On the direct capture to unbound states

Nuclear Phyalu A283 (1977) 521-525 ;© NartH-HoUwd Pbliddna Co., Amsterdam Not to be rproduced by photoprint or microfm without wrlttm pamiudon hom the publidw

ON THE DIRECT CAPTURE TO UNBOUND STATES

a. BAUR lmiltutfrr 8ermphyalk. Kernforadwmnlape hitch, D-51701aüd, West Germany Received 10 December 1976 Alsttract: Recently a discrepancy between the theory of direct radiative capture to unbound states and axpmimental results for the reaction 12C(p, yi)13N (2366 keV, unbound) has arisen at low yumvies (150 keV < By < 500 keV). It is shown that this discrepancy can be removed by a careful evaluation of the theoretical expression for the usual theory of direct capture to unbound states.

1. Introduction The direct radiative capture process to resonant states can be formulated in close analogy to the radiative capture to bound states. In this treatment care has to be taken to ensure the convergence of certain matrix elements. This problem has been solved by Faessler 1) for E1 radiation in the long wavelength limit ; this theory has been very successful in explaining the existing experimental data. Recently, the work of Rolfs and Azuma ') led to an apparent discrepancy between theory and experiment for low y-energies (150 keV < E, < 500 keV) in the direct capture reaction 12 C(p, 7)13N to the 2366 (1 + ) keV unbound state. It was speculated in ref. 1) that this discrepancy is related to the well-known infrared problem in bremsstrahlung theory 3). It will be shown here that a careful evaluation of the theoretical expression of ref. 1) removes the discrepancy, the infrared problem appears only at much lower 7-energies . Aftera shortreview of the theory 1) the cross sectionfor radiative capture 12C1.,, lY 7)13N (2366 keV, unbound) is studied in sect. 2 with 2 different parameter sets of a square well model for the p-12C interaction. Both square well potentials reproduce the resonant phase shift in the sk channel at Ep = 457 keV. Therefore there is at present no discrepancy between experiment and theory in the radiative capture to unbound states, and it is not necessary to consider higher order radiation processes to explain the experimental results of ref. 2). 2. Summitry of the theory, results and conclusion Matrix elements of the dipole operatorr between eigenfunctions ofthe Hamiltonian H=

_A2

2m

A+V(r),

521

522

G. BAUR

where the potential V commutes with r, can sometimes be evaluated conveniently by applying the double commutation relation (see also ref. a )) : z [H, [H, r]] = #' grad V(r) . (2) m With the eigenfunction property of Ii> and If> and the hermiticity of H we obtain z = 1 = h (3) z . (Er-E1)z m(Er - E1) By means of a partial wave expansion in eq. (3) the result of ref. 1 ) is obtained . As it is clear from the derivation, eigenfunctions Ii> and If> of the same Hamiltonian H have to be used to calculate the matrix element according to eq. (3). The total El capture cross section to an unbound state with the width l' is given by 1,2) Q

,,

= 13.52

° L ZEP Mt ZJ/

z /

íf

1

l

E, EiEr Er

(V=~-V

2 li .%(l,Ol0l

zzz 9p1e ir

lr0)z

~b] ,

(4)

where the mass and charge of the projectile and target are given by Mp, Zp and M respectively . The energy of the proton in the initial and final state is given by E, and Er. In the special case considered here we have l = 0.039 Mev, l, = 1, k = 0, Jr = }, Jp = I and J, = 0. This p -. s transition with y-energy E,, shows a pure sin23 angular distribution, therefore the 90° cross section, measured in ref. z) can be expressed by c,,, as 4

du o __ _3 (900) dQ 81r ~

`~

In order to calculate the matrix elements V.., and Vc,, the interaction V(r) between the proton and the target nucleus 1zC has to be specified. For simplicity, a square well potential is used, the parameters of which are determined by the requirement that the s.* phase shifts in the 457 keV resonance region are reproduced . The same potential is used to determine the (non-resonant) p-wave function in the energy region 500 keV < Ep < 1500 keV. In terms of these wave functions, the interaction matrix elements are given by V..., = uJR0)Vouk(RO), VC ., _ -ZpZ,ezj

RO

druk(r) t2 uk(r).

(6b)

The radius and the depth of the square well potential are given by Ro and- Vo respectively, for r > Ro we have the usual Coulomb potential. The radial wave

u C (p . p)

50

C

81

I2

elastic scattering phase shifts Pot. I

3.0

Pot. II

0

0

2.0

-0.01

-0.03

I

1

.4

1

.6

I

I

.8

1

1

1.0

Fig. 1. The phase shifts do

I

1 -I -

1.2

-0.05 1.4 Ep

MeV

and d1 as calculated with square well plus Coulomb models I and lI. (Note that the left hand scale applies to do and the right-hand scale to dl.) 10' V MeV

100

1d'

1Ií2

.5

t0

1.5

M Ep

Fig. 2. The interaction matrix element V.., (Pot A and the difference (V..,- Ya1) (Pot I and 11).

52 4

G. BAUR 45 ~ IEsâu ' NCE

ßam,P'14CI81F9°°

10'2

a5

1A

1.5

EIPSW Fig. 3. Comparison of the experimental results of ref.') with the theory, using potentials I and II respectively.

functions are normalized like

uj = Fin, qr) cos b,+GXq, qr) sin 61,

r > Ru.

(7)

It must be noted that the exact p-wave phase shifts are not known in the energy region considered here . They are consistent with being smaller than the corresponding hard sphere phase shifts (see ref. 1)). It would not be sufficient here, however, to use these hard sphere phase shifts, because this would leave expression (6a) undetermined (Vo -+ oo, u,,(RO) = 0). The following two parameter sets, which reproduce satisfactorily the 4 phase shifts in the Ep = 457 keV region, are used I:

Vo = 26 .32 MeV,

Ro = 4.48 fm,

Il:

Vo = 69.93 MeV,

Ro = 4.5 fm.

(8)

The phase shifts S, (1= 0 and 1) determined with these potentials are shown in fig. 1. The interaction matrix element VQ,o, (Pot I, for Pot If the curve is almost coincident) and the difference (Va,-Vc,) (Pot I and II) are shown in fig. 2. The Coulomb

matrix elements Vc w have been integrated numerically. With these values, the 90° cross section for capture to the 2366 keVunboundstatein "Nis calculated and compared with the experimental results of ref. 2) in fig. 3. It can be seen that both models reproduce very well the experimental results down to ED = 600 ktV. At the lower energies, the two potentials give somewhat different predictions for the capture cross section. Therefore the measurements of ref. 2) could serve to rule out certain choices of potentials, which may even fit the s* resonance in elastic p+ t2C scattering . The 11R, divergence appears only around Ep s:ts 500 keV. At this energy, theanalytical integration over the resonance energy will no longer be allowed because the high energy tail of the resonance cannot be populated any more because of energy conservation. In conclusion the present calculations have removedthe apparent discrepancy between theory t) and experiment 2) in the direct capture to unbound states. I want to thank Profs. A. Faessler and C. Rolfs for interesting discussions.

1) 2) 3) 4)

Bef

A. Faaaler, Nucl. Phys. 65 (1%5) 329 C. Rolfs and R $. Anima, Nucl. Puys . A227 (1974) 291 W. Heitler, The quantum theory of radiation (Clarendon Press, Oxford, 1954) p. 242 I.. C. Biederharn, Plays. Rev. 102 (1956) 262