Direct radiative capture of p-wave neutrons

A. Mengoni,1,2 T. Otsuka,2,3 and M. Ishihara2,3

1 ENEA, Applied Physics Section, V.le G. B. Ercolani 8, I-40138 Bologna, Italy
2 RIKEN, Radiation Laboratory, 2-1 Hirosawa, Wako, Saitama 351-01, Japan
3 The University of Tokyo, Department of Physics, Hongo, Bukyo-ku, Tokyo 113, Japan
arXiv:nucl-th/9508018v1 10 Aug 1995

(February 9, 2008)

Abstract

The neutron direct radiative capture (DRC) process is investigated, high-

lighting the role of incident p-wave neutrons. A set of calculations is shown

for the 12 C(n, γ) process at incoming neutron energies up to 500 keV, a cru-

cial region for astrophysics. The cross section for neutron capture leading

to loosely bound s, p and d orbits of 13 C is well reproduced by the DRC

model demonstrating the feasibility of using this reaction channel to study

the properties of nuclear wave functions on and outside the nuclear surface.

A sensitivity analysis of the results on the neutron-nucleus interaction is per-

formed for incident s- as well as p-waves. It turned out that the DRC cross

section for p-wave neutrons is insensitive to this interaction, contrary to the

case of incident s-wave neutrons.

PACS number(s): 25.40Lw,21.10Gv,23.40.Hc

Typeset using REVTEX

1
 The direct radiative capture (DRC) process of neutrons in the keV energy region has some
peculiarity recently revived by theoretical analysis [1] as well as by new experimental results
[2–4]. Because of the non-resonant nature of the DRC process, the complications related to

the calculation of the compound nucleus wave function in the entrance channel are removed.
This is a general feature of all the direct capture processes, including those induced by
charged particle reactions. However, because of the lack of the Coulomb interaction, the
(n, γ) reaction has salient features which makes it a unique probe for investigating nuclear

structure information. In fact, the neutron capture process can be explored in the very
low neutron energy region where the reaction mechanism may be fully decoupled from the
resonance process. In this way, precise information can be obtained for the structure of the
capturing orbit and the relative contribution of the various l-wave components to the cross

section can be examined separately.

Because the DRC process is essentially taking place on the nuclear surface and in the
external region, it has been recently proposed [1] to use this reaction channel to study the
properties of nuclear wave functions, in connection with the discovery of the neutron halo of

light drip-line nuclei [6]. The same kind of information can also be derived from the inverse
reaction channel (Coulomb dissociation) where a strong enhancement of the low-lying dipole
mode has been observed [5] and treated as an inverse DRC process [1].
While the DRC of protons and alpha particles have been widely investigated in the energy
range from a few hundreds of keV up to several MeV [7], the DRC process of neutrons has

been mainly examined at thermal (En = 0.0253 eV) energies where s-wave neutrons are
captured into bound p orbits. The DRC formalism for thermal neutrons has been revised
by Raman et al. [8]. They have shown in detail how the neutron-nucleus potential strongly
affects the capture mechanism of s-wave neutrons in light nuclei, whereas no reference to high

energy extension nor to higher partial-wave (including p-waves) contributions was given. On
the other hand, we may expect that, as the incoming neutron energy increases the capture
of p-wave neutrons into bound s and d orbits comes into play and, under proper conditions,
it can be regarded as the dominant capture process.

2
 The extension of the capture models required to include p-waves and higher partial
waves into the neutron DRC process is the main task of this note. In addition, a sensitivity
analysis of the DRC process to the neutron-nucleus potential for energies in the keV region

will be performed. We stress that applications of DRC models have been rarely extended to
energies higher than thermal. In nuclear astrophysics there have been several applications
at neutron energies of interest in the r-process nucleosynthesis [9] and for inhomogeneous
big-bang theories [10]. For such kind of applications it is necessary to assess quantitatively

the DRC prescriptions in a range of energies from a few up to several hundreds of keV.
Here we will briefly revise the DRC model for s- and p-wave neutrons and apply it to
12
the calculation of the C(n, γ) cross sections for transitions leading to all the four bound
states of 13 C . In particular we will consider realistic wave functions for the initial scattering

state and we will focus the attention on the influence of the initial l-wave character on the
capture cross section.
In the early works on neutron capture reactions [11–14] it was recognized that a capture
mechanism, in which the incoming neutron is scattered directly into a final bound state

without forming a nuclear compound state, might take place for nuclei where the final state
is dominated by a strong single-particle configuration. There have been several formulations
of the DRC mechanism differing considerably among each other in the way the incoming
channel and the final state are described [13,15,8,1]. In general, because the direct capture
process is alternative to the compound nucleus (CN) formation mechanism, we can separate

the collision matrix into two components

Ui→f = Ui→f (CN) + Ui→f (DRC) (1)

where all the quantum numbers necessary to define the initial and final states have been

lumped into the notation i and f , respectively. The reaction cross section is given by

π
σi→f = |Ui→f |2 (2)
k2

where k is the wave number of the relative motion in the entrance channel. Here we will

3
deal only with the DRC part of the collision matrix. The capture cross section for emission
of electric dipole radiation (E1) in the transition i → f is given by

16π 3 2 (1 ) 2
σn,γ = k ē |Qi→f | (3)
9h̄ γ

where kγ = ǫγ /h̄c is the emitted γ-ray wave number corresponding to the γ-ray energy ǫγ
and ē = −eZ/A is the E1 effective charge for neutrons. The cross section is, therefore,

essentially determined by the matrix elements

(1 )
Qi→f =< Ψf |T̂ E1|Ψi > (4)

where T̂ E1 = rY (1) (θ, φ) is th electric dipole operator. Here, the initial state wave-function

Ψi is given by a unit-flux incoming wave in the entrance channel, scattered at the origin
by the neutron-nucleus potential. The final state wave-function Ψf is given by the residual
nucleus (bound) final state. The radial coordinate r denotes the distance of the incoming
neutron with respect to the target nucleus.

The entrance channel wave function can be decomposed into spherical (l-wave) compo-
nents

Yl,m(θ, φ)
Ψlm (r) ≡ wl (r) (5)
rv 1/2

where wl (r) depends also on the wave number k and is written, as usual, as
√
i π√
wl (r) = 2l + 1il [Il − Ul Ol ]. (6)
k

Here, the common notation for the asymptotic forms of the incoming and outgoing waves,
respectively Il and Ol , has been adopted

1 1
Il ∼ exp(−ikr + ilπ) and Ol ∼ exp(+ikr − ilπ). (7)
2 2

Ul indicates the collision matrix for the scattering process in the entrance channel, v is the
incoming neutron velocity and k the corresponding wave number.
(1 )
The matrix elements can be decomposed into the product of three factors Qi→f = Iif ·
√
Aif · S.

4
Radial part: Indicating with ulf (r) the radial part of the final state wave function, the
radial overlap integral is given by
Z ∞
Ili lf ≡ ulf (r)rwli (r)dr. (8)
0

If the final state wave function is dominated by a single-particle configuration with a long
tail outside the nuclear radius (halo), there will be a strong effect on Ili lf [1]. This will be
shown below in the case of incoming s- and p-wave neutrons. Analytical expressions for Ili lf

can be derived for specific assumptions on the initial and final state wave functions. If a
hard-sphere model for the scattering wave function and a crude square-well model for the
bound p orbital are assumed for the initial and final state respectively one recovers the very
well known [13] expression for the hard-sphere capture cross section

HS 32π 3 ē2 R5 3 + y 2
σn,γ = kγ ( ) (9)
3 h̄v y 4 1 + y

where y ≡ χR. Here, χ is the reciprocal attenuation length of the wave function tail
√
given by χ = 2µSn /h̄, where µ is the reduced mass of the system and Sn the neutron

separation energy from the residual state. More elaborate expressions corresponding to
different assumptions on the initial and final state wave functions, can be derived for this
overlap integral and they can be found in the literature [13,8,16].

Angular part: For a general li → lf transition, the angular factor which takes into account

the magnetic degeneracy of the final state and is averaged over the initial magnetic substates
is given by

1 3
A2li lf ≡ |hlf kŶ1 kli i|2 = (li 010|lf 0)2 (10)
2li + 1 4π

where (....|..) is a Clebsh-Gordan coefficient. The orbital angular momentum, l, can couple
with the intrinsic spin, s, to give the angular momentum j = l + s. The corresponding ji and
jf can be combined with the target spin I to obtain the initial and final state total angular
momenta Ji and Jf . The angular-spin coefficient for the full coupling is given by

5
 |hJf , I, jf kŶ1 kJi , I, ji i|2 3
A2if ≡ = (2li + 1)(2ji + 1)(2jf + 1)(2Jf + 1) ×
2Ji + 1 4π
ji I Ji li 1/2 ji
× (li 010|lf 0)2 { }2 { }2 (11)
Jf 1 jf jf 1 lf

where {...} are Wigner-6j coefficients.

Final state strength: The single particle strength, S, of the final-state orbit is usually derived
from (d, p) stripping reactions (spectroscopic factor). We note here that if the radial part

of the matrix elements, Eq. (8), is calculated with reliable wave functions (see below) the
spectroscopic factor can be derived from a DRC analysis of the experimental cross section.
This technique, proposed and applied in proton capture reactions [7], could be applied in
the neutron capture channel with the advantages already noted above.

For thermal neutrons, the main contribution to the capture process is due to incoming
s-wave neutrons captured into p-wave orbits and emitting E1 radiation. At higher neutron
energy, however, the incoming p-wave neutrons come into play as they can be captured into
13
bound s and d orbits. In C both these even parity orbits are available. Moreover their

respective levels have the characteristics of being loosely bound and with large spectroscopic
strengths: they are good example of halo structure in excited states of stable nuclei. The
13
nuclear structure information on C are summarized in Table I.
Before showing the result of the full 12 C(n, γ) cross section calculations we will show here
how the s- and p-wave neutron capture is affected by the neutron-nucleus potential.

For the purpose of the present study it is sufficient to consider single particle wave
13
functions for the bound-state orbits of C. They have been calculated using a Wood-Saxon
potential with a radius parameter r0 = 1.236 fm, a diffuseness d = 0.62 fm and a spin-orbit
potential strength Vs = 7 MeV. The potential well depths were adjusted so as to reproduce

the correct binding of the four bound states in terms of the corresponding single particle
orbits. In the case of the 2s1/2 state this gives rise to V0 = 59.23 MeV.
In the present investigation we have treated the incoming neutron channel with the
following approximations:

6
 (a) plane wave (PW) approximation

Ul = 1 for all l

(b) hard-sphere (HS): scattering by an infinitely deep potential well of radius R

Ul = e−2ikR for l = 0
1+ikR
= e−2ikR × 1−ikR
for l = 1

(c) a general case in which the collision matrix is calculated numerically for a given po-
tential of Wood-Saxon form.

In order to see the effect of these three different treatments on the radial part of the
matrix elements we show in Figs. 1 and 2 the calculation of the integrand of Iif in Eq.
(8) at En = 200 keV. Because of its dominant contribution to the integral, only the real
(imaginary) part is shown in Fig. 1 ( 2) for s- (p-) wave capture. The cross section is

proportional to the squared area under the curves shown in the figures.
From Fig. 1 it is evident that for an initial s-wave state the result is strongly dependent
on the neutron-nucleus potential adopted, namely on the different collision matrix of the
scattering channel used in the calculation. Moreover, the behavior of the wave function

inside the nuclear radius may result in a significant cancelation of the radial matrix element.
This result is in full agreement with the conclusions of a detailed study of the thermal
capture in light nuclei by Lynn et al. [17]. We believe that this sensitivity is the main source
for discrepancy observed between the calculated and experimental thermal cross sections
in DRC model analysis. The influence of compound nucleus components in the collision

process is, naturally, another source of uncertainty.

On the contrary, for incoming p-wave neutrons, the radial matrix elements are essentially
insensitive to the different collision matrices considered. This is evident, from Fig. 2, for
both the p → s and the p → d transitions. In other words, in a situation where the

capture is dominated by a DRC process of p-wave neutrons, the cross section is not sensitive

7
to the neutron-nucleus potential. Hence, the capture process is essentially determined by
the structure of the final state wave function. In particular, the component of the wave
function outside the nuclear radius plays the principal role in the determination of the

capture strength. This is the main result of the present investigation.

The results of the full calculation of the capture cross section for transitions leading to
13
the four bound states in C are shown in Figs. 3 and 4.
In Fig. 3, the capture cross sections for transitions leading to the ground state (upper

part) and to the level at Ex = 1.26 MeV (lower part) are shown. The curves labeled by
(a), (b) and (c) refer to three different scattering wave functions calculated according to the
three treatments (PW, HS and diffused Wood-Saxon potential) described above. For these
two transitions, incoming s- and d-wave neutrons are involved in composing the initial state

wave function. The large discrepancy between the experimental values and the calculation
is removed only if a realistic neutron-nucleus potential is employed in the calculation. In
our case, such an agreement was obtained when the same potential used for the bound-
state calculation was employed (curve (c) in the figure). This is an important point because

the obtained wave functions might have had unphysical overlaps and should have been
orthogonalized if different potentials would have been used.
We have extended the calculations down to thermal energy (En = 0.0253 eV) using
the same Wood-Saxon potential and the results are shown in Table I. Considering that
no adjustment of any parameter was performed, the results of our DRC calculation can be

considered satisfactory.
The relative contribution of incoming d-waves can be seen in Fig. 3. There, the dotted
lines show the contribution of the s-wave capture component only, whereas the full line
(c) show the contribution of both s- and d-wave components. Thought not decisively, the

experimental results seem to follow the increasing trend due to the onset of the higher partial
wave component.
Finally, the cross sections for transitions leading to the s1/2 and d5/2 orbits are shown
in Fig. 4. These are the transitions due to the capture from p-wave neutrons. A very

8
good agreement is found for both the reaction channels, implying reliability of the DRC
mechanism in the energy region under consideration. In this figure the conclusion drawn
above from the radial matrix elements calculations can be verified explicitly. The results

of the calculations obtained using the PW, HS or the full Wood-Saxon potential are barely
distinguishable. No major influence of the neutron-nucleus potential used is revealed by the
calculations, very well supported by the experimental results.
In summary, our calculations have shown that, while the DRC process of s-wave neu-

trons is strongly biased by the incoming-neutron interaction with the target, the DRC of
p-wave neutrons is essentially insensitive to the details of this interaction. This fact can be
used to derive important nuclear structure information on the residual nucleus like “exotic”
components of the neutron wave function outside the nuclear radius (i.e. neutron halo) or

single particle strength (spectroscopic factor) of the final capturing state.

We acknowledge many fruitful discussions with Y. Nagai and C. Coceva. This work

has been supported in part by Grant-in-Aid for Scientific Research on Priority Areas (No.
05243102).

9
 REFERENCES

[1] T. Otsuka, M. Ishihara, N. Fukunishi, T. Nakamura and M. Yokoyama, Phys. Rev. C

49, R2289 (1994).

[2] Y. Nagai, M. Igashira, N. Mukai, T. Ohsaki, F. Uesawa, K. Takeda, T. Ando, H.

Kitazawa, S. Kubono and T. Fukuda, Ap. J. 381, 444 (1991).

[3] T. Ohsaki, Y. Nagai, M. Igashira, T. Shima, K. Takeda, S. Seino and T. Irie, Ap. J.
422, 912 (1994).

[4] M. Igashira, Y. Nagai, K. Masuda, T. Ohsaki and H. Kitazawa, Ap. J. 441, L89 (1995).

[5] T. Nakamura et al., Phys. Lett. B 331, 296 (1994).

[6] I. Tanihata, H. Hamagaki, O. Hashimoto, Y. Shida, N. Yoshikawa, K. Sugimoto, O.

Yamakawa and N. Takahashi, Phys. Rev. Lett. 55, 2676 (1985); I. Tanihata, Nucl.

Phys. A522, 275c (1991).

[7] J. D. King, R. E. Azuma, J. B. Vise, J. Görres, C. Rolfs, H. P. Trautvetter and A. E.

Vlieks, Nucl. Phys. A567, 354 (1994); references therein.

[8] S. Raman, R. F. Carlton, J. C. Wells, E.T. Jurney, and J. E. Lynn, Phys. Rev. C 32,
18 (1985).

[9] G. J. Mathews, A. Mengoni, F. K. Thielemann and W. A. Fowler, Ap. J. 270, 740

(1983).

[10] M. Wiesher, J. Görres and F. K. Thielemann, Ap. J. 363 340 (1990).

[11] R. G. Thomas, Phys. Rev. 84, 1061 (1951). [This is the oldest reference that we have
found on the subject.]

[12] H. Morinaga and C. Ishii, Prog. Theor. Phys. 23, 161 (1960).

[13] A. M. Lane and J. E. Lynn, Nucl. Phys. 17, 563 (1960); 17, 686 (1960).

10
[14] J. E. Lynn, The theory of neutron resonance reactions (Clarendon Press, Oxford, 1968).

[15] A. M. Lane and S. F. Mughabghab, Phys. Rev. C 10, 417 (1974).

[16] A. Mengoni, T. Otsuka and M. Ishihara, in Proceedings of a Specialists’ Meeting on

Measurement, Calculation and Evaluation of Phototn Production Data, Bologna, 1994,

edited by C. Coceva, A Mengoni and A. Ventura, (NEA/DOC/95/1), p. 185.

[17] J. E. Lynn, S. Kahane, and S. Raman, Phys. Rev. C 35, 26 (1987).

[18] F. Ajzenberg-Selove, Nucl. Phys. A523, 1 (1991).

[19] S. F. Mughabghab, M. Divadeenam and N. E. Holden, Neutron Cross Sections, (Aca-

demic, New York, 1981), Vol. 1.

11
 FIGURES
25

20 (a)

15

10 (b) s -> p
u(r) r w(r)

5
(c)
0

-5

-10

-15
0 5 10 15 20 25 30
r (fm)

FIG. 1. Real part of the integrand of Eq. (8) for incoming s-wave neutrons of 200 keV and for the

1p3/2 orbit bound by 4.96 MeV. The scattering wave function w(r) has been calculated according

to (a) PW, (b) HS and (c) Wood-Saxon potential prescriptions. See the text for parameters and a

detailed explanation.

12
 10

5
p -> d

0
u(r) r w(r)

-5
p -> s
-10

-15

-20
0 5 10 15 20 25 30
r (fm)

FIG. 2. Imaginary part of the integrand of Eq. (8) for incoming p-wave neutrons of 200 keV

and for the 2s1/2 and 1d5/2 orbits bound by 1.86 and 1.09 MeV respectively. The scattering

wave functions w(r) have been calculated according to a PW (dashed line), HS (dotted line)

and Wood-Saxon potential prescriptions (solid line). See the text for parameters and a detailed

explanation.

13
 −2
10
s+d −> p1/2 (Sn=4.95 MeV)
−3
10

cross section [b]

−4
10 (a)

(b)
−5
10

−6
(c)
10

−7
10
−4
10
s+d −> p3/2 (Sn=1.26 MeV)
cross section [b]

−5
10
(a)
(b)
−6
10
(c)

−7
10
0.0 0.1 0.2 0.3 0.4 0.5
Neutron energy [MeV]

FIG. 3. Neutron capture cross section of 12 C for transitions leading to the ground state of
13 C (upper part) and to the level at Ex = 3.684 MeV (lower part). The experimental values are

from the reference [2]. The curves labeled with (a), (b) and (c) correspond to the three different

assumptions concerning the collision matrix for the scattering channel. See Figs. (1) and text for

explanation. The dashed-line represents exclusively s-wave contribution to the capture.

14
 30
p −> s1/2 (Sn=1.86 MeV)

cross section [µb]

20

10

0
8
p −> d5/2 (Sn=1.09 MeV)
cross section [µb]

0
0.0 0.1 0.2 0.3 0.4 0.5
Neutron energy [MeV]

FIG. 4. Neutron capture cross section of 12 C for transitions leading to the s1/2 orbit (upper part)

and to the d5/2 orbit of 13 C (lower part). The experimental values are from Ref. [2]. The dashed-,

dotted- and solid-line correspond to the three different assumptions concerning the collision matrix

for the scattering channel. See Fig. (2) and text for explanations. In this case, the capture is due

to incoming p-wave neutrons only.

15
 TABLES
TABLE I. Nuclear structure information of 13 C and thermal neutron capture cross section of
12 C.

Ex (MeV) Sn (MeV) ln Jπ Sdp a th (mb)

σn,γ

0.0 4.946 1 1/2− 0.77 2.10

3.089 1.857 0 1/2+ 0.65 -

3.684 1.262 1 3/2− 0.14 1.10

3.854 1.093 2 5/2+ 0.58 -

Total 3.20

Experimental 3.53 ± 0.07 b

a From Ref. [18]

b From Ref. [19]

16