PHYSICAL REVIEW C MARCH 1997VOLUME 55, NUMBER 3

Spectroscopic factors for bounds-wave states derived from neutron scattering lengths

P. Mohr, H. Herndl, and H. OberhummerInstitut fur Kernphysik, Technische Universita¨t Wien, Wiedner Hauptstraße 8-10, A-1040 Wien, Austria

~Received 20 November 1996!

A simple and model-independent method is described to derive neutron single-particle spectroscopic factorsof bounds-wave states inA11Z 5 AZ^n nuclei from neutron scattering lengths. Spectroscopic factors for thenuclei 13C, 14C, 16N, 17O, 19O, 23Ne, 37Ar, and 41Ar are compared to results derived from transfer experimentsusing the well-known disorted wave Born analysis and to shell model calculations. The scattering length of14C is calculated from the15Cg.s. spectroscopic factor.@S0556-2813~97!04503-2#

PACS number~s!: 21.10.Jx, 24.10.Eq, 25.40.Dn, 27.30.1t

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Spectroscopic factors~SF’s! are an important ingredienfor the calculation of direct transfer reaction cross sectionthe distorted wave Born approximation~DWBA! and capturereaction cross sections in the direct capture~DC! model.Usually, SF’s can be determined experimentally by the raof the measured transfer reaction cross section to the csection calculated with the DWBA,

C2Si5s iexpt/s i

DWBA , ~1!

for each final statei . In the case of neutron transfer main(d,p) reactions were analyzed to determine the neutsingle-particle SF. This determination has relatively laruncertainties because the optical potentials of both thetrance and exit channels have to be known accurately freliable DWBA calculation. Usually one obtains SF’s wiuncertainties of up to 20%. However, in many cases systatic deviations exceeding the claimed uncertainties canfound when the results of various experiments@differenttransfer reactions like (d,p), (4He,3He!, (7Li, 6Li !, etc., atdifferent energies# are compared~see, e.g., Table 8 of Ref@1# or Table II of Ref.@2#!.

Recently, our group showed that a model-independmethod exists to extract SF’s from the thermal neutron cture cross section@3#:

C2Si5s iexpt~nth ,g!/s i

DC~nth ,g!. ~2!

This method has very limited uncertainties because at tmal energies the neutron optical potential can be adjuproperly to the scattering length. However, because the tmal (n,g) cross section is dominated by incomings wavesandE1 transitions, this procedure works well only for bounp waves in the residual nucleus.

In this work we present a simple and model-independprocedure for the extraction of SF’s of bounds waves fromthe scattering lengthb. In this work we use the free nucleascattering lengthb, which is related to the bound scatterinlength by b5(bbound2Zbne)•A/(A11) with the neutron-electron interaction length bne5(21.3860.03)31023

fm @4,5#. These SF’s are very important for the calculationthe (n,g) cross section at astrophysically relevant energiethe order of several keV where transitions from incomingp

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waves to bounds and d waves become comparable to thtransitions from the incomings wave to boundp waves@6,7#.

The method can be applied to light and intermediateclei with only one bounds wave or a strongs-wave stateclose to the neutron separation threshold. In these casescattering length can be interpreted as the very brpositive-energy wing of thes-wave subthreshold state. Thcomparison of the calculated width assuming a singparticle configuration and the experimental width of this suthreshold state leads to the SF

C2S5Gexpt/Gspcalc. ~3!

This calculation is performed in the following way.First, the wave function of the subthreshold state is cal

lated using a neutron-nucleus optical potential. The potenstrength~parametersV0 or l; see below! is adjusted to re-produce the binding energy of the bound state~taking intoaccount the Pauli principle byq52n1 l whereq, n, and lare the oscillator, radial node, and angular momentum qutum numbers!. In this work both Woods-Saxon~WS!

VWS~r !5V0@11exp~r2R/a!#21, ~4!

with R5R0AT1/3, R051.25 fm, anda50.65 fm, and folding

potentials@8–10#

VF~r !5lE E rP~r P!rT~r T!veff~s,r,E!d3r Pd3r T ~5!

were used; the results practically do not depend on the csen parametrization of the optical potential. In this sensemethod is model independent.

Second, we calculate the single-particle scattering lenbspcalc and the widthGsp

calc from the optical potential which wasadjusted to the bound state energyEB ~note thatEB,0). Thescattering phase shiftd l50(E) is related to the scatteringlength b and the width of the resonance by the followinwell-known equations:

k•b52sin@d l50~E525 meV!# ~6!

and

1591 © 1997 The American Physical Society

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1592 55BRIEF REPORTS

TABLE I. Spectroscopic factors of bounds-wave states of13C, 14C, 16N, 17O, 19O, 23Ne, 37Ar, and41Ar derived from the scattering length, from different transfer experiments, and from the shell mode

Nucleus Jp Ex ~keV! q52n1 l C2S a C2Sexpt Ref. C2Scalc Ref.

13C 1/21 3089 2 0.9666 0.015 0.65 – 1.2 @18–20# 0.85 a14C 12 6094 2 0.8946 0.020 0.43 – 0.87 @21,22,1# 0.76 – 0.85 a,@23–25#14C 02 6903 2 0.9316 0.020 1.02 @21# 0.96 – 1.00 a,@23–25#16N 02 120 2 1.0126 0.020 '0.46 @26# 0.95 @16#16N 12 397 2 0.9696 0.020 '0.52 @26# 0.96 @16#17O 1/21 870 2 0.9896 0.010 0.45 – 1.96 @27–32,2# 1.0 a,@2#19O 1/21 1472 2 0.9196 0.020 0.86 –'1 @2,33# 0.7 – 0.9 a,@2,34#23Ne 1/21 1017 2 0.6986 0.030 0.37 – 0.70 @35–37# 0.654 a37Ar 1/21 8789 4 0.5306 0.010 – –41Ar 1/21 6098 4 0.1806 0.010 – –

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tan@d l50~E!#5G~E!

2~EB2E!, ~7!

wherek is the wave number of thes wave atE525 meV.Third, the experimental widthGexpt is calculated from

Eqs.~6! and~7! using the experimentally determined scatting lengthbexpt @11,12#. The SF which is a measure of thsingle-particle strength is calculated from Eq.~3! by the ratioof Gexpt andGsp

calc at the thermal energyE525 meV.Our new results are listed in Table I. The main uncerta

ties in this procedure are given by the experimental unctainties of the experimental scattering lengths. The uncertties from different potential parametrizations are practicanegligible. The results agree well with different transfer eperiments.

The theoretical SF’s were calculated from the shell mowith the codeOXBASH @13#. Since we need the spectroscopfactors for a 2s1/2 transition, one-particle one-hole excitations have to be taken into account for the C isotopes.used the interaction WBN of Warburton and Brown@14# forthis purpose. For the16N states we took the interaction ZBMI @15#; the results for16N were already published in Re@16#. The spectroscopic factors for the O and Ne isotowere calculated with the Universalsd-shell interaction ofWildenthal @17#. The shell model SF’s agree well with thexperimental SF’s derived from scattering lengths.

In the case of 14C5 13C^n the SF’s for two bounds-wave states (Jp502,12) can be determined, because thprocedure can be applied to both channel spinsS50 andS51. The relevant scattering lengths can be derived frthe coherent and incoherent scattering lengths on13C. The

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same arguments hold for the case16N5 15N^n. However,for the nucleus16N the agreement between the experimenSF’s derived from our method and from a (d,p) transferexperiment is quite poor whereas the theoretical SF’s agwell with our new SF.

In the cases of37Ar5 36Ar^n and 41Ar5 40Ar^n sub-threshold resonances atE5210 keV (Ex58778 keV! andE521 keV (Ex56098 keV! @4# determine the scatteringlengths. Unfortunately, the relatively small SF’s of thestates were not determined experimentally@38,39#; a calcu-lation of these SF’s is very difficult because the neutronlocated in the 3s1/2 shell.

Finally, for the system15C5 14C^n we can invert theprocedure to predict the experimentally unknown scatterlength of 14C from the SF’s of the 15C ground state(1/21). The SF is well known both from transfer experments @40,41# and from the shell model: We adopC2S51.060.05. The resulting scattering lengthb57.25760.369 fm. An experimental verification of thiprediction is desirable.

In conclusion, this method for the calculation of SFworks well for several light and intermediate nuclei. Becauof the model independence, the SF’s presented in this wcan be used as a benchmark for SF’s derived from tranreactions or determined by shell model calculations.

We would like to thank Dr. H. Beer, Dr. G. Staudt, and DV. Kolle for stimulating discussions during the preparatiof the paper. This work was supported by Fonds zForderung der wissenschaftlichen Forschung~FWF ProjectNo. S7307-AST! and Deutsche Forschungsgemeinsch~DFG Project No. Mo739!.

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