Li 1 + states [ 1 ]. The assumptions on which standard R-matrix approximation; one would expect this to be possible because of the known structure o f the 6Li 1 + states [ 1 ]. The assumptions on which standard R-matrix theory [ 6 ] is based are not satisfied for reactions involving photons, or for beta decay. Lane and Thomas [ 6 ] dealt with the inclusion o f photon channels in R-matrix theory and showed that photons play a role in nuclear reactions similar to that o f heavy particles, except that the external region may contribute to the electromagnetic transition matrix elements. In a similar way beta decay may be described in R-matrix theory, with additional contributions to the beta decay matrix elements coming from the external region [ 7 ].

0370-2693/94/$ 07.00 © 1994 Elsevier Science B.V. All rights reserved. SSDI 0370-2693 (93)E1 555-C

17

Volume 322, number 1,2

PHYSICS LETTERSB

2. Formulae

From the many-level, many-channel standard Rmatrix formula for beta decay, as given for example by Barker and Warburton [ 8 ], we write the deuteron energy spectrum in the one-level (6Li ground state), one-channel (4He + d, l = 0) approximation, with GT transitions only, as

Here E i ( r ) = u i ( r ) / u i ( a ) , where ui(r) is the exponentially-decaying function corresponding to the energy of the aHe ground state in the 4He + 2n channel ( - 0 . 9 7 5 MeV). The 6Li final state ~f(E) is taken as pure T=0; in the internal region it is ~U(l+ 00), independent orE, and in the 4He + d channel 1/2

N(E) =fa(E)Po(E)

(r>~a) .

2 2 Bao(E)yad X [Ea - y~d{So(E) -Bo} - E ] 2 + [y~aPo(E)]2, (1) where the energy dependence is shown explicitly. Here Ea is the eigenenergy and y2d the deuteron reduced width of the 6Li ground state,fa is the integrated Fermi function, Po and So are the penetration factor and shift factor, both depending on the channel radius a, and Bo is the boundary condition parameter. In standard R-matrix theory, as used for example by Riisager et al. [ 1 ] and Barker and Warburton [ 8 ], the G T matrix element Bao is taken as energy-independent; here it is energy-dependent because a contribution from the external region (r> a) is included, and this uses wave functions with the correct asymptotic forms. In the same approximation, the number of decays to the 6Li ground state is [ 8 ]

~f#(Eg)B~ (Es)

(2)

Ng = 1 +y~d(dSo/dE)E, ' where E~= - 1.475 MeV is the ground-state energy. The GT matrix element is given by [ 7 ] ( ~ g f ( E ) II2 6=, a(k) t+ (k) I[~i ) Bxo(E)=c [(7,f(E)I~.,r(E))(7,iI~)]I/2,

(3)

where c is a constant, the value of which is not of concern here since we are interested only in the shape of the deuteron spectrum and the branching ratio. The 6He initial state ~i is taken as pure T = 1, and is written in the internal region as ~( J~TMr= 7"(0 + 1 - 1 ) (normalized over the internal region) and in the 4He+2n, l = 0 channel as [6,7] ( 2 ) 1'2

~i =

1

10 February 1994

(5)

At the energy of the 6Li ground state, uf is t h e / = 0 Whittaker function corresponding to the energy E v and for E > 0 it is

uf(E, r) =Fo(E, r) cos ~o (E) + Go(E, r) sin 8o (E) (r>~a) ,

(6)

where Fo and Go are the regular and irregular Coulomb functions and ~o is the 4He+ d s-wave phase shift. 0i and Of in eqs. (4) and (5) are constant dimensionless reduced width amplitudes; Of is related to the reduced width Y~d by y2d

h2 t~2

(7)

= ma 2 vf ,

where M is the reduced mass in the 4He + d channel. Eq. (3) can now be written in a form similar to that appearing in photon reactions (see eq. (40c) of Thomas [9], and eqs. (24) and (26) of Barker and Ferdous [ 10 ] ):

Bao(E) =c[NiNf(E) ] - l / 2 [ M i f + 2OiOfJif(E)

18

(8)

where Ni = (~il ~i) is independent of E and Nf(E) = ( ~ f ( E ) I ~f(E) ). For E = E v one has from eq. (IV.7.4)ofref. [6] Nf(E) = 1 +7~d alSo dE '

(9)

or from eq. (XIII.3.31) ofref. [6] oo

Nr(E) = 1 + _202 / E](E, r) dr . a

d a

(10)

These are equivalent because [ 6,10 ]

+

OiEi(r) r g.q/2.(O 1 -- 1 ) i u2(r) dr= ( l + y2--~-) i u2(r) dr

(r>~a) .

],

(4)

0

0

(11)

Volume 322, n u m b e r 1,2

PHYSICS LETTERS B

for a bound state. From the similarity between beta decay and radiative transitions, and the formulae derived for the latter in ref. [ 10 ], we also use eq. (9) for E > 0. Also 6

J~if = (

~V(l+0 0 ) 1 1

10 February 1994

Table 1 Parameter values from one-level R-matrix fits to 4H¢-k d s-wave phase shift and 6He beta decay deuteron branching ratio.

a (fm)

r~o (MeV)

O~

c2

tr(k)t+(k)llW(O+ 1 --1))ira ,

case (a)

case (b)

0.521 0.537 0.340 0.098

3.63 1.42 0.720 0.243

k= 1

(12) which is independent of E, and oo

J i f ( E ) ~-~m i f a1 ~

Ei(r)Ef(E, r) dr

3.0 3.5 4.0 4.5

7.50 2.48 0.914 0.213

2.15 0.969 0.466 0.138

(13)

a

where 2

m i f = ( g d ( 1+

00)11

a(k)t+(k)ll~u2n(O+ 1 - 1 ) ) . k=l

(14) Then the GT matrix element can be written in the form

BxG(E)=c~

1+c2(1/a) f~Ei(r)Ef(E, r) dr [l+y2ddSo(E)/dE],/2 ,

mental values of Jo, which have been given by McIntyre and Haeberli [ 11 ] for E ¢ 6.7 MeV. Since we require wave functions that are accurate at low energies (E~<2 MeV), we choose 7~d to fit j~xp (2.0 MeV) = 109 ° [ 11 ]. The constant Cl in eq. ( 15 ) is determined by fitting the experimental value ofNg [ 2 ], taken as 362/(7.6X 1 0 - 6 ) = 4 . 7 6 X 107, and the pa-

(15) 180

where c~ and c2 are constant parameters, with ~ mif C2 = ZUiUf M i f •

160

(16)

~x~x

The branching ratio for emission of deuterons with energies greater than the cut-off energy E~o ( = 360 keY) is

BR= ~f N(E)dE Ng

140120

xx~

xx

(17) N

1.5Eco

The phase shift Jo, which appears in eq. (6), is given by 72dPo(E) Jo (E) = arctan Ea -- 9'2d{So (E) - B o } - E -~o (E),

~x

80

40

(18)

where - 0 o is the hard-sphere phase shift, which is also a function of a.

4.5 ~ - .

20

i 1 3. Procedure and results

We choose Bo=So(E~), so that E~=E~. For given a, the value of y2d can be obtained by fitting experi-

J 2

i 3

i 4

i 5

i 6

E {MeV) Fig. 1. The 4 H e + d s-wave phase shift Jo as a function o f channel energy E. The points are experimental values [ l l ] and the curves are one-level R-matrix fits with the indicated values of the channel radius a.

19

Volume 322, number 1,2 200

,

100

/"4.5 /

50

/

PHYSICS LETTERS B ,

10 February 1994

trum N corresponding to the parameter values of table 1 are shown in figs. 1 and 2 respectively, for a = 3.0 and 4.5 fm; for intermediate values of a, the calculated values tend to lie between those shown.

-,,

3.0/-~ ~ '\,

/ 4. Discussion

i 20 / iIiI// 10

\\

I/

0

0.2

0.4

0.6

0.8

1.0

1.2

E~ (MeV)

Fig. 2. The deuteron spectrum N following 6He beta decay as a function of deuteron energy Ed. The points are experimental values [ 2 ] with statistical error bars added [ 5 ], and the curves are one-level R-matrix calculations that fit the measured branching ratio [2]. The value of a is indicated. Solid curves are case (a), dashed curves case (b). rameter c2 is adjusted to fit the observed branching ratio [2] o f 7.6X 10 -6. Parameter values so obtained are given in table 1. Galonsky and McEllistrem [12] had previously found that the channel radius in such one-level Rmatrix fits to Jo was restricted to the range 3-5 fm, because smaller values of a led to y2o values exceeding the Wigner limit and larger a required negative values o f 72d. For a = 3 . 5 fm, they obtained y2d= 1.96 MeV. For each value o f a, table 1 shows two values of c2, corresponding to the internal contribution to B ~ ( E ) for E > 0 being larger (case ( a ) ) or smaller (case ( b ) ) than the external contribution. The values of c2 are not very sensitive to the value of the branching ratio fitted; e.g. for a = 3 . 5 fm, the ___0.6 × 10-6 experimental uncertainty in the branching ratio [2 ] leads to uncertainties in the values o f c2 o f -T-0.014 and _+0.024 for cases (a) and (b) respectively. The fits to the phase shift Jo and deuteron spee20

The agreement of our calculated deuteron spectrum shape with experiment, shown in fig. 2, is comparable with that obtained in previous calculations [2,5 ] that also fitted the measured branching ratio. There are other similarities between these calculations. Although our internal and external contributions to the G T matrix-element have the same sign for decay to the 6Li ground state, there is considerable cancellation between them in the continuum region. This is because the overlap integral a - l f a E i ( r ) E f ( E , r)dr is positive for E = E 8, when both initial and final states are bound, but negative for E > 0, provided that uf(E, r) is given by eq. (6) with Jo fitting the experimental values. This cancellation is similar to that found by Baye et al. [ 5 ] in their semi-microscopic model. Their best fit (with interaction V2) corresponds to our case (b), since their halo contribution is dominant. Also our external contribution could be described as the beta decay from an initial state in which the 6He is broken up into 4He + 2n, to a final state of 4He + d; it therefore resembles the additional contribution o f Borge et al. [2]. We note that the formula at the top o f p . 672 o f ref. [ 2 ] can be written, using our notation, 0i

mif = ~ i f 0 f

•

( 19 )

F r o m this equation and eq. (16), we therefore have ~ . ~ 0i

mif

C2

(20)

Of M i f - - 203 "

Borge et al. fitted the data with ~ 0 . 6 5 , which corresponds to our case (a) because their standard contribution is larger than their additional contribution. The values in table 1 give ~<0.4 for case (a) and ~ 0.8 for case (b); however, the parameter ~ enters the bcalculations of Borge et al. through a formula of different type from eq. ( 15 ) that involves c~ here. The quality of our fits to the data does not give a clear-cut preference for a particular value of the

Volume 322, number 1,2

PHYSICS LETTERS B

channel radius a. The low-energy fits to the phase shift (fig. 1 ) are acceptable over the range a = 3.0-4.5 fm, with a tendency to favour the smaller values coming from the high-energy behaviour. The spectrum shape (fig. 2) suggests that case ( b ) with the larger values o f a should be rejected. The p a r a m e t e r value that give the best fits for each value o f a are also o f interest. I f the a + d description o f the 6Li g r o u n d state is c o n t i n u e d into the internal region, with radial wave function r - l u f ( r ) , then one has [6,10] 2 r

(21)

~ u2(a)(a/2) '

with the spectroscopic factor S f = 1. I f uf(r) is taken as a 2s (one n o d e ) wave function in a W o o d s - S a x o n potential with reasonable p a r a m e t e r values, one finds 02 ~ 1. F r o m table 1, this favours a ~ 3.5 fm. I f 6 H e is similarly described in its internal region as a + 2n with S i = 1, one has 0i~ Of. In this potential m o d e l description, as discussed for e x a m p l e by D e s c o u v e m o n t a n d Leclercq-Willain [ 3 ], strong correlation between the two neutrons to form a d i n e u t r o n would also give rnif ~Mif, leading to ~ 1 a n d supporting case ( b ) . O n the other hand, if the two neutrons are weakly correlated so that mif<
10 February 1994

favouring case ( a ) . G r e a t e r precision in the deuteron spectrum shape is p r o b a b l y required before a clear preference for case ( a ) or case ( b ) can be established.

References [ 1] K. Riisager, M.J.G. Borge, H. Gabelmann, P.G. Hansen, L. Johannsen, B. Jonson, W. Kurcewicz, G. Nyman, A. Richter, O. Tengblad and K. Wilhelmsen, Phys. Lett. B 235 (1990) 30. [2] M.J.G. Borge, L. Johannsen, B. Jonson, T. Nilsson, G. Nyman, K. Riisager, O. Tengblad and K. Wilhelmsen Rolander, Nucl. Phys. A 560 (1993 ) 664. [ 3 ] P. Descouvemont and C. Leclercq-Willain, J. Phys. G 18 (1992) L99. [4] M.V. Zhukov, B.V. Danilin, L.G. Grigorenko and N.B. Shul'gina, Phys. Rev. C 47 (1993) 2937. [ 5 ] D. Baye, Y. Suzuki and P. Descouvemont, Universit6 Libre de BruxeUespreprint PNT/8/93 ( 1993 ). [6] A.M. Lane and R.G. Thomas, Rev. Mod. Phys. 30 (1958) 257. [7] F.C. Barker, NucL Phys. A 537 (1992) 134, 147. [ 8 ] F.C. Barker and E.K. Warburton, Nucl. Phys. A 487 (1988 ) 269. [9] R.G. Thomas, Phys. Rev. 88 (1952) 1109. [ 10] F.C. Barker and N. Ferdous, Aust. J. Phys. 33 (1980) 691. [ 11 ] L.C. McIntyre and W. Haeberli, Nucl. Phys. A 91 (1967) 382. [ 12] A. Galonsky and M.T. McEllistrem, Phys. Rev. 98 (1955) 590.

21