Nuclear Physics A368 (1981) 189-200 @ North-Holland Publishing Company

THE 01* AND THE NEUTRON SCATTERING LENGTHS OF 3He

M. BAUMGARTNER, H. P. GUBLER, M. HELLER, G. R. PLATTNER, W. ROSER and I. SICK

fnstitut fir Ph.vsik, &&emit& Base& CH-4056 Basei, Switzerland

Received 4 May 1981

Abstract: We have investigated the excitation spectrum of 4He around E, = 20 MeV by inelastic scattering of 64 MeV a-particles from 4He. From an R-matrix representation of our data we have extracted level parameters for the a*. the first excited state of ‘Hc: EA = 20.29(2) MeV, To = 0.89(4) MeV and (f$, = 0.4(l). We are able to predict reliable low-energy P+~H and n + 3He s-wave phase shifts. Our results help to resolve a long standing ambiguity in the neutron scattering lengths of 3He, for which we obtain a, = 7.0(4)+i4.4448(9) fm and a, = 3.4(2) fm. We find the (x* to be an instructive example for the influence of thresholds on level parameters and line shapes.

NUCLEAR REACTIONS ‘He(a, a’), E = 64 MeV; measured a(&). 4He first excited E state deduced level parameters; 3He deduced neutron scattering lengths; 3He, 3H deduced

s-wave nucleon scattering phase shifts.

1. Introduction

A renewed interest in accurate level parameters for the CC*, i.e. for the first excited state of the 4He nucleus (excitation energy E = 20 MeV, J” = O’, 2’ = 0) is currently apparent. On the one hand, nuclear reactions resulting in unbound tl* ejectiles are beginning to be investigated [e.g., refs. l* 2)]. The accuracy of such studies depends on a good knowledge of the line shape and thus of the level parameters of the a* in the spectra. On the other hand, a recent investigation of the neutron scattering length of 3He [ref. 3)] has emphasized the fact that the available empirical informa- tion 3 - ‘) does not allow an unambigous separation into spin triplet and singlet contributions. Since the a* contribution dominates the singlet scattering length, an accurate set of level parameters will help to resolve this ambiguity. Last but not least, it must be realized that the a* is the first metastable nuclear few-body state which can be investigate by modern theoretical methods of the Faddeev type. Such “exact” calculations are progressing to the point where a description of the structure of the 01* is attempted 6). A comparison of reliable theoretkal with accurate empirical information promises to be very fruitful ‘).

189

scptcmbcr 1981

190 M. Baumgartner et al. / The u*

Since its discovery more than 25 years ago *), the CC* has been investigated several times with the aim of determining its excitation energy EL and width r. Studies of inelastic scattering of protons 9), cc-particles lo* 11) and electrons 12) from 4He as well as of several reactions ’ 3, have yielded conflicting values for these parameters, particularly for the width. This may be explained by the poor energy resolution of most of these experiments (i.e. of the order of the apparent width of the state). At a more fundamental level, however, the very method used to determine the parameters from the data is doubtful. In all of the cases where the CX* line was well delineated from the background, a symmetrical line shape was assumed, equating E2 with the energy of the maximum and r with the full width at half maximum of the peak observed to sit on an energy-dependent background. This procedure is entirely inadequate, because the level overlaps the two-nucleon decay thresholds of 4He at E(P+~H) = 19.815 MeV and E(n+ 3He) = 20.578 MeV. Thus the a* state density is strongly affected by threshold and barrier penetration effects, which lead to a pronouncedly asymmetric line shape 14). Part of the apparent “background” must in fact be interpreted as the high energy tail of the state.

In view of this we have undertaken ls) a new investigation of the excitation spectrum of 4He around E = 20 MeV by inelastic scattering of 63.97 MeV a-particles from 4He at Qlab N 15”. This process is quite selective in that neither the 2nd excited O-(T = 0) state nor any of the T = 1 levels can be formed. In addition, it has been shown in ref. lo) that the 3rd excited 2- (T = 0) state is weakly produced, if at all, by this process at this energy.

We have made a special effort to determine the a* state density with sufficient energy resolution and a careful energy calibration. We have extracted the a* level parameters from the data by an adequate method (R-matrix analysis), which takes the effects of both nucleon decay thresholds properly into account. We show that our results help to resolve the triplet-singlet ambiguity in the neutron scattering lengths of 3He and yield a reliable prediction for the low-energy s-wave p+ 3H and n + 3He scattering phase shifts.

2. Experiment

The experiment was performed at the SIN injector cyclotron. A large scattering chamber, equipped with a 2 10 pg/cm’ Ni entrance foil situated 11 cm upstream from the center, was filled with 50 Torr 4He. The incident beam (* 100 nA) was carefully aligned and positioned, so that the angular uncertainty was less than 0.03”. No beam integration was attempted, since absolute yields were of no interest to us. Spectra of inelastically scattered a-particles were recorded by a 1000 pm position-sensitive Si detector with 0.1 mm position resolution. This detector (9 mm wide, 6.5 mm high) was fixed at &,, = 14.6” and at a radius r = 230 mm. A front slit of 0.4 x 10 mm at radius r = 50 mm defined the target volume. A removable 150 pm AE Si detector

M. Baumgartner et al. 1 The u* 191

(15 mm diameter) situated before, and a 1000 pm Veto detector behind the position sensitive detector served to identify the scattered a-particles. The total energy resolution of the system for a-particles was calculated for the experimental conditions encountered during both the calibration and the actual a* data runs. For the calibra- tion runs (elastic scattering of a-particles from 4He and from 12C at 39.54 MeV, see below), this predicted energy resolution could be compared to the observed peak widths. Excellent agreement within a few keV of - 100 keV was obtained. As a consequence we trust the predicted energy resolution of 150 keV FWHM (table 1) for the a* data runs to be correct to within + 15 keV. This resolution compares very favourably with the apparent width of the prominent part of the a* state density, which our experiment shows to amount to - 450 keV.

For our measurements, we studied the energy region E - 19-24 MeV in 4He, corresponding to E, - 33340 MeV. The analog electronics was set up in duplicate to reduce the consequences of element failure during a data run after calibration. Two dimensional spectra were recorded in a PDP 1 l/45 on-line computer. Both the angle and the energy scales corresponding to the 40 position and 120 energy channels of the complete detection system were calibrated by methods relying solely on an accurate knowledge of nuclear masses. For both scales, a reference value and the (linear) dispersion were determined.

The angle calibration made use of the fact that for 63.97 Me incident a-particles the reactions 4He(a, ‘Li)p and 4He(a, ‘Be)n yield heavy particles only within a forward cone limited by tilab = 14.85O and 13.99”, respectively. These turning points of the kinematical loci could be observed within the f 1.5” angular acceptance of the position-sensitive detector (- channels 12 and 23), when the AE detector was removed. Their exact location and their separation calibrated the angular scale to +o.o3o.

The energy calibration was obtained by observing a scattering at E, = 39.54 MeV both from 4He and from a 12C foil. At this bombarding energy and at the scattering angle of our system, a-particles from the processes 4He(a, a) 4He, “C(a, a)r2C and

TABLE 1

Calculated contributions to the energy resolution and their estimated errors for the reaction “He(a, a’) a* at E, = 63.97 MeV and a scattering angle O,,, = 14.6O, in keV (FWHM)

dE of a-beam (foil + gas) Al9 of a-beam (foil + gas) energy straggling of Q’ (gas) angle straggling of a’ (gas) angle straggling of a’ (150 pm Si) detector and electronics (pulser) angular acceptance (slits + position detector)

overall resolution (lab) overall resolution (c.m.)

231 2 117*11

9* 1 46k 4 72+ 8

120*10 82+ 8

210*20 150+15

192 M. Baumgartner et al. 1 The c(*

“C(ct, d)12C*(4.44) fall into the observed energy range (- channels 75, 100 and 2( respectively). The location of the 4He line and the separation between the two 12( lines calibrated the energy scale to + 15 keV in excitation energy for the c1* run:

The absolute energy scale of our data was related to the accurately know energy 16) of the lowest T = 3 state in i3N E = 15.065 MeV. For this we calibrate1 the 110” analyzing magnet of the beam ;raLsport system by studying the T = resonance in 12C(p, p)“C, 12C(p, p’)i2C(4.44) and 12C(p, cr)‘B, initiated wit1 protons of - 14.23MeV and H: ions of - 28.46 MeV. The necessary magnet setting correspond to - 40 MeV and rr 57 MeV cr-particles, respectively, and are close to thl settings actually used in our experiment. We achieved a precision of - 10-4, whicl for the tl* data translates into an absolute uncertainty of f 3 keV in excitation energy

The accumulated counts in the calibrated angle-energy matrix containing the CI’ data were finally converted into an excitation spectrum of 4He by projecting the counts in each matrix element onto a 150-channel excitation energy scale. Thi: transformation also contained the proper conversion of the measured cross section: from the laboratory into the center-of-mass system. The resulting final excitatior spectrum is shown in fig. 1, where the (gaussian) experimental resolution and the nucleon decay thresholds of 4He are also indicated. The solid line is the best-fit R-matrix representation (see below) of the CI* peak. It is seen to be sitting on a slowly decreasing background which must be of instrumental origin, since there can be no excitation of 4He below the p+ 3H threshold at E = 19.815 MeV. The background is thought to be due to slit-edge scattering of the huge elastic contribution.

19 20 21 MeV E

Fig. I. The experimental excitation spectrum of “He. The solid line is the best-tit R-matrix representation, the dashed line shows the background contribution. The experimehtal resolution LIE,,, and the nucleon

decay thresholds of “He are also indicated.

M. Baumgartner et al. 1 The cc* 193

A pronounced asymmetry of the c(* is evident from fig. 1, vividly illustrating the inadequacy of fitting a simple lorentzian to the data. Taking the peak energy and the width of the prominent part (after unfolding the experimental resolution) as represen- tative parameters, we find EA - 20.2 MeV and r N 450 keV (FWHM). in good agreement with ref. lo). H owever, it is clearly impossible to define a level energy and width without a detailed understanding of the line shape.

3. R-matrix parametrization

A reasonable description of the CI* excitation spectrum is possible in the framework of R-matrix theory 17, i8), if the dec ay of the CI* is independent of its formation in the primary inelastic collision and if the c1* decay products have no final state interaction with the observed (inelastically scattered) a-particle. This assumption is justified, since for E, = 64 MeV and for an assumed GY* width of r = 0.9 MeV (i.e. r = 7x 1O-22 s), the separation between the CC* and the observed cl’ is about 25 fm at the instant of decay. In addition, the decay products have very small relative velocities and are therefore confined to a narrow forward cone around the a* mo- mentum.

At a given angle, the a* excitation spectrum can be factorized is) into a term f(Q, E,, E) describing the inelastic excitation of the CC*, and an c(* state density function p(E) as

N(R E,, E) = I./-(& E,, E)12p(E). (1)

Since other nearby levels of 4He are not appreciably excited by inelastic a-scat- tering at E, = 64 MeV at forward angles, as mentioned in the introduction, we shall henceforth assume that the observed spectrum is entirely due to this a* contribution except for an incoherent, linear, instrumental background. This assumption cannot be rigorously justified and must be judged a posteriori on the merits of its success.

At E, = 64 MeV we are far above the inelastic scattering threshold, so that j(lJ, E,, E) may be taken as independent of E over the width of the CC+ resonance. It may be replaced by an (unknown) constant, which is absorbed into the unknown absolute normalization of the empirical spectrum. The CC* state density function p(E), on the other hand, is given i7,r8) by an R-matrix expression of Breit-Wigner type, which describes the decay of the LX* (J” = O+) into the two channels with “+rZJ = ‘S,(p+ 3H), ‘S,(n+ 3He). These channels have their thresholds just below and just above the LX* energy. We obtain, for a fixed a-energy and angle,

N(E) = [E,+&f;!+$ri(E) +(c2+c3E)*

The constants cl, c2, c3 describe the normalization and the linear background. The

194

width T(E) is given by

M. Baumgariner et al. / The ct*

T(E) = &(E)+ 1”,(E) = ?I:[qE)+P,(E)]. (3)

Here P,(E) and P,(E) are the penetrabilities in the “S,(p+ 3H) and ‘S,(n+ 3He) channels. We have chosen the reduced nucleon widths to satisfy y% = y: = 3~; as an approximate way of expressing charge independence of the two mirror decay channels.

The denominator contains the c1* level energy EL and the level shift

A(E) = -+&[S,(E) + S,(E)- B]. (41

The boundary condition B is chosen such that b(E,) = 0, i.e. the shift factors S are just compensated at EA. Thus, the level shift depends on the energy derivatives of the shift factors only.

The ‘S,(d +d) channel opens at E = 24 MeV, so that in our energy interval it cannot contribute to T(E), but only to d(E). However, so far below threshold the shift factor S,(E) is essentially constant. Therefore, we have neglected the deuteron channel despite the sizeable partial (d+d) width, which the c1* might conceivably have 19).

As usual it is true that the calculated excitation spectrum is virtually inde~ndent of the choice of the “singlet” interaction radius rs, provided that appropriate values are chosen for E, yi and cI. However, the formal dependence of our “one level, two channels” expression on this choice still presents some problems. in R-matrix theory, the wave functions are normalized to unity in the interior region (r c rJ. The wave function of a state like the c1*, sitting close to the threshold of a decay channel (n+ 3He) with a low barrier (1 = 0, Z,Z, = 0), extends very far beyond rs. Therefore, the II* wave function will be normalized to a value much greater than unity. This leads to very large values for the widths yi and I’ as well as for the ratio Sk of yi to the “Wigner limit” 3i?/2& (p reduced mass). Lane and Thomas “‘) have already shown that this can be taken into account by renormalizing the widths according to

b4)o = Yiw ‘7

r,(E,) = w,w I, (7)

The no~alized widths correspond to a wave function no~ali~tion of unity in a volume which includes the open p+ 3He channel out to the classical turning point,

M. Baumgartner et al. / The a* 195

and all of the bound n + 3He channel. Their values are found to be much less depen-

dent on the choice of interaction radius rs, so that they represent the physical proper-

ties of the a* (lifetime, nucleon spectroscopic factor) in a more realistic manner.

For a comparison with existing data, we also want to calculate the p+ 3H and

n+ 3He elastic s-wave scattering amplitudes in the framework of our single-level

R-matrix “model”. In such calculations the singlet interaction radius Y, enters in yet

another fashion, since the scattering from a hard sphere with radius rs must now be

added to the c1* contribution. Expressed in terms of the IS, phase shift, this amounts

to adding the corresponding real hard sphere phase shift, leaving the imaginary

part unaffected. At the n + 3He threshold, it is customary to describe the ‘S, elastic

neutron scattering amplitudes by the complex singlet scattering length a,, for which

we obtain :

Rea, = y, l- ------

[

+I@, + W”) - E”) (E, + W,) - -q2 +$@,2(E,) 1

= r-,+/l, (9)

(10)

where En denotes the n+ 3He threshold energy. The real part of the singlet scattering

length [eq. (9)] will be very sensitivie to the choice of interaction radius due to the

hard sphere term r,, which must be added to the single level term /1.

4. Results for the a*

We have used the formalism outlined above to interpret our data on the a* state

density. We have restricted the fits so that they reproduce the accurate empirical

information on the imaginary part of the singlet n + 3He scattering length, available

from absorption experiments with thermal neutrons on 3He [refs. “* 22)]. This

requires that 22)

Im a, = 4.4448 fm. (11)

The uncertainty of this value lies between kO.06 (admitting spin triplet contributions)

and f0.0009 (our choice, assuming that there is no absorption in the spin triplet

state, i.e. Im a, = 0).

After folding in the gaussian experimental resolution of 150 keV FWHM, we have

been able to reproduce the data with interaction radii in the range 2.5 fm 5 rs s 5.5

fm without appreciable differences in the quality of tit. The results for E,, for the

widths and for the contribution /i of the a* state to Re a, are shown in table 2. Even

with so large a variation of the interaction radius, the level energy E,, the renorma-

lized width To and the contribution LI of the a* to the real part of the singlet scattering

196 M. Baumgartner et al. / The u*

length are well defined. The sizeable variation of the dimensionless reduced width (Q, is of no consequence in view of the uncertainty commonly associated with the interpretation of this quantity.

In view of this insensitivity of the parameters, we take the standard choice for the interaction radius

rs = 1.4(Aj+At) & 3.4 fm (12)

and determine the final set of level parameters of the CI*. The errors are taken to include the uncertainty of both the experiment and the interpretation. We obtain

E, = 20.29 f0.02 MeV, (13)

To = 0.89 f 0.04 MeV, (14)

(e& = 0.4fO.l (15)

We emphasize the fact that the width To is an averaged quantity. Due to the peculiar, asymmetric line shape (figs. 1 and 3), a large fraction of the total a* density lies far above the apparent a* position (see also sect. 6).

5. Results for p+ 3H and n + 3He scattering

From our parametrization we can obtain predictions for low-energy p+ 3H and n + 3He s-wave scattering, if we add a reasonable hard-sphere scattering term, i.e. choose a realistic interaction radius as outlined in sect. 3.

This choice is best discussed in terms of empirical information by studying the s-wave hard-sphere scattering contribution where it dominates. Since they are not affected by the u* contribution (J” = O+, T = 0) nor by any other known excited states, all s-wave nucleon-trinucleon scattering processes with J = 1 and/or T = 1 are suitable for that purpose, if charge independence is approximately valid. A review of the corresponding empirical low-energy phase shifts C3S, p + 3H, refs. 23, 24) ; 3S, n+3He, refs. 24, 25); ‘S,, 3S, p+3He, refs. 26* 27); ‘So, 3S, n+3H, ref. 28)] and of the empirical IS, and 3S, n+ 3H scattering lengths 29) allows the very con- servative conclusion that

2.5 fm < rs, r, < 4.5 fm, (16)

0.8 < (r,/rJ < 1.2. (17)

Recent theoretical values 6, agree with these limits, which also encompass the standard choice r, = rs = 3.4 fm [eq. (1 O)].

Combining these empirical limits for the hard-sphere terms with our results (table 2) for the contribution n of the a* level to Re a,, we obtain limits on the real

M. Baumgartner et al. 1 The rl* 197

at

fm

5

4

3

1 3 5 7 fm a,

Fig. 2. Empirical limits on the real parts of the n+3He scattering lengths. (a): CT&, [ref. ‘)I;

(b): gel [ref. 4)]; (c): a, [ref. 3)]; (d): this work, 0 best compromise (see text).

parts of the n+ 3He scattering lengths. This is shown in fig. 2, where previous empiri- cal information 3- 5, is also indicated. We find excellent agreement between the loci allowed by the various sources of information. Evidently our results completely resolve the long standing ambiguity given by the almost tangent position of the three previous empirical loci. Of course it has been realized earlier 22) that the presence of the a* level below the n+ 3He threshold requires Re a, > Re a,. but no accurate determination of the contribution A has been available until now.

The standard choice for the interaction radii leads to the values indicated by the full circle in fig. 2. These constitute a very reasonable compromise between the various empirical limits, so that we choose them as current best values for the n + 3He scattering lengths. Together with the information [eq. (1 l)] on Im a, we obtain

a, = 7.0(4) + i4.4448(9) fm, (18)

a, = 3.4(2) fm. (19)

the two quantities by the accurate value 3, for the coherent must be taken into account when they are used together. Our the errors with those of the first set of ref. 3, and of set A of

The correlation of scattering length Q, values agree within ref. 6), but our uncertainties are considerably smaller.

With the standard choice for the interaction radii and our a* level parameters (table 2), we can now calculate as a function of energy the is, and 3S, phase shifts for both p+ 3H and n+ 3He scattering as well as the s-wave amplitudes for the 3He(n, P)~H reac tion. In the 3S, states we have an S-matrix element of unit modulus,

198 M. Bawngartner et al. 1 The CY*

TABLE 2

Results of the R-matrix fit to the a* state density

r s E* ri (Y,& (@d r,(4) A

2.5 20.28 350 3.1 0.28 0.85 3.75 3.5 20.29 55 2.6 0.38 0.90 3.60 4.5 20.30 26 2.0 0.48 0.91 3.45 5.5 20.31 16 1.5 0.57 0.92 3.35

All energies are in MeV of 4He excitation. The interaction radius rS and the quantity A (see text) are given in fm.

i.e. a real phase shift. The rS, states are coupled via the reaction channel, so that they must be represented by a unitary and symmetric 2 x 2 matrix

S,, = qezidl, (20)

S,, = ylezis2, (21)

S,, = Jwjei(d1+62) = S,,, (22)

which is specified by the three real parameters 6,, 6, and q. In fig. 3 we show these phase-shift parameters as well as the tl* state density.

In the energy interval from the p + 3H threshold up to 2 MeV proton bombarding energy, the sum of CI* plus hard-sphere contributions should provide a reliable description. Our results agree well with what little is known from direct experimental observation of the corresponding s-wave scattering and reaction processes at non- zero energies 23-2s). In view of the fact that they have been deduced by consideration of all the available experimental information, they represent the best empirical description of the J” = O+, l+ and T = 0 parts of the A = 4 system around 20 MeV excitation.

6. Conclusions

We have performed an accurate measurement of the state density of the CC*, the first excited state of 4He. By fitting our data to an R-matrix expression, we have for the first time been able to deduce reliable level parameters for this state.

The influence of the nearby nucleon decay thresholds completely dominates the CC* state density, which falls off very slowly with increasing excitation energy. The “no barrier” ‘S, n + 3He channel leads to such an asymmetric line shape that a large fraction of the total CC* strength lies far above the apparent a* position. In fact, our best-fit R-matrix parameters predict that only 8% of the total strength is found below E = 21 MeV, only N 50% below E = 100 MeV excitation! These values are rough

M. 3aumgartn~~ et al. / The a* 199

0 a5 1.0 1.5 Mev Ej,,,a,

Fig. 3. The state density p(E) for the tl* and the s-wave nucleon-%ie scattering phase shifts, deduced from the best-fit R-matrix representation.

estimates because of the limited energy range of our data and because of the doubtful validity of the R-matrix representation over so large an energy interval. However, they show that quantitative (a, a*) spectroscopy ‘,2) must be viewed with caution.

From our a* level parameters we have also calculated reliable low-energy s-wave p+ 3H and n-t 3He scattering phase shifts. In particular, we have been able to resolve the long standing empirical ambiguity concerning the separation of the neutron scattering length of 3He into spin triplet and singlet contributions. Our data confirm that the real part of the scattering length is just about twice as large in the singlet as in the triplet spin state.

We gratefully acknowledge the support by SIN and by the Swiss National Science Foundation.

200 M. Baumgartner et al. 1 The a*

Note added in proof: In two recent publications (Alfimenkov et al., Yad. Fiz. 33 (1981) 891 and Borzakov et al., JINR preprint P3-81-305, Dubna, 1981), accurate new data on the total, the elastic and the reaction cross sections of neutrons on 3He for 0.2 < E,, < 140 keV are reported. Our predictions agree quantitatively with these data and reproduce them as well as do the author’s own calculations.

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