Angular distribution effect on the integrated cross section for radiative capture of 14 MeV neutrons

NUCLEAR PHYSICS A ELSEVIER

Nuclear Physics A 645 (1999) 262-269

Angular distribution effect on the integrated cross section for radiative capture of 14 MeV neutrons F. Cvelbar, A. Likar, T. Vidmar "J. Stefan '" Institute and Faculty of Mathematics and Physics, University of Ljubljana, 61111 Ljubljana Slovenia Received 2 October 1998; revised 29 October 1998; accepted I 1 November 1998

Abstract In the past, many reported integrated cross sections for the 14 MeV neutron radiative capture were measured at 90 ° relative to the neutron beam direction and multiplied by 47r, so as to obtain a measure of the angle-integrated cross sections. In such a procedure, an isotropic angular distribution of y-rays is assumed. We calculated this distribution using the consistent direct-semi-direct model, in which the need for the model-free parameters has been eliminated. The result is that the al Legendre polynomial, averaged over the bound state transitions, is practically zero (distribution is forward-backward symmetric) and that the a2 coefficient is a smooth function of the mass number with the values between - 0 . 4 and - 0 . 6 , indicating anisotropy of the distribution. Reported integrated cross sections are therefore for about 20% to 30% too high relative to the properly angle integrated cross sections. (~ 1999 Elsevier Science B.V. PACS: 25.40.Lw; 24.30.Cz Keywords: Radiative capture of fast neutrons; Giant dipole resonance; Direct-semi-direct model

1. Introduction Study o f the fast neutron capture y - r a y transitions to the bound states has initially been in m a n y cases oriented towards the m e a s u r e m e n t o f y - r a y spectra at 90 ° relative to the projectile direction [ 1 - 6 ] . To c o m p a r e these results o f p r o m p t m e a s u r e m e n t s with the activation cross-section O-act data [ 7 - 1 1 ] obtained in the off-time procedure, the d o u b l e ( e n e r g y and a n g l e ) integrated cross sections (O'in t ) have to be extracted. This, in practice, involves a s u m m a t i o n o f the spectral intensities o v e r all the b o u n d state transitions ( i ) , m e a s u r e d at a certain polar angle 0 and a subsequent integration o v e r the solid angle, i.e., 0375-9474/99/$- see front matter (~) 1999 Elsevier Science B.V. All rights reserved. PH S 0 3 7 5 - 9 4 7 4 ( 9 8 ) 00609-5

E Cvelbar et al./Nuclear Physics A 645 (1999) 262-269

o"int = J ~io"i(O) d O .

263 (1)

TO get the o"int data one should therefore measure the angular distribution of the capture y-rays. This, however, has only been done for the capture to a few well-isolated low energy final states. Reported O'int values [ 1 - 6 ] were, conversely, deduced from the spectral intensities measured at 90 ° multiplied by 47r O'int ~ 47"r2io'i(90 °) .

(2)

This procedure is based on the assumption that the angular distribution is not much different from the isotropic one. Measurements of the capture transitions to the isolated states mentioned above showed that this is far from being true. The capture differential cross section is usually represented by the Legendre expansion coefficients ak defined by do -- = ao(1 + 2akPk(cOsO)) dO

.

(3)

Isotropic distributions are characterised by the fact that all the coefficients ak are zero. For distributions symmetric around 90 °, the odd coefficients are zero and the anisotropy is described by the values of even ones. For the distributions of the form of sin 2 0 and 1 + cos 2 0 the values of the coefficient a2 are is - 1 and +0.5, respectively. Ao has the meaning of the average cross section with respect to angle 0, so that

f do"arian= {J o-(0)ds2 = 47rA0.

(4)

In the case of pure dipole transitions, only the a2 coefficient is different from zero and the following relation holds f

o_(900 ) o " ( 0 ) d ~ = 477 1 - a 2 / 2 "

(5)

This means that for the distributions of the form of sin 2 0 and 1 + c o s 2 0 the denominator in Eq. (1) is 1.5 and 0.75, respectively. Ideally, the angular distribution of the complete spectral intensity of all bound states should be measured. Such a task, however, has not yet been carried out. To get an estimate of the anisotropy, we calculated it on the basis of the consistent direct-semi-direct (DSD) radiative capture model [ 12]. To demonstrate the model's predictive power, we first calculated angular distributions for transitions to the isolated states in (a) 4°Ca (f7/2) [13,14], (b) 28Si (d3/2) [15], (c) 89y (d5/2) [16,17], (d) 2°spb (g9/2) [18] and (e) 2°spb (i11/2), [18] and compared the results to the cited experimental data. Then the calculation was done for radiative capture transitions to all known and es18 • 32 52 56 80 88 timated single particle levels populated in 14S1, 16S, 2~oofa,24Cr, 26Fe, 348e, 38Sr, 89 39Y, 1381~ 140C',~ 2081~1a and 238rl 56~ , 58~ , 82L°, 92~"

264

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2. The direct-semi-direct model In the DSD model [ 19-23] the transition matrix element M is the sum of the term describing the direct capture of the bombarding nucleon to the (single particle) final state, accompanied by the emission of a y-ray, and the resonant term, resulting from the two step process in which first the giant (multi-pole j ) resonance mode is excited and subsequently de-excited by the emission of a photon

Tif ~ (~9fldjlx +) +

(g'flDjfXd) (XdlHrjlX +) E - - ER,j -k iFi/2

= Mdir + M r e s .

(6)

Here, (~f[ is the wave function of the captured nucleon in the bound state, IX+) is the optical model continuum wave function, and di and D r stand for the radial parts of the single particle and the collective electro-magnetic operator for radiative transition of multi-polarity j, respectively. ER,j and F i refer to the position and the width of the (multi-pole j ) giant resonance of the combined (target plus nucleon) system, respectively. Several forms have been proposed for the radial part of the incident nucleon-target-nucleus vibration coupling interaction Hr,.i. The origin of the rather adequate complex volume form [21 ], in which the real and imaginary strengths Vj and W~ were treated as free parameters, has recently been explained in such a way that their interdependence is fixed by the consistency requirement [ 12], i.e., the requirement that the gradient of the nuclear current equals the time derivative of the nuclear density (~Tj = - 3 p / O t ) . This requirement is satisfied automatically if the same potential is used in the calculation of the initial and of the final state wave functions. This is not the case in the DSD model since the initial optical wave function is generated by the complex potential and the final wave function is the result of the real potential. The remaining parameter Vi is taken from general theory. The DSD model has initially been formulated only to treat the dipole giant resonance contribution that comes into play at the excitation energy of around ERI

=

77A U3 MeV

(7)

(see e.g. Ref. [24], also for other than GDR resonances). Recently, it has been upgraded so as to account also for higher multi-pole giant resonances [25], e.g., the iso-vector giant quadrupole resonance (IVQR), appearing at EVR2 ~

130A U3 MeV

(8)

and the iso-scalar giant quadrupole resonance (ISGQR) observed at the excitation energy of EIR2 ---- 65A 2/3

MeV.

(9)

In this formulation of the DSD model, a sum of the dipole transition matrix elements and those of quadrupole resonances is considered. As the parity of the initial state is not defined, we sum up coherently the contributions of different parities. Due to the

E Cvelbar et al./Nuclear Physics A 645 (1999) 262-269

265

interference effect, the odd Legendre coefficients al and a3 appear, as well as the a4. In the original (dipole) version [25,26] of the DSD model, only the a2 Legendre coefficient is different from zero. In the transition amplitude the direct (DIR) and the resonant (RES) terms contribute. It is worth mentioning that due to the low neutron quadrupole effective charge, intensities of the direct neutron quadrupole capture transitions are very small in comparison with other terms. If we limit ourselves to the dipole and quadrupole processes, the DSD neutron capture matrix element is, therefore, the following sum: Zif : TDIR-DIP -I- TRES-DIP At- TRES-SQGR -]- TRES-IVQGR .

(10)

The resulting angular distribution is in general forward-backward asymmetric and the odd Legendre coefficients, different from zero, offer the possibility of extracting information on the quadrupole giant resonances. Till now, the contribution of the iso-vector quadrupole resonances in different nuclei has been considered [ 13,17,27], along with an contribution of the iso-scalar quadrupole resonance, which was possibly observed in case of 41Ca [13].

3. Guidelines from experimental prompt V- ray data From the first angular distribution measurements of the 14 MeV neutron radiative capture it was evident that an experimentally clear picture can only be obtained for a well-separated ground state and eventually for the low excited state transitions. Angular distributions of these transitions were then measured rather intensively at different bombarding energies [ 16,18]. To measure the cross section o'i(O) for the transition to the individual low energy levels, the sample is irradiated [16-18] by a collimated neutron beam and y-rays are detected by a large high quality NaI(TI) spectrometer, usually embedded in the anticoincidence shield. Pulses belonging to the neutrons hitting the NaI(T1) crystal directly are eliminated by the time of flight discrimination technique. For this purpose the neutron source is nanosecondly pulsed. The energy resolution of such systems is between 1% and 2%. This is good enough for the treatment of the transitions to the well-isolated final states. In the region of the states near to the neutron binding energy this resolution limits the accuracy of the spectra belonging to the prompt radiative capture process to the bound states. The much more intense y-rays from the inelastic scattering, which appear at energies below 14 MeV, interfere with the capture spectrum due to the finite resolution of the measurement system as well. The high-energy tail of the inelastic y-ray spectrum overlaps in part with the lowenergy part of the capture spectrum. Its contribution to the O'int is in some cases difficult to estimate and so the experimental uncertainty is increased. This is probably also one of the reasons why the experimental data on the angular distribution we are considering here are not yet available, though they would be desirable from the point of view of

266

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the consistency between the O'int and O'act . In addition, the experimental capture y-ray spectra contain small contributions of secondary (cascade) y-ray transitions to bound states, following the primary low energy transitions within the unbound region. This contribution is small, though, due to the factor of E~-3 and due to the particle emission competition from the unbound region, which increases at the lower bombarding energies.

4. Results

In the 14 MeV neutron capture process of the nucleus having 8 MeV neutron binding energy, the intermediate system is excited to 22 MeV and the average energy of the prompt y-rays is about 16 MeV, which roughly corresponds to the typical energy of the giant dipole resonance. As follows from Eqs. ( 7 ) - ( 9 ) , the iso-scalar and iso-vector quadrupole resonances appear at lower and higher energies, respectively. This means that for 14 MeV neutron capture the angular distribution of y-rays is symmetric around 90 °, i.e., a~ and a3 ,~ 0. Such an expectation is confirmed in all cases treated experimentally and also theoretically (independently of the optical potential used in the calculation), i.e., (a) 4°Ca (f7/2) [13,14], (b) 288i (d3/2) [15], (c) 89y (d5/2) [16,17], (d) 2°8pb (g9/2) [18] and (e) 2°spb (i11/2), [18]. In addition, the coefficient a4 is very small. At higher bombarding energies the al coefficient increases rapidly to the value of al ~ 1. Angular distribution, being therefore symmetric around 90 ° at 14 MeV neutron energy, becomes more and more forward-backward asymmetric. In all the cases studied (see also Ref. [28] ), the energy dependence of the al coefficient is reproduced by the DSD calculations well. We have therefore paid attention to the analysis of the a2 coefficient. We first analysed the angular distributions as a function of neutron bombarding energy for some isolated states in the region between 4 MeV and 14 MeV (Figs. la-e). From these figures one concludes that at low bombarding energy (E < 11 MeV) the a2 coefficient is positive, indicating that the angular distribution is of the type of cos 2 0 + 1, while at higher energies its value is negative and the angular distribution is described by sin 2 0. Taking into account the fact that in the low energy part of this energy interval statistical processes are of importance, that the measuring points are rare and the experimental errors are rather high, the agreement between the experimental and calculated data is relatively good for all different optical model potentials sets used. The agreement is still better in the region around 14 MeV neutron energy (see Figs. la-e). The exception is the capture in the 2°spb high spin state (i11/2) (see Fig. le), though one should take into account the fact that there exists only one, not very precise measurement. The Legendre polynomial coefficient a2, averaged over primary transitions to the " 32 40 52 56 80 88 89 1381:1~ 140¢'~ 208Dh and bound states of 28 1481, 16 S, 20Ca, 24Cr, 26Fe, 348e, 38Sr, 39Y, 56~, 5 8 ~ , 82-u, 2381 I 92,.,, was calculated (see Fig. 2). The same optical model potentials [29-32] as in the study of transitions to the isolated states were used for the calculation of the initial wave functions. The interval of the resulting a2 values is in Fig. 2 represented by the error bars. The average value (of the values obtained with different optical potentials),

E C v e l b a r et al./Nuclear Physics A 645 (1999) 262-269

(a) ~=

267

(b) ~.° l 0.8

0.64

-9.1 ~

\

~

298i 1 d3/2

\

-0.2.9.3 i

\\\~\~

-0.4

lf7/2

410a

-o.~" -0.6 ~ 4

5

~ 6

7

i 8

~ °q 9 10

~ 12

11

~ - i 13 14

_l.0f

--

15

16

, 4

5

.... 6

7

8

9

--

9.6-

11

12

13

14

l 15

16

En [MeV]

En [MeV]

(c) 96'°t -

i 10

(d) ~ •

209

Oo.~~\.

2d5/2

9oy

!

Pb 299/2

i

I

0.4 q

0.0 i ~-2

oo

-0 2

\

\

__

"~'. ~'~~.

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-04

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l ~-

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~

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-°6l -1 .o * 4

, 5

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T 9

I

,

i

19

11

12

•

13

~ - 14 15

-°91

/

-1.9-~-16

~-r

6

7

8

9

En [MeV]

r

10

11

@

12

,

,

13

14

15

16

En [MeV]

(e) o..i 0.7

1

06

a2

,

I

02

0.0 -0.1

t4

-0.2 2°9pb 1 i 11/2

*

-0.3

[

7

8

9

10

11

12

13

14

15

16

En [MeV] Fig. 1. Bombarding neutron energy dependence of the a2 Legendre coefficient for radiative capture in (a)

4°Ca (f7/2) [13,14], (b) 28Si (d3/2) [15], (e) 89y (d5/2) 116,17], (d) 2°spb (g9/2) [18] and (e) 2°8pb (il 1/2) [ 18]. Curves are the result of the consistent DSD model calculation using different optical models - Rosen 129 ] (solid line), Becchetti-Greenlees [ 30 ] (dashed line), Wilmore-Hodgson, [ 311 (dash-dotted line), and Holmqvist [321 (dotted line).

E Cvelbar et al./Nuclear Physics A 645 (1999) 262-269

268

I 0.4

0.2

14 MeV (n,7)

0.0

a2

-o2 -04

-06

-0.8

50

150

100

200

250

A Fig. 2. The a2 Legendre polynomial (averaged over all bound state transitions) for the radiative capture of 14 MeV neutrons in 28 14$1'. 32 16S' 40(',~ 20~a, 52 24Cr' 56 26Fe' 80 34Se' 88 38Sr, 89 39 Y, I38R~ 56 oat, 140("° 58~ , 208ph 82 -~, a n d 23811 92~" The solid curve is the corresponding smoothed mass dependence. Error bars indicate the interval of the values obtained by using the four different optical, a~s indicated in Figs. l a m e .

indicated by open circles, is a smooth function of the mass number ranging from of - 0 . 6 for A ~ 30 to the rather constant value of - 0 . 4 in the mass region A > 80. Details of the (bound) single particle levels treated in the calculation can be found in Refs. [33,34].

5. Conclusion To evaluate the procedure of obtaining the integral cross section (O'in t ) for the radiative capture of 14 MeV neutrons from the prompt y-ray spectra measured at 90 ° relative to the neutron direction, the angular distributions (i.e. the corresponding Legendre coefficients) averaged over all transitions to the bound states, were calculated, using the consistent DSD model. The result is that the odd coefficients al and a3 are very small (al < 0.1) and that the coefficient a4 is negligible. The coefficient a2 is a smooth function of the mass number having the value of - 0 . 6 for A ~-, 30 and decreasing to the rather constant value of - 0 . 4 in the mass region A > 80. The corresponding angular distribution is therefore rather anisotropic, but symmetric around 90 °. Asymmetry is detected at higher bombarding energies. Values of the integrated cross sections obtained from the intensities of the prompt y-ray capture spectrum measured at 90 ° and multiplied by 4~- are therefore for about 20% to 30% too high relative to the properly angle integrated cross sections. The calculated values of a2 coefficient as a function of the mass number (along with a smooth curve, drawn to guide the eye) are given in Fig. 2.

E Cvelbar et al./Nuclear Physics A 645 (1999) 262-269

269

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