Phonon-particle doorway states in (n, γ) reactions on nuclei with N = 28 and N = 82

Nuclear Physics A194 (1972) 458---462; (~) North-Holland Publishin# Co., Amsterdam N o t to be reproduced

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P H O N O N - P A R T I C L E D O O R W A Y STATES I N (n, 7) R E A C T I O N S O N N U C L E I W I T H N -- 28 AND N = 82 v. A. KNAT'KO and E. A. RUDAK Institute of Physics of the Byelorussian Academy of Sciences, Minsk, USSR Received 14 February 1972 Abstract: The role of phonon-particle doorway states is studied in the de-excitation of the capture state following thermal (n,~,) reactions on 5°Ti, 52Cr, 54Fe, 13aBa, 14°Ce and 142Nd. The reduced probabilities of E1 ~,-transitions from the capture state to the low-lying p-levels in

~lTi, 53Cr, 5SFe, 139Ba, ~4°Ce and 14aNd are calculated and compared with the experimental ones.

1. Introduction

The even-odd nuclei with N = 29 (SlTi, 53Cr, 55Fe) and N = 83 (139Ba, 141Ce, 4aNd) are of considerable interest for the investigation of the phonon-particle doorway-state influence on the primary 7-radiation resulting from the (n, 7) reaction. The properties of the low-lying levels of a given nucleus were studied within the framework of the intermediate coupling approach in the unified model 1- a). It was shown that the model provided a reasonably good description most of the properties of the states up to 2.5 MeV excitation. Therefore, it can be assumed that using the capture-state wave function in form of an expansion in the basic functions of the unified model it is possible to get information on the contribution of the phonon-particle doorway states in the de-excitation of the capture state, and in this way to explain the features of the neutron capture primary 7-ray spectra of the above-mentioned nuclei. Analogous calculations of the primary E1 transition probabilities in thermal (n, 7) reactions on the even isotopes of Fe, Ni, Zn, Ge and Se were made in terms of the pairing plus quadrupole model in ref. 4). The results show that in the most cases the experimental data can be explained by taking into account contributions from the single-phonon doorway states to the capture-state wave function. In the present paper the spectra of the low-lying levels and their wave functions were calculated in the framework of the intermediate coupling model for 51Ti, 53Cr, 55Fe, 139Ba, 141Ce and 14aNd. The p-level wave functions obtained were used to evaluate the reduced probabilities of E1 transitions from the capture state. Comparing the latter with the strengths of the primary E1 transitions I J E 3, where Ir is the relative intensity of the 7-line of energy E~, we found the expansion coefficients of the capturestate wave function in the basic functions of the unified model. The energy values of the excited levels, the orbital and total angular momenta and the reduced neutron 458

PHONON-PARTICLE DOORWAY STATES

459

widths used in our analysis are listed in refs. ~-3' 5 - 9 ) ° The experimental results for the 7-ray energies and intensities were taken from refs. 5, 6, ~o-12). 2. Estimation of E1 transition matrix elements The probabilities of the single-neutron E1 ~-transitions can be determined by the formula r(E1)

_

,t~, i-*f

16/rZ2e2E3

A,t(E1) 2

9A2hch2c 2

~ri

i~f

(1)

"

The matrix element is given by M ( i~t E1)

= ( J , Jf; IfIlf(Et)lIJ, Ji;

[i)

=

F323 _Ji3fb7 *

(1) -

-

{Jr Ji

x ii

1t/If

If JJl.ji

Jf

li

L 4re .J ½t(lf 1 ~) l J\0

0

(fllrlli),

(2)

where li, Ji and 1r, jf denoted the orbital and total angular momenta of a valence neutron in the initial and final single-particle states respectively, li and If the spins o f the initial and final states o f a nucleus, J the angular m o m e n t u m o f a core; a is defined as fi = 2a + 1. The radial part of the matrix elements is written as (fl[r[li) =

foR..f,

rR.,t,i, r z dr,

(3)

where R.tj is the radial single-particle wave function of a neutron. It should be noted that as the physical nature of the core states is not taken into account in the matrix element MeEI)i-~f,formula (2) can be used to study doorway states of any type. I f the doorway states of the phonon-particle type are considered, as in our case, the capture functions Ti and the wave functions of the final levels 7re should be presented as an expansion in the basic vectors INR, j; IM):

T m = ~ A~e[NR, j; IM),

(4)

N,R.j

where N denotes the number of phonons, R the spin of a vibrational state,j the angular moment for the last odd nucleon, and I and M the resultant moment of the level and its projection; A]w are the amplitude (expansion) coefficients. In the present paper the expansion coefficients for the low-lying levels of 5~Ti, 53Cr, 5 s Fe, 13 9Ba ' 14 t Ce and ~43Nd were calculated in terms of the intermediate coupling model. Values of the single-particle energies and the parameters h09 and ~ were taken from refs. ~-3). As the contributions from the coupling to three-phonon states are negligible [see, e.g., ref. 3)] a two-phonon approximation was used to diagonalize the model Hamiltonian. The results of the present calculation are in good agreement with the results of refs. i-3). The expansion coefficients corresponding to the low-lying p-levels in 53Cr and 139Ba are listed as an example in table 1.

460

V. A. K N A T ' K O A N D E. A. R U D A K TABLE 1 Energies and wave functions of the low-lying p-levels in 5aCr and 139Ba 5aCr

~

139Ba

Spin

"-.~eV)

~-

½-

~-

~-

½-

.~-

0.00

0.54

2.15

0.63

1.09

1.97

--0.176

--0.161

Configuration 00, p~ 12, p~ 20, P~I22, p~ 00, P,I12, p~ 20, p~ 22, p~ 12, f~ 22, f~. 24, f~_ 12, f..]_ 22. f~ 24, f] 24, h~

0.632 --0.235

0.820 0.173

0.149 0.081 0.867 --0.387 0.095 0.043 0,131 --0.004 --0.098

0.639 --0.221 0.330 --0.110

0.084 --0.233 0.371 0.841 -- 0.202 0.102 --0.009 --0.131 0.028

0.032 0.747 --0.201 0.095 0.051 0.089 --0.024 --0.043 0.572 --0.134 --0.079 0.031

0.416 --0.080 0.308 --0.062

0.187 0.080

--0.062 --0.487 0.295 0.077 0.041 --0.059 --0.036 0.066 0.746 --0.240 0.104 0.008

The expansion coefficients for the capture state were considered as parameters. In general the initial configuration 100, s½; ½+), the single-phonon doorway states 112, d~; ½+) and 112, d~; ½+), and the two-phonon doorway states 120, s~; ½+), 122, d~; ½+) and 122, d~; ½+) were taken into account in the capture-state wave function. Ignoring y-decay through doorway states we have ordinary direct capture. Our analysis shows that in most of the considered nuclei used to estimate the reduced probabilities of the primary y-transitions it is sufficient to take into account only the initial configuration and the single-phonon doorway states. The radial part of the matrix elements were calculated by using both the harmonicoscillator wave functions (hco = 41A -~ MeV) and the wave functions appropriate to the Woods-Saxon potential with the parameters V0 = 52 ( 1 - 0 . 6 3

(N-Z)/A)

MeV,

a = 0.6 fm and R o = 1.25 A * fm. In both cases practically identical results were obtained. 3. Discussion

In order to describe the spectra of hard y-rays from (n, y) reactions on 5°Ti, 52Cr, 54Fe ' 13aBa' 140Ce and 142Nd it is sufficient to analyse four or five intense E1 transitions from the capture state to the low-lying p-levels. In the case when the singlephonon doorway states are taken into consideration in the capture-state wave function, the matrix element of the primary transition will be a linear combination of the param-

PHONON-PARTICLE D O O R W A Y STATES

461

i-+ -,I-+ eters Ao0, A t~-+ 2 and Al2."

MtE1) i'-*f

=

-~+ -~+ -~+ ~ f A o o -[-fffA12 + ' y f A I 2 ,

(5)

where cq, flf and yf are expressed in terms of the calculated coefficients Ao~o, Ado ~+ I-+ A~I2, .412 ,~- Al~2, and A~2. Taking Aoo = 1 and changing the parameters `412 and ~.+ A 12 the theoretical reduced probabilities came into agreement with the experimental ones. Calculated and experimental reduced probabilities of the primary y-transitions compared to the (2•+ 1)S values in the (d, p) reaction for levels with 1. = 1 in 51Ti, 5SFe, 53Cr, 139Ba, 141Ce and 143Nd are shown in fig. 1. The magnitudes of the 1-+ and A12 ~+ for 5~Ti, SSFe, 139Ba and 141Ce are listed in table 2. It can parameters -4~2

j:

::j:

r

0 Fig. 1. Comparison of calculated (hatched columns) and experimental (unhatched columns) reduced El ~'-transition probabilities with the reduced neutron widths of final p-levels (black columns). TABLE 2 Magnitudes of the coefficients Al2 ~-+ and AI2~-+ in the capture wave function Nucleus

Ax2~r+

S~Ti 5SFe

2.50 --0.20 0.00 0.11

1a 9Ba

x41Ce Aoo'l -+ = I.

A12~ + 0.00 --0.20 0.00 0.13

462

V . A . K N A T ' K O AND E. A. R U D A K

be seen that the y-decay of the capture state in (n, 7) reactions on S4Fe, 13SBa and 14°Ce proceeded primarily through the initial state 100, s~; ½+). The contributions of the doorway states are small and account for 10-20 % of the initial-configuration contribution. Unlike the above-listed reactions, in the case of the 5°Ti (n, 7)5~Ti reaction the single-phonon doorway states make the predominant contribution to the de-excitation of the capture state. It is interesting to note that the experimental data show poor correlation between E1 neutron capture and 1, = 1 stripping intensities to the same set of final states (see fig. 1) in 51Ti. In order to describe the high-energy primary 7-ray spectrum from the 52Cr (n, ~,) reaction the two-phonon doorway states had to be taken into account. Fair agreement for the calculated and experimental reduced probabilities was obtained by using the parameters A~2 = 0.4 and A12 = 0.7. Analysis of the a42Nd(n, 7) reaction also showed that the two-phonon doorway states contribute to the primary 7-transitions. However, in the ~4aNd case there is an ambiguity in the parameters in that two sets ~+ ~+ ~+ = ~+ ~+ o f the coefficients A22 = 0.7, A12 = 0.0, A22 - 5 . 0 and A12 = 1.0, AI2 = 0.0, A22 = 3.0 can describe the primary transition probabilities quite well. It should be noted that from the data ~42Nd(n, 7) reaction are incomplete and further experimental study is needed to made conclusive parameter assignments. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)

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