Threshold effects in nuclear reactions and the line shape of resonances

Nuclear Physics A297 (1978) 237-253 ; © North-Holland Publishing Co ., Amsterdam Not to ba reproduced by photoprint or microfilm without written permission from the publisher

THRESHOLD EFFECTS IN NUCLEAR REACTIONS AND THE LINE SHAPE OF RESONANCES 1 . ROTTER and H . W . BARZ Zentralinstitut .lür Kernforschung Rossendort, DDR-8051 Dresden and J . HÖHN Technische Universität Dresden, DDR-8027 Dresden Received 5 October 1977 Abst acl : Threshold effects in the cross section of nuclear reactions are investigated in the continuum shell model using the "N+n and ' 6 0+y reactions as examples . The level density near particle emission thresholds is almost not enlarged as a threshold effect . Thresholds for particle emission have also almost no influence on the correlation of the resonance levels via the continuum or on their decoupling . The function r(E) influences the line shape of resonances . Threshold effects in the functions r (E) may lead to a cusp in the cross section instead of a resonance.

1. Introduction In the neighbourhood of particle emission thresholds peculiarities in nuclear reactions are expected since the radial extension of the nuclear system is larger at this energy than at other energies. The effects are expected to be visible especially near neutron decay thresholds. They have been investigated for a long time but until now have not been cleared up. In theoretical studies, threshold effects were investigated in special models neglecting nuclear structure effects. Recently, Lovas and Dénes') have reduced the problem to a three-body problem which can be solved exactly. But in realistic cases, the resonances arise from compound nucleus states with a complicated nuclear structure. Therefore, the problem needs further investigation. In other recent papers 2), threshold effects are used to account for the appearance of the pygmy resonances observed in y-absorption processes. The problem of threshold effects is a part of the question of how the continuumcontinuum coupling and the reaction channels influence the nuclear reaction cross section. The best way to investigate threshold effects is therefore to study them together with other continuum effects such as the energy dependence of widths and positions of the resonances and correlation effects caused by the mixing of the resonances via the continuum. All these effects may be investigated by comparing the data of neighbouring nuclei having a similar nuclear structure. Here, the states with a similar structure as well as the positions of the thresholds are changing from one 237

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nucleus to another so that threshold effects should become visible. Theoretically, such a situation may be simulated best by changing independently the positions of the resonances and ofthe thresholds in one nucleus. In such a procedure the variation ofthe average nuclear field in going from one nucleus to another is excluded so that different continuum effects can be studied unambiguously. Investigations of such a type are possible on the basis of the continuum shell model described in refs. "). The nuclear structure calculations are performed in this model like in usual structure calculations . They provide the wave functions and energies of the quasibound states embedded in the continuum (QBSEC) which are used in the coupled-channels calculations . The number of resonances in the reaction cross section is determined by the number of QBSEC. The positions and widths of the resonances are calculated by a diagonalisation procedure. In this model we have the possibility to perform calculations of two types. First the energies of the QBSEC are changed leaving their wave functions and the positions of the thresholds fixed. Secondly, the positions of the thresholds are changed but the positions and wave functions of the QBSEC remain unchanged. The single steps of the variation are separated and can be investigated independently from each other. This means that threshold effects and the mixing of the resonances via the continuum can be studied separately . The aim of this paper is to investigate threshold effects. The calculations are performed for the reactions 1 sN + n and 160 +y. The effects are discussed by means of the 1 - resonances with lp-lh nuclear structure and of the 0+ resonances with 2p-2h structure. The reaction channels taken into account correspond to levels of the target and residual nuclei with lh and 2h-1p structure. Details of the model are given in sect. 2. In sect . 3 the question is considered of whether the level density near particle emission thresholds is enlarged . The influence of particle emission thresholds on the correlation of resonance levels via the continuum is investigated in sect. 4. Here, the cross section of photoabsorption processes is considered . In sect . 5 we give our results on the line shape of resonances for the case when there is no interference with the direct reaction part or with other resonances. The results are summarized in the last section. 2. Details of the calculations The coupled-channels method in the framework of the continuum shell model is described in refs . 3.'). Here, we repeat only those formulas which are needed in the following calculations . The whole function space of the continuum shell model is divided into two parts by means of the projection technique. The Q-space is defined by the set of discrete QBSEC while the P-space contains the wave functions with one particle in the continuum . It is defined by P = 1- Q. The QBSEC are obtained by a usual nuclear structure calculation including the narrow single-particle resonances into the set of

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239

bound single-particle wave functions by means of a.cut-off technique . The QBSEC are characterized by their wave functions OR and energies ERA, being the eigenvectors and eigenvalues, respectively, of the operator HQQ _- QHQ. The coupled-channels equations are solved for the continuous wave functions ~ as well as o)R. The functions are defined as solutions of the Schròdinger equation in the P-space while in the functions o), -- GV)HpQOR [G(P) = P(E+-Hpp)-'P is the Green function in the P-space and Hp. = PHQ] the coupling to the discrete wave functions OR is taken into account. Thewave functions as well as the energies and widths ofthe resonance are obtained from the eigenvectors & and eigenvalues ÉR - jiPR of the operator Ha = HQQ +HQpGP )HpQ,

(2.1)

which is effective in the Q-space after the coupling to the P-space has been taken into account. The energies ER of the resonance are obtained from the condition ËR("R) = ER, while the widths rR follow from the equation

(2.2)

rit = f'R(&). (2.3) If all narrow single-particle resonance are included into the set of basic states defining the Q-space then the number of resonance states is equal to the number of QBSEC contained in the Q-space. This has been shown in ref. 3) for the 1 - states of 1110 . Almost all of the wave function with large amplitude inside the nucleus is contained in the Q-space. Therefore the resonance parameters depend only weakly on the magnitude of the continuum-continuum coupling, as has been shown a) numerically for the case ofneutron scattering on "N. Thus, the concept ofa resonance state defined by eqs. (2.2) and (2.3) and by the wave functions (2.4) k(ER) = &(ER) +&R(ER~ where cúR = Gp+)HpQ& is the continuation of the function & into the P-space s seems to be reasonable. The functions GR and PIt are energy dependent. The function PR(E) must vanish as E -. 0 because Ha becomes Hermitian as E -. 0. For E -+ oo, Hn -. HQQ holds so that the function PR(E) also vanishes for E -* oo. The conditions formulated here for the functions PR(E) are the same as those found by Lane') on the basis of quite another theory . The energy dependence of the functions L`R and ` R is modified by the coupling of the resonance state considered to different reaction channels as well as to other resonance states via the continuum. The dependence on the reaction channels can be investigated easily from the behaviour of the functions ER(E) and PR(E) . The coupling to otherresonance states viathecontinuumcan be studied best by comparing the functions PR(E) and PR(E) with the corresponding functions VR(E) and PR(E)

24 0

I . ROTTER et al.

of the isolated resonance. These functions are provided by the diagonal matrix elements OR- l RR = «RIH`IOR>. (2.5) In order to describe photonuclear reactions the electric dipole strength DR will be defined by 160 1 : . e2 M 0 2. DR ( (2.6) = DR ER) = 9 2Jo + 1 fitc E,

Heré, JR is the total spin of the resonance state, JO the total spin of the target state, e2/hc the fine structure constant and E., the photon energy. The reduced matrix element <9)RIIrYi.ljOO> describes the dipole transition from the resonance state OR to the target state 0 0. The dipole strengths $R are complex due to the coupling of the resonance to the continuum. They are connected with the resonance part of the dipole absorption from the target state 00 : (2.7) As in the Feshbach theory, the cross section is obtained as a sum of the resonance reaction part and'the direct reaction part including the interferences. The direct reaction part of the nucleon scattering is obtained from the solutions ~,(`°) of the Schrödinger equation in the P-space in the coupled channels representation (c stands for the channel). The wave functions ~(`°) are normalized asymptotically like sin kr in the entrance channel co. In thecase of the photonuclear reaction the direct reaction part follows from the solutions tc of the Schr6dinger equation with the dipole part rY1 0o of the electromagnetic field as a source term F. The whole cross section follows from the total wave function a)

v = Z+

E-~r

+ Í%R <&IHQg+F>,

(2 .8)

with F - 0 for nucleon scattering. The partial widths FR , for the decay of the resonance R into the channel c are obtained from rR.C

= 4- I<~CIHpQI10R>I 2 .

(2.9)

The numerical calculations reported in this paper are performed for the 1 sN + n reaction as well as for the 160 + y reaction . In the first case, the QBSEC with negative parity are calculated in the Ip-Ih configuration space. All configurations with one particle in the (1d,}, 2s*, Id.,) shell and one hole in the (1p,,, 1%) shell are taken into account. The QBSEC with positive parity are obtained in the c2p-2h configuration space. The 2p-2h configurations are (lp*)-2[(íd1)2, (1d}, 2s*), (ldf, ld.1), (2s})2], (lpj)-'(Ip})-'[(ldt)2, (Id,, 2s,)] and (Ipj)_ 2(Id1)2. The, coupled-channels

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24 1

calculations are performed with 0 + and 1 - resonannces representing the features of the two types of resonances. The positive parity states of the target nucleus t eN are obtained by diagonalizing the shell model Hamiltonian in the full lp-2h configuration space with one particle in the (1dß., 2s ,1d,) shell and two holes in the (1p },1p }) shell. The two negative parity states }- and J - are assumed to be the l p. and lp,. hole states . The shell model calculations have been done using a Woods-Saxon potential for the single-particle states and a 8-force for the nucleon-nucleon interaction V(rt - r2) _ - Vo(a+bP1Art - r2).

(2.10)

The parameters are Vo = 500 MeV - fin, a = 1 .0 and b = 0.5 while the parameters . of the Woods-Saxon. potential are the same as in ref. 6). The single-particle d} resonance is cut off at 8 fm: In the 160+ y case only 1 - resonances with 1 p-1h structure have been considered which are the dominating states in the 16 0+y reaction . Like in the '6N case particles in the (ld # , 2s*, ld t) shell and holes in the (lpt., lp*) shell are taken into account. Isospin mixing is allowed already in calculating the QBSEC. The spurious state with T = 0 has been excluded by the method described in ref. '). The parameters of the Woods-Saxon potential, the cut-off radius and the parameters of the interaction (2.10) are the same as in the '6N case. The shell model energies E' o f the resulting nine QBSEC as well as the energies ER , widths rß, dipole strengths DR, isospins TR and leading configurations of the corresponding resonannces are listed in table 1 . The isospin Tß of the resonances is not a good quantum number, but the impurity is small in most cases. The only exceptions are the resonance nos. 4 and 5 . The Shell model energies E:", positions Fz , widths 1'a, dipole strengths Dt, isospins Ti and leading configura tions of the nine 1 - resonances with 1p-1h structure in '60 (energy E' of the neutron threshold is 6.16 MeV)

AL

P.

Resonance no.

Er (MeV)

(MeV)

(keV)

1 2 3

10 .81 13.59 16.46

9.32 13 .22 15 .35

244 178

4.94 2x 10 -'

4

16.84

16 .83

676

-0 .03

5 6 7

17.57 20.11 22.37

17 .35 19 .61 22 .35

632 346 456

22 .55

1386

4.95 12 .30 0 .01

161 .10

1 1 0 1

25 .30

24 .41

2012

35 .56

1

8 9

23.92

TR

Leading config.

0 1

lpi,2 2s 1y 4 1 P' 22 25,12 lo 31'22s, f 1Pi121d3 12 llp31=1d, 1= lp 1-1ild3i2 1P3~i s 21 i2 lp3-112 1d si2 Í1P3131ds~z , P312 d3i3 1 1p-1 `~P31'lds,2

o

o

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I . ROTTER et al.

leading configurations given in table 1 are obtained by diagonalizing the shell model operator H. .. They remain the leading configurations also after diagonalizing the continuum shell model operator Hjg . The resonance no. 8 is the giant dipole iesonance. The resonance no . 6 is distinguished by its (1p..)-'2s.ß structure and T& = 1 . The four channels included in the calculation correspond to the hole states in the 1 p., and 1 p* shells of the "N and ' s0 nuclei . No further approximations are made in solving the coupled-channels equations after the nucleon-nucleon interaction (2.10), the size of the configuration space and the number of channels have been fixed. 3. Level density near partide emission diresholds In the neighbourhood of thresholds thecross section of neutron scattering reactions may have a resonance-like behaviour which is caused by the enlarged radial extension of the nucleus in the neighbourhood of a threshold'). The question whether those threshold states can be seen in the real cross section is not clarified up to now. In fig. 1 the direct part of the elastic cross section of the reaction 'N(n, n)' 5N corresponding to J= = 0+ is shown. The cross section is reduced at all energies where a new channel opens. The effect is however very small. No effect at all could be seen

à

Ee

52

W

0

GA

5e

E~ (~ArJI Fig. 1 . Direct reaction part and cross section of the elastic neutron scattering on "N (J` = 0*) with six 0* resonances between 9 and 10 MeV . The three inelastic channels correspond to the I' and I' states of "N at 3 .30 MeV and to the }- state at 6.32 MeV .

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243

Fig . 2. Cross section of the elastic neutron scattering on "N with two 1 - resonances . The energy of the threshold corresponding to the }- state of ' °N has been chosen to be 6 .32 MeV (calculated value, full line) as Well as 5 .66 and 7 .44 MeV (dotted lines) .

in the inelastic channels of the same reaction and even in the elastic channel eorrespónding to J'° = 1 - . The thresholdeffect inthedirect reaction part will be diminished or amplified by threshold effects in the function P(E) . In fig. l the elastic cross section of the reaction 1 sN(n, n) t °N is shown also for the case in which both threshold effects add with the same sign . In this case six 0 + resonancelevels with 2p-2h structure exist which all lie between 9 and 10 MeV. A resonance-like behaviour of the cross section results which is not connected with a QBSEC but is a pure threshold effect ("threshold state"). It is however very small and will be covered by the resonances arising from the QBSEC in realistic cases. Thus, threshold states will hardly lead to an enlargement of the level density near particle emission thresholds in realistic cases. An enlargement of the level density near particle emission thresholds could be caused by another threshold effect s) : by an additional shift of the compound nucleus states to thresholds lying in the neighbourhood . In order to investigate this effect the elastic cross section of the 1sN+n reaction with two 1 - resonances is shown in fig. 2. The energy of the threshold corresponding to the j - level of t °N is varied . The resonance at 6.4 MeV is shifted to lower energies ifthe threshold is at 5 .66 MeV and to higher energies if the threshold is at 7 .44 MeV (dotted lines). The shifts are small however and the widths are almost unchanged although the reduced width of the 6.4 MeV resonance to the channel corresponding to the I- level of 'IN is not

244

I. ROTrER et al.

small. The reason is that the partial width to the channel considered is still small in the neighbourhood of the threshold because of its energy dependence, in spite of the large reduced width. Thus, the additional shift of compound nucleus states to particle emission thresholds will also not lead to an enhancement of the level density near threshold which is well beyond the statistical uncertainties. These results are in agreement with the following fact. In some cases an enhancement of states with relative large cluster widths has been observed experimentally in heavy ion reactions near the corresponding cluster emission thresholds and has been interpreted as a threshold effect. In the cases wherean analysis of cluster reduced widths has been done') on the basis of the simple shell model an enhancement of the cluster reduced widths has been found in the same manner as in the experimental data. That means the observed enhancement of states with relatively large cluster widths near threshold is not a threshold effect but follows from the microscopic nuclear structure. This result obtained earlier') is confirmed by the continuum shell model calculations given here. 4. TóreshoM effects in the correlation of resonance levels via the-oond~

As has been shown in refs. 4- 1 °) for the case of 0+ and 1 - resonance in "N both the energy dependence as well as the absolute value of the functions 1`R are determined by the number of channels taken into account in the calculation. The absolute value of rR is influenced strongly also by the mixing of the resonances via the continuum ("external mixing ) 11). The functions PR(E) contain the correlation of the resonance via the continuum while this correlation is excluded from the functions PR(E) . Decoupling ofresonance as a threshold effect can therefore be investigated by comparing the curves AR(E) and 1'ß(E) . Such a comparison is shown in fig. 3. The calculations are performed for six ix, T = 0+ , 1 resonance levels of 16N which all have a 2p-2h structure. Four channels corresponding to the four low-lying levels of 1 'N are taken into account. In fig. 3, the curves A(E) and 1`(E) of the resonance no. 1 are shown. The shell model energies Er of the other levels are chosen in such a manner that EsM < Er (dashed line), and g < Elm < EIK (dash-dotted lined respectively . The distances between the shell model energies E:m are chosen to be 200 keV. For the case Elsm < E.' the positions ER and widths rR are given in table 2. The resonance no. l does not overlap with the other resonances . Nevertheless, the values VR and rR differ from each other by about a factor of two. In all curves the width enlarges when a new channel opens. However, there cannot be seen any enhancement of 1`1 in comparison with P1 in the neighbourhood of the thresholds which would be larger than at other energies . That means, threshold effects can be observed in the energy dependence of the functions l R as well as PR but not in the mixing of the resonance via the continuum. Further, of the curves rR do not depend on the positions ER the QBSEC if the

THRESHOLD EFFECTS

245

Fig. 3 . Functions &(E) and P,(E) of the J`, T = 0 + , 1 resonance state no. 1 in ' 6 N . Sequence of the five surrounding 0 + , 1 resonances : 1, 2, 3, 4, 5, 6 (dashed) ; 3, 2, 1, 4, 5, 6 (dash-dotted).

relative distance of the different QBSEC remains constant. The curves 1`a do not alter also if, at such a shift of the shell model energies E:" of all the QBSEC, the threshold lies between the resonances or at the energy of one of the resonances . The threshold effect is of the same type in the different curves rR(E) obtained by T~ 2

Shell model energies

ERI, positions 4 and widths Fit of the six resonances shown in fig. 55El" :5 6MeV

E:m

Resonance no .

(MeV)

F.,t (MeV)

1 2 3 4 5 6

5 .00 5.20 5.40 5 .60 5 .80 6.00

4.97 5 .03 5 .30 5 .53 5.72 5.99

l'a (keV) 23 27 88 5 4 0.5

3 for the case

246

1 . R01TER er al.

varying the threshold energy . Consequently, the correlation of the resonances via the continuum ("external mixing") is almost not influenced by the existence of a particle emission threshold between the resonances . The question of whether threshold effects in the dipole strengths could lead to a decoupling of dipole resonances is investigated in fig. 4. The nuclear structure of the resonance level no. 6 in 160 is almost pure l pi 12st. Threshold effects in the correlation of this resonance with the giant resonance no . 8 should therefore be large. In fig. 4, the positions ER and the dipole strengths DR of the resonance levels nos. 6, 8 and 9 are shown as a function of the energy of the second neutron threshold corresponding to the 1 p., hole state in "0. They are calculated with all nine resonances (see table 1) correlated via the continuum. If the threshold is in the neighbourhood of the energy of the resonance no . 6 the dipole strength does not change very much, i.e. threshold effects in the dipole strengths are not very pronounced . A decoupling ofresonances may also occur if the nuclear structure of the different resonances and the threshold behaviour of the basic single-particle states differ strongly . Since a decoupling of such a type appears already in the shell model without continuum ("internal mixing") it will not be discussed here. The "O(y, n) and 160(y, p) cross sections calculated with all nine 1 - resonances are shown in fig. 5 from the lowest neutron threshold up to 27 MeV. The two lowlying resonances nos. 5 and 6 have smaller shell model dipole strengths DRSI than the two resonances nos. 8 and 9 (see table 1). According to this they are excited with a smaller probability in the photonuclear reaction than the two high-lying resonances. The second neutron threshold lying between the resonances nos. 5, 6 and nos. 8, 9 at E`h` = 6.16 MeV hardly changes this ratio. This can be seen by comparing the cross sections calculated with different threshold energies : E" = 6.16 MeV between the resonances nos. 6 and 8, El` = 7.4 MeV above the resonance no. 8, E" = 0.4 MeV below the resonances nos. 5 and 6 and Ell' = 4.0 MeV at the energy of the resonance no. 6 (fig . 5). The resonance no. 6 at 19.6 MeV has an almost pure nuclear structure 1 p . 12s.. The ground and excited - states ofthenuclei 1 sN and ' 10 have the structure l 1 and lpj 1, respectively. The partial width of p; the resonance no. 6 is therefore larger in the nucleon channels corresponding to the I - states [r6 r = o> +1'br=l2) x ru~o) = 281 keV for protons, E' = 6.16 MeV] than in the channels corresponding to the ground states ofthe target nuclei [r6'0) _ 48 keV, r6r. -ó) = 7 keV for protons, E" = 6.16 MeV] . According to these values, the resonance no. 6 is excited with a different probability in the different partial cross sections (fig . 5). That means, that the relative small excitation probability of the two low-lying resonances nos. 5 and 6 in the 160 ty reaction is aboveallconnected with the nuclear structure of the two levels determining the position UR .as well as the widths VR and the dipole strengths DR of the QBSEC. The different height of the giant resonance peak for different threshold energies E" is above all connected with the larger decay width at lower threshold energies (0.4 S Ell' S 7.4 MeV). Therefore, it must not be considered as a threshold effect.

THRESHOLD EFFECTS

247

E' " E Da - Vt ( ~ MM~

1 - ,1

nesononces in %

2

3

40

D

0 25 24 23 12 21 20 19

1

f.

S

Cl*

SI V" 7

E*' [M&V] Fig. 4. Positions F, and dipole strengths of the three J`, T - 1 - , 1 resonances nos . 6, 8 and 9 in ` 60 versus the energy El of the neutron threshold corresponding to the J - level in' ° 0. The calculations are performed with all nine l' resonances.

4

248

1 . ROTTER et al.

'so

ne)

32 E th --- E hr_-_ Or... .... E a-.-

24

24 MW a .1smav &0 MW OA MW

'

v

1 1

a

l

a 24

1

po )

>s

Ì Ì

11t

16 0(pn,)

15

: v

w

n

i

16

w

_ _

~I

w

a

E *W-04MsV

,.

2a E*C ZMW EIk61GMW

2c ZiMal

26 EI IMrUI

Fig. 5 . Cross section of the photonuclear reactions ' 6 0(7+ n) and 160(7, p) with all nine 1 - resonances having lp-lh structure. The energy of the neutron threshold E'' has been varied. The value E°' = 6 .16 MeV corresponds to the calculated value.

THRESHOLD EFFECTS

249

The results obtained for the 160 +y photonuclear reaction suggest that the pygmy . resonaaces observed in heavier nuclei 12) are caused not only by threshold and correlation effects but mainly by the nuclear structure of the resonance levels . The observation of the pygmy resonaaces in the (n, y) channels but not in the (p, y) channels 13) of some nuclei suggests that for these nuclei the pygmy resonance have a simple structure with a large neutron partial width to the ground state of the one target nucleus and a small proton partial width to the ground state of the other target nucleus. Therefore, they can be excited well from the neutron channel but hardly from the proton channel. 5. The Hue shape of resonances In the continuum shell model, the energy and width of a resonance are defined by eqs . (2.2) and (2.3). The cross section is however calculated not with these constant values but with the energy dependent functions ÉR and rR. In the Breit-Wiper formula the constant values ER and FR are both multiplied by &Eß and rv/r t, respectively . Therefore, the line shape of resonance may be non-Lorentzian also in cases where interference effects with the direct reaction part or with other resonaaces do not play a role . In most cases, the functions rR still increase with energy in the giant resonance region . For this reason most dipole resonance calculated in the continuum shell model for light nuclei have â longer tail at higher energies . An example is the giant dipole resonance 1", T = 1 - ,1 in 160 at 22.6 MeV having the dominant (1 p.1) -1 (ld t) nuclear structure. In this case, the direct reaction part is small and the interference with other resonance can be excluded for illustration by performing the coupledchannels calculations with only one QBSEC . The remaining longtail of this resonance at higher energies. is connected with the increase of the function rR for increasing energies . Another example is the J =, T = 1 - , 1 resonance in 160 at 19.6 MeV having the dominant (lp.,)-1(2s.í) nuclear structure. This resonance has a stronger fall-off at higher energies due to the fact that in this case the function f, decreases with energy . At higher energies all the functions r,, decrease of course with energy . For calculations with one channel this decrease has been found numerically in ref. 1°). In realistic cases the position of the maximum in the function PR(E) depends on the energies of the thresholds for channels the reduced widths of which are not small compared to the sum of the reduced widths for all channels. The threshold effects in the functions PR(E) may lead to extremely strong deviations from the Lorentzian line shape. In fig. 6 the energy dependence of the function rR is shown for a 0+ resonance in the 16Nnucleus. The coupled-channels calculations are performed with only one QBSEC in order to investigate the properties of this resonance without any effect of overlap with other resonance .Three channels are taken into account which correspond to the two states I' at 0 MeV, I- at 6.3 MeV

250

1. ROTTER et at.

111

172 -

173

17A

Fig. 6 . The function P. versus energy E for a 0 + resonance of ' 6 N with 22p-2h] nuclear structure. The encrgy E°' of the inelastic neutron channel corresponding to a I + state in "N with lp-2h nuclear structure has been changed .

and to a high-lying I + state in tsN, the threshold energy of which is changed from 16.07 to 16.22 MeV around the energy of the resonance in order to investigate the threshold effect . The reduced width of the resonance considered relative to the two states with negative parity in t ° N is small. The opening of the new channel corresponding to the I + state in 1 'N has therefore a relatively large influence on the total width f ,t. The results of calculations with slightly changed energy of the threshold E?'` (fig . 6) confirm the conclusion that the microscopic behaviour of the functions .r,, at threshold can be explained by the general condition 1`R, JE) -. 0 for E, = E-E:' -+ 0, which follows in a similar manner as the condition 1 R -~ 0 for E -" 0. The function ER(E) shows â peculiarity of the same order of magnitude as the function R(E) near threshold. It is however much less pronounced since ÉR is much larger than 1 R" p~ The behaviour of the function ` R near threshold is reflected by the line shape of the resonance. In fig. 7, the line shape of the 0* resonance in the elastic neutron scattering on ' - N is shown in a small energy region around the threshold of the channel corresponding to the I + state in tsN. Here, the interference with the direct

r

THRESHOLD EFFECTS

25 1

reaction part is of little importance for the line shape. The different calculations . have been performed as in fig. 6 by varying slightly the energy E~r of the inelastic threshold but leaving unchanged all other values including the energy ERA and the wave function O of the QBSEÇ. In fig. 8 the elastic t 3N +n cross section with another isolated 0 + resonance in t 6N is shown, the function AE) ofwhich shows an unusual behaviour near the I+ neutron threshold at El` = 5.299 MeV (fig . 8). The peculiarity in the function 1`(E) is connected with the large reduced width of the 0 + resonance considered, . = 5.299 MeV. The partial width to .' in relation to the channel which opens at E° the elastic channel considered is nearly constant (3.8 keV) in the energy region shown in fig. 8 . The shell model energy E' of the QBSEC is chosen in such a manner that ER < E`h ", ER > E`h`, and ER x E11r . The resonance is not fully symmetric in spite of 1`(E) const. in the first case because of the interference with the direct reaction part . In the second case the resonance has a longer tail at higher .energies corresponding to the increasing function T(E). For ER x El a cusp appears in the cross

idytl~ I

& Uwl ts.sst E7~ ó.K2 nz,

123

114

E [ MwI Fig. 7 . Cross section of the elastic neutron scattering on "N with excitation of the 0* resonance, the function PR(E) of which is shown in fig. 6 . For Eü' see fig . 6.

25 2

1. ROTTER et al.

Fig. 8. Cross section of the elastic neutron scattering on "N with one 0" resonance. The shell model energy E:m of the QBSEC his been varied. The function r,,(E) is shown in the upper part of the figure. The threshold energy of the }+ neutron channel is denoted by El and the direct reaction part by "direct" .

section instead of the resonance. For Asx = 5.4001 MeV it looks very similar to the cusp observed experimentally in the 7Li(p, p) reaction 14) . The cusp obtained for ER "= 5.4000 MeV is similar to that obtained by McVoy et al. t s) in a general case without specifying the structure of the resonance, and by Lovas and Dbnes 1) in solving a three-body problem for a single-particle resonance near threshold. The results shown in fig. 8 lead to the conclusion that a cusp in the cross section will appear only rarely in realistic cases since it is strongly correlated with the barrier properties of the system as well as with the nuclear structure of the resonance and of the channel (residual nucleus) which opens, i.e. with the behaviour of the function P(E) in the neighbourhood of a threshold. 6. Conclusions

In the foregoing sections the influence of the particle decay thresholds on the cross section of neutron scattering on 1sN and on the photonuclear cross section 160+y were investigated . Threshold effects in the reaction cross sections are caused above all by the energy dependence of the functions Pa which changes if new reaction channels are opening. The energy dependence of the functions PA is determined for the most part by the nuclear structure of the resonance level considered and of the channels (states of the residual nucleus) which are open. The magnitude of the threshold effects depends therefore on the nuclear structure including the position of the resonance.

THRESHOLD EFFECTS

25 3

The results of the calculations have shown that the level density of resonances near particle emission thresholds is almost not enlarged as a threshold effect . This is explained in the main by the fact that r1!,, -" 0 as E, -" 0 also for resonances with a large reduced width. Further, the correlation of resonances via the continuum is practically uninfluenced by the appearance of a threshold between the resonances. Besides the distance in energy between the resonances, the nuclear structure of the resonances plays the dominant role also for the correlation of the resonances via the continuum ("external mixing") . The energy dependence of the function Pa may lead to measurable deviations from the Lorentzian line shape of resonances. A cusp in the cross section may appear instead ofa resonance ifa QBSEC lies near a threshold and has a large reduced width in relation to the channel which opens. The cusp obtained in such a way is nothing but a resonance with a special line shape . In most cases the influence of the energy dependence of the function 1 R is of course less pronounced but nevertheless lion-negligible. Energy dependent lifetimes of compound-nucleus fine-structure resonances have been measured in the presence of an isobaric-analog resonance using the crystal-blocking method for the inelastic scattering of protons in germanium r6). This energy dependence is caused by the presence of the isobaric-analog resonance which shortens the mean lifetime of the compound-nucleus fine-structure resonances . It would be very interesting to measure the lifetimes in a similar manner also for othercompound-nucleus resonances in order to see whether they also show an energy dependence. References 1) I. Lovas and E. D6nes, Phys . Rev. C7 (1973) 937 2) B. Gyarmati, A. M. Lane and J. Ziményi, Phys. Lett. SOB (1974) 316; L. P. Csernai and J. Zimdnyi, Int. Symp. on nuclear structure, Balatonf ired (Hungary), 1975, vol. 2, p.259 ; Central Research Institute for Physics report KFKI-76-16, Budapest 1976 3) H. W. Barz, I. Rotter and J. H8hn, Nucl . Phys. A275 (1977) 111 4) I. Rotter, H. W. Barz, R. Wünsch and J. H61m, Particles and Nucleus (USSR) 6 (1975) 435; Zentralinstitut far Kernforschung report ZfK-278, Rossendorf 1974 5) A. M. Lane, Proc. Int. Symp. on highly excited states in nuclei, Jillich 1975, Jül-Conf.-16 (2), p. 95 6) B. Buck and A. D: Hill, Nucl. Phys . A95 (1967) 271 7) A. I. Baz, Adv. in Phys. 8 (1959) 349 8) D. R. Inglis, Nucl . Phys. 30 (1962) 1 9) V. V. Balashov and I. Rotter, Izv. Akad. Nauk USSR (ser . phys.) 30 (1966) 479 10) I. Rotter, H. W. Barz and J. HSIm, Yad. Fiz. 24 (1976) 513 11) I. Rotter, H. W. Barz and J. H8hn, to be published 12) G. A. Bartholomew, Advances in nuclear physics, vol. 7, ed . E. Baranger and E. W. Vogt (Plenum Press, NY, 1973) p. 229 13) H. U. Gersch, E. Hentschel, D. Hinke and H. Schobbert, Zentralinstitut für Kernforschung report ZfK-295, Rossendorf 1975, p. 18 14) P. R. MaImberg, Phys . Rev. 101 (1956) 114; L. Brown, E. Steiner, L. G. Arnold and R. G. Seyler, Nucl . Phys . A206 (1973) 353 15) K. M. McVoy, Nucl . Phys. A115 (1968) 481, 495 ; C. J. Goebel and K. W. McVoy, Nucl . Phys . Aí15 (1968) 504 16) W. M. Gibson, Y. Hashimoto, R. J. Keddy, M. Maruyama and G. M. Temmer, Phys. Rev. Lett . 29 (1972) 74