The role of coincidence experiments in studying the nuclear continuum with high-energy electrons and protons

Nuclear Physics A345 (1980) 367-385

@ North Holland Publishing Co., Amsterdam

Not to be reproduced by photoprint or microfilm without written permission from the publisher

THE ROLE OF COINCIDENCE EXPERIMENTS IN STUDYING THE NUCLEAR CONTINUUM WITH HIGH-ENERGY ELECTRONS AND PROTONS V.V.

BALASHOV,

E.F.

KISLYAKOV

and

V.L.

KOROTKIKH

Institute of Nuclear Physics, Moscow State University, USSR and R. WtiMCH Zentralinstitur fiir Kemforschung Received (Revised

Rossendorf, DDR 8051 Dresden, PSF 19 13 September 21 November

1978 1979)

Abstract: In the framework of the unified theory of direct and nuclei caused by high-energy electrons and protons is excitation. As is shown, the coincidence experiments information about the nuclear continuum, in addition to

resonance processes the disintegration of investigated in a wide region of nuclear (e, e’N) and (p, p’N) can give important the (e, e’) and (p, p’) experiments.

1. Introduction Nuclear

structure

in the continuum

is being studied

on an ever increasing

scale.

The continuum is inhomogeneous. A striking manifestation of this non-uniformity is the presence of giant resonances of various multipolarities and spin and isospin characteristics. The use of high-energy particles can be a promising tool for the spectroscopy

of nuclear

states

in the continuum

‘). However,

many

important

theoretical questions about applying this method have not been developed sufficiently so far. Among these is the question about the relation between the direct and resonance processes of nuclear disintegration by high-energy particles. In the spectra of inelastically scattered electrons and protons (as well as other high-energy particles) the inhomogeneities of the nuclear continuum associated with the collective states manifest themselves against the background of intense direct transitions due to the quasi-elastic scattering of the incident particle on individual nucleons of the nucleus and possibly (especially in the case of strongly interacting particles) on nucleon clusters as well. It is commonly assumed that the quasi-elastic scattering and giant-resonance excitation are caused by different mechanisms, their contributions to the nuclear excitation probability being additive. This additivity hypothesis is widely used in current experimental investigations when extracting information about the giant resonance excitation from the energy loss spectra in (e, e’), (p, p’) and ((Y,(Y’), as well as from the photon spectra in the 367

368 (K,

V. V. Balashoc et al. / Coincidence experimenrs y)

reaction

as a prerequisite approach

refs. ‘“)I. It IS ’ used also in many theoretical papers to overcome the disadvantages of the single-particle

[see, for example in attempting

and to describe

the collective

giant-resonance

phenomena

simultaneously

with the quasi-free direct processes. In this case one sees that it is just the additivity hypothesis that leads to serious double counting defects 5). In the present paper we continue the line of our preceding papers 6-11) giving an unified description of the giant resonances and the quasi-free phenomena in the (e, e’) and (p, p’) reactions on the basis of the continuum shell model. The main idea of ref. 6, is that, in calculating the continuum excitation, a final nuclear state vector & is used taking

into account

the channel

coupling,

where the uncoupled basic vectors cp,, are eigenstates of an independent particle hamiltonian HO, and V”” is a residual interaction. Such a multi-configurational method allows one to trace, without any change in the model, the transformation of the nuclear excitation picture from one with dominating resonance excitation into one with dominating quasi-free scattering. No additivity assumption is necessary. This unified approach has been used on the basis of the lp-lh continuum shell model in refs. 7V8)t o calculate the nuclear response function of the (e, e’) reaction on the “C nucleus as well as the angular correlation characteristics of the (e, e’p) and (e, e’n) coincidence experiments on the same nucleus. The possibilities of the coincidence experiment in studying the multipole giant resonances were specially discussed in refs. 19879).In ref. lo) this approach was used to analyze the (e, e’) experimental data on r2C. The nuclear isoscalar response function which is an important ingredient of the description of the nuclear excitation by hadrons was w ere it also was compared with the corresponding charge considered in ref. ‘I), h response function. Coincidence experiments, such as (e, e’N), (p, p’N) etc., where the scattered high-energy particle is detected simultaneously with an ejected nucleon N, may yield much more information than the conventional experiments on nuclear scattering of electrons or strong interacting particles. The characteristics of the angular correlation between the scattered and the ejected particles are determined not only by the intensities of the transitions of vari&s multipolarities but also by the phase relation between the transition amplitudes. In the theories (both microscopic and collective), if they are based on the idea of additivity of the contributions from the direct and resonance transitions, the questions of such phase relations have not been given any consideration so far. In contrast, in the unified theories the fulfillment of these relations is assured automatically. The advantages of the unified theories should be borne in mind also in connection with the possible use of coincidence experiments to study the branching ratio for various channels of nuclear disintegration

especially

in the case of broad resonances.

V. V. Balashov et al. f Coincidence experiments

369

Thus the problem of using coincidence experiments to study the structure of the nuclear continuum with high-energy electrons and hadrons and the problem of a further development of an unified theory are immediately related to each other. It is also noteworthy that coincidence experiments as a method for studying the resonance structure of the continuum have been proposed not only in nuclear physics but also with reference to electron-atom collisions 12-15);the first experiment of this kind has been carried out just in atomic physics 13).Now the coincidence technique is intensively used also in nuclear physics to study the character of the nuclear disintegration via giant resonance excitation 16) [see also refs. 2*3)and references therein]. In the present paper we investigate the question to what extent the electron and proton coincidence experiments (e, e’N) and (p, p’N) may supplement each other in getting information about the nuclear continuum. The calculations of the (p, p’N) reaction have been performed for 1 GeV protons. In calculating the (e, e’N) reaction only the longitudinal component of the electromagnetic interaction has been considered. In order to generalize the theory of nuclear disintegration by electrons to the case of strongly interacting particles, one should simultaneously take into account both the individual structural features of the nucleus and the multiple collision between the projectile and the nucleons of the nucleus. So far as the latter point is concerned, we draw upon the experience of describing the excitation of discrete nuclear levels in the reactions (p, p’), (r, r’) etc. ‘**19) using the single inelastic collision approximation i9) of Glauber theory. All the calculations in this paper have been carried out for the 12C nucleus within the framework of a version of the continuum shell model developed in refs. 2”*21). Only lp-lh configurations have been taken into account. Other than one-nucleon disintegration channels have not been considered. The parameters of the WoodsSaxon potential and the residual interaction used are the same as in ref. “), where the calculated single-particle energies and resonance parameters are also given. A preliminary publication of the results has been made in ref. **). 2. Formalism,

notations,

parameters

2.1. KINEMATICS

The geometry of the coincidence experiment is illustrated in fig. 1, where k and k’ are the momenta of the incident and the inelastically scattered particle, respectively, and 4 = k -k’ is the transferred momentum. The momentum of the ejected nucleon N = p, n is denoted by kN, while the unit vector I, = kN/]kNI indicates its direction. It is described by the polar angle BNand the azimuthal angle (PNrelative to the direction of the vector 4. The values (PN= 0”/180” correspond to the scattering plane, where (pi = 0” denotes the half-plane on the side of the incident particle.

V. V. Balashov et al. / Coincidence experiments

370

Fig. 1. Geometry

2.2. THE

NUCLEAR

RESPONSE

of the coincidence

experiment.

FUNCTION

The nuclear response function R(E, 4) corresponding to an energy and momentum transfer E and 4, respectively, was given long ago 23). To describe a coincidence it is useful to introduce the correlated response function dRk to the ejection of a nucleon in channel N into the (~5 4, &)/dk^ N w h’tc h corresponds direction iN. If the nuclear transition amplitude can be factorized into an elementary one and a nuclear matrix element, the correlated response function is simply related to the triple differential cross section of the nuclear disintegration into the channel experiment

N:

(2) where (do/dL),;; is the elementary differential cross section for the reaction on a single nucleon. The index i denotes the type of the reaction: charge response function (i = CH) or generalized isoscalar response function (i = GE). At small scattering

angles the (e, e’N) reaction

is described

by the charge response

function

neglecting the transversal components of the electromagnetic interaction. The vector IE, I&) describing the final nuclear state with a nucleon in channel N into the direction iN is normalized as (&N, EIE’, ik,) = S(E - E’)S(iN - cam)&,,. The initial state of the target nucleus consisting of A nucleons is denoted by IO). The charge operator i(l + i3(j)) is equal to 1 for protons and equal to 0 for neutrons. The sums %i & mean averaging and summing over the spin projections of the initial and final states, respectively. In calculating the (p, p’N) reaction we consider only isoscalar transitions to states with “normal” parity n = (- l)‘, bearing in mind that at 1 GeV incident energy the spin and isospin terms of the elementary NN amplitude are small. So we have function. In the present paper we calculated in ref. 11) th e isoscalar response introduce the generalized isoscalar nuclear response function (i = GIS) taking into

371

V. V. Balashor; et al. / Coincidence experiments

account

the distortion dR”lS(E,

effects of the passing

proton

beam,

(I, &,J = -

dkN

T T J(E, &NJj$, S(b) exp {iq . ~j}lO)J’.

The distortion factor S(b), which is a function from the nucleon profile function,

of the impact parameter

6, is calculated

r(b-4=&J exp{id . @-s W(d) d*q’, according

to S(b) = [ 1 -A-’

Here f(q) and p(r) denote

1 r(b -s)p(s, z) d*s dr]*’

the elementary

projectile-nucleon

(6) scattering

amplitude

and the nuclear ground-state density. Within the approximations made in this paper the nuclear response functions (3) and (4) contain all the information about the nuclear structure which can be obtained from (e, e’N) and (p, p’N) reactions. Comparing (3) and (4) one finds a principal difference in the symmetry behaviour of the charge and the generalized isoscalar response functions. At fixed E and 4 the charge response function depends only on the polar angle I&, i.e. it is axially symmetric with respect to the direction 4 of the transferred momentum. In the (p, p’N) reaction, however, this symmetry is disturbed by the multiple scattering of the incident proton on the target nucleons. So the coplanar and non-coplanar (p, p’N) coincidence experiments give, in principle, different information about the excitation mechanism of the nuclear continuum states and their properties. The specific possibilities of the non-coplanar coincidence experiment were studied in the case of the direct (p, 2p) knock-out reaction [see e.g. ref. ‘“)I. In the case of interfering direct and resonance processes of nuclear disintegration this question is quite open and we make in this paper some preliminary estimations on this line (subsect. 3.5). Integrating (3) and (4) over all directions one gets the usual response function

kN and summing

over all open channels

N

(7) which is an additive sum of contributions of various multipolarities J”. With the same restrictions as in the case of eq. (2) it enters as factor into the double differential cross sections of the (e, e’) and (p, p’) reactions: R’(E, 4).

03)

To investigate the relative contributions of the proton and neutron disintegration channels to the cross section (8) we introduce the partial response functions Rk (E, q)

372

V. V. Balashoc et al, / Coincidence experiments

which are obtained neutron

by summing

and integrating

either

over all proton

or over all

channels

(9)

Neglecting

other than one-nucleon

disintegration

channels

one has

R’(E,q)=R;(E,q)+Rh(E,q).

2.3. NUCLEAR

WAVE

(10)

FUNCTIONS

The ground state of the “C nucleus is considered as a shell-model state with closed l.sr,z and 1p3,2 proton and neutron shells. The final state IE, lN) is calculated according to eq. (l), taking into account all the lp-lh configurations with a hole in the 1~~12 state and a particle in a bound or unbound state above the 1pj12 shell. All the single-particle states are calculated as eigenstates of a real Woods-Saxon potential with state-dependent parameters reproducing the experimental singleparticle binding energies. The various particle-hole states are mixed by a residual interaction approximated by a S-force. with spin-exchange term. All the parameters are taken from ref. ‘l). Usually the average potential includes an imaginary part simulating the effect of the configurations and channels neglected in the calculation. In our model, the strength of this absorptive term should be smaller than in the optical model, since the coupling of several channels is explicitly taken into account. We shall not deal with the problem of the determination of the imaginary part of the average potential. We neglect it completely being interested only in relative cross sections which are weakly influenced by the strength of the imaginary part. The mathematics of the continuum shell-model version used is the same as in refs.

11.20,2 1 ).

2.4. PARAMETERS

In calculating parametrization

OF THE HADRON-NUCLEUS

the generalized isoscalar of the NN amplitude

INTERACTION

response

f(q)=gc(l-iLY)exp{fpq*}.

function

(4) we use the usual

(11)

‘. The parameters are taken from ref. 25): g = 44 mb, (Y= -0.3, p = 5.45 (GeV/c) The distortion factor S(b) is calculated with the nuclear ground-state density p(r) of the standard oscillator form (rO = 1.64 fm).

373

V.V. Balashov et al. / Coincidence experiments

3. Results and discussion

In the framework of the model described above we investigated the continuous part of the spectrum of ‘*C from the one-nucleon disintegration threshold up to 60 MeV. Three subregions with considerably different features can be roughly defined. The first one (E s 20 MeV) is characterized by quite well separated narrow resonances. In the medium region (20 MeV s E 6 3.5 MeV) the excitation of giant multipole resonances, while at higher excitation energies the direct quasi-free process dominate.

3.1. COMPARISON SCALAR

OF THE

RESPONSE

GENERAL

PROPERTIES

OF

THE

CHARGE

AND

ISO-

FUNCTIONS

Fig. 2 shows the generalized isoscalar response function RGIS(E, q) presented for fixed values of the momentum transfer 4, as a function of the energy E, together with the charge response function RCH(E, q) calculated earlier ‘I). The genera1 character of the changes in the excitation spectrum of the (p, p’) reaction caused by varying the q = 0.5fti’

q = 1.0 ft-6’

f--7-

q -1.5 fro-’

r---7-

t--71-----1

5 1 3 2 1 Cl l7l92125

35

Is

55 E(MeV)

n

19 21 25

35

45

55 E(MeV)

l719a25

35

L5

55 E(MeV)

Fig. 2. The ‘*C response functions RCH (E, q) and RG’“(E, q) for three fixed values of the momentum transfer q. Upper row: proton RzH (E, q) (solid curves) and neutron Rj;‘” (E, q) (dashed curves) contributions to the charge response function. Middle row: Charge response functions RC”(E, q) and their components Rc”(E, q; J”) corresponding to the indicated multipolarities J”, summed over all proton and neutron channels. Lower row: The same as in the middle row for the generalized isoscalar response function RG’S(E, q). Note that the energy scale is exaggerated between 17 and 21 MeV.

374

V. V. Balashov et al. / Coincidence experiments

momentum when

transfer

4 is consistent

the momentum

function

shifts towards

For a more detailed contributions The reactions

transfer

with that what is known

increases,

larger excitation comparison

the “centre

in the case of electrons:

of gravity”

of the response

energies.

of the electron

and proton

data, fig. 2 shows the

of the various multipolarities to the corresponding response function. (e, e’) and (p, p’) are differently sensitive to different resonances in the

nuclear excitation spectra. The structure of the energy-loss spectra in the (e, e’) reaction, to which the selection rules AT = 0, 1 correspond, is much more complex than that in the (p, p’) reaction, which (according to the assumptions made about the properties of the elementary NN amplitude) obeys the selection rule AT=O.

3.2. THE

PROPERTIES

NEIGHBOURHOOD

OF

THE

ANGULAR

OF NARROW

CORRELATION

FUNCTION

IN

THE

RESONANCES

In this subsection, we discuss the possibilities of investigating the properties of separate narrow resonances analyzing simultaneously both electron and proton coincidence experiments. The angular correlation function is given by the 6, dependence of the correlated response functions (3), (4) mentioned above, the charge response function depends only while the generalized isoscalar response function depends angle (PN. In this subsection we consider only its behaviour ((Pi.,= 0’/180”).

at fixed E and 4. As on the polar angle BN, also on the azimuthal in the scattering plane

The calculation shows (fig. 2) that in the scattered particle spectra the narrow resonances 2 ’ at E = 17.85 MeV, 3- at E = 18.67, and 3 at E = 20.24 MeV are best isolated. In the (p, p’) reaction, the first two of the resonances the T = 1 dominance in the corresponding nuclear

are absent, which testifies to wave function ‘l). The cor-

responding correlation functions for the (e, e’p) reaction are shown in fig. 3. The angular distribution of the protons corresponding to the resonance at E = 18.67 MeV is considerably asymmetric with respect to the plane perpendicular to the momentum transfer vector q. The asymmetry follows from the fact that this 3resonance lies on a rather intensive background, to which a close-lying resonance of the opposite parity 2’ provides a large contribution. The angular distribution of ejected protons at E = 17.85 MeV is on the contrary quite symmetric. This would seem to indicate a high purity of the 2+ resonance. The calculation, however, proves the relative contribution from the E2 tranthe contrary. Whereas at q = 1.5 fm-’ sition to the nuclear excitation probability is indeed large, over 95%, at q = 0.5 fm-’ it amounts to merely about 75%, the remainder being almost entirely due to the El transitions. Such a relation between the E2 and El transitions usually leads to a high asymmetry in the angular correlation function. The fact that, in the given case, the calculation practically does not show any interference between these transitions is

K V. Balashov et. al. / Coincidence experiments

375

a =0.5fm-l

E =17.85 MeV

E = 18.67 Me

Fig. 3. Polar diagrams of the angular correlation function dRgH(E, q, k&/d& (in arbitrary units) for the *‘C(e, e’, p)llB reaction as a function of the polar angle 0, of the ejected nucleon. The direction of the momentum transfer q is indicated by the arrows (@, = 0”).

associated with the pecularities in the structure of the 2+ resonance being considered and of the adjoining continuum. The resonance 3-, T = 0 at 20.24 MeV is well pronounced in the spectra of nuclear excitation by both electrions and protons at larger values of the transferred momentum. Due to the T = 0 selection rule in the proton scattering the background under this resonance is much smaller in the (p, p’) than in the (e, e’) reaction. This background depression turns out to be an advantage of the high-energy proton experiment for identifying the multipolarity of a resonance. Figs. 4 and S show the angular correlation functions for the four reactions (e, e’p), (e, e’n), (p, 2p) and (p, p’n) at E = 20.24 MeV. There is a significant difference between the shape of these correlation functions in the (e, e’N) and (p, p’N) cases. In the latter it is much more symmetrical due to the background suppression. We note also that the signs of the asymmetry of the correlation function in the (e, e’p) and (e, e’n) reactions are opposite to each other. .While protons are preferably emitted into the hemisphere directed to the vector q, neutrons prefer the opposite hemisphere [in this connection see also ref. “)]_

3.3. EXCITATION

AND DECAY

OF GIANT RESONANCES

A combined analysis of the data presented in fig. 2 for electrons and protons enables one to gain an insight into the general resonance structure of the continue of the nucleus. Between 21 and 25 MeV the giant dipole resonance is dominant. The isoscalar E2 resonance is situated near E = 30 MeV and, as was suggested long

376

V. V. Balashov et al. / Coincidence experiments

q = 0.5 ‘5’

q =1.5fm+

E -202LMeV

E=30

MeV

Fig. 4. The same as in fig. 3 for three larger values of the excitation energy E. Here, the solid curve describes the ejection of protons and the dashed curve the ejection of neutrons.

ago 26), is well separated from the main group of isovector E2 transitions. The transition strength corresponding to the isovector E2 resonance and to the giant resonances of higher multipolarities are distributed over a very wide excitation energy region. These transitions do not form, at least at small 4, distinct maxima. The conclusion that the isoscalar E2 giant resonance in the “C nucleus is located near 30 MeV is in line with the inferences drawn from experimental studies 27-2g) according to which, in this nucleus, only a small fraction of all E2 transitions is concentrated at E < 30 MeV. We emphasize however, that the present calcula .ions are based on the simple lp-lh model and they should not be used for a quantitative analysis of the experimental data. Fig. 6 shows the distribution of the probabilities for the isoscalar E2 transitions calculated by two different methods. The solid curve corresponds to the data of fig. 2 obtained in the lp-lh model. The bins indicate the probabilities for the transitions /(l~)~(lp)*; O’)+ lo’; 2+) calculated by using the intermediate coupling model 30). In terms of the ph representation, this calculation corresponds to the inclusion of the vacuum polarization of the “C ground state and of the spreading of the lp-lh excitations over more complicated states. Of course, the data in fig. 6 which have been obtained with two different models, should

V. V. Balashoc et al. / Coincidence experimenls

377

a= l.Sfd

a = 0.5 fni’

\ IE-M.XMeV i

/

\ E=3OMdr

/ I

E= 60 MeV

Fig. 5. Polar diagrams of the angular correlation function dR :“(I?, q, &t.,)/d~x (in arbitrary units) for the r’C(p, p’N) reaction with 1 GeV protons as a function of the polar angle 0~ in the scattering plane of the high-energy proton. The solid curves describe the ejection of protons, the dashed curves the ejection of neutrons. The arrow indicates the direction of the momentum transfer q. On all the polar diagrams the incoming particle momentum lies in the left half-plane.

?X ’

20

30

Lo

E(MeV)

Fig. 6. The 2’ component of the generalized isoscalar response function for the ‘*C(p, p’) reaction with 1 GeV protons at q = 1.0 fm-‘. The solid curve corresponds to the data of fig. 2, the bins showing the contributions from the transitions Ins; O’)+l(l~)~(lp)s; 27 within the lp shell.

378

V. V. Balashov

et al. / Coincidence TABLE

Relative

1

contribution XL (E, q. J”) of various multipolarities (in ‘%) to the charge generalized isoscalar response functions of 12C at E = 30 MeV 12C(e, e’)

J”

experiments

CH XP

r2C(p, p’); EP = 1 GeV CH X”

GIS XP

q=OSfm-’ 0' 1234Z(J”)

2.3 27.9 31.9 0.7 0.2

37.0

1.8 8.8 47.0 1.1 0.8 59.6

q=l.Ofm-r 0'

1.5 18.8 40.4 3.3 5.2 0.2

4.6 11.5 11.9 2.1 0.5 -

VJ”)

69.4

30.6

0.4 2.5 52.7 1.4 5.4 0.2 62.6

q = 1.5 fm-’

12’ 34’ -+ I;J-i

1.3 9.6 24.9 13.8 21.7 0.4 71.7

1.8 8.6 28.7 0.9 0.4 40.4 q=l.Ofm-’

1 23+ 4’ 5’

0'

GIS

X”

q = 0.5 fm-’ 2.5 27.2 7.0 0.2 -

63.0

and

0.4 2.4 30.2 1.2 3.1 0.1 37.4 q = 1.5 fm-’

3.1 3.8 11.4 7.0 3.0 28.3

2.9 6.2 20.6 2.1 29.5 0.4 61.7

2.8 6.2 12.3 2.0 14.8 0.2 38.3

not be just added, because in part they reflect the same effects. Their quantitative comparison, however, gives an indication of the scale of the effects ignored by the lp-lh model. Table 1 presents the relative contributionsXk (E, 4; 7) = RN@, q, J”)/R’(E, q) from the transitions of the various multipolarities to the total excitation probability at E = 30 MeV. For the excitation by electrons, the maximum contribution of E2 transitions slightly exceeds 50%. Note that the branching ratio between the proton and neutron disintegration channels in the 2’ contribution to the (e, e’N) reaction varies considerably with the transferred momentum. Such a behaviour, quite untypical for an isolated resonance, is due to the admixture of a small 2’, T = 1 component at 30 MeV. In the coincidence study the E2 transition strongly interferes with transitions of other multipolarities, largely with the tail of the giant dipole resonance. Consequently, in the (e, e’p) and (e, e’n) reactions near the giant E2

V. V. Balashov et al. / Coincidence experiments

379

resonance the angular correlation functions have a rather complicated shape, asymmetric with respect to ON= 90”, which greatly changes (especially, in the neutron channel) with the changing momentum transfer (figs. 4 and 5). In the (p, p’) case, the giant E2 resonance is much purer, because an intensive background of El transitions is here absent. In the (p, 2p) and (p, p’n) reactions, the asymmetry in the correlation function is weaker than in the reactions with electrons. We note, that at the largest momentum transfer considered, the bump at 30 MeV in the (p, p’) excitation spectrum is determined mainly by E4 transitions. The situation in the quadrupole giant resonance region enables one to treat the nuclear disintegration process, neither in the (e, e’) nor in the (p, p’) reactions, as a simple additive superposition of the resonance E2 excitation and a direct quasielastic scattering. Firstly, the E2 excitation concentrated in the region of 30 MeV is formed predominantly from transitions to unbound single-particle states, mostly from those leading to the wide If shape resonance. On the other hand, shape resonance effects are taken into account automatically in the framework of a usual direct disintegration model. So, a double counting arises if one tries to treat resonance and direct E2 transitions separately. Secondly, our calculations show that, in the ‘*C case, the summed contributions of non-quadrupole transitions to the (e, e’) and (p, p’) energy loss spectra in the E2 giant resonance region can hardly be reproduced by a smooth and monotonic function of the excitation energy, contrary to that what is often made “by eye”. 3.4.

ON THE

E-DEPENDENCE

OF THE

CORRELATED

RESPONSE

FUNCTIONS

Up to now we have demonstrated the properties of the correlated response functions presenting them depending on the direction ih’ of the nuclear emission at fixed E and 4. Figs. 7 and 8 present the charge and the generalized isoscalar response functions calculated at some fixed 4 and l& as functions of the nuclear excitation energy E (we shall call these quantities the coincidence excitation spectra). a=1.0fm4

q =1.5fm-l -I

l!!Y ’

o”

,’

/

-

90”

180°

Fig. 7. Coincidence excitation spectra in the “C(p, 2p)“B (solid curves) and the 12C(p, p’n)“C curves) reactions calculated in the scattering plane at 8, = 0”, 90” ((Pi = 0”) and 180”.

(dashed

V. V. Balashov ef al. / Coincidence experiments

380 q -

0.5 fm-’

q - 1.0 fm-’

q = 1.5fn-i’

0 t

I

i‘3l

Fig. 8. Coincidence

Boo

40

50

E (MeV)

excitation spectra in the ?(e, e’p)“B (solid curves) and the “C(e, curves) reactions at 0~ = O”, J5”, 90”, 135” and 180”.

e’n)“C

(dashed

In the (p, 2p) and (p, p’n) reactions, the shape of the coincidence spectrum is largely determined by the interference of the collective E2 transition and a coherent contribution of higher multipolarity transitions. At energies above the E2 giant resonance this coherent contribution manifests itself, in the direction close to the vector q(tlN + O’), as the direct quasi-elastic knock-out of a nucleon. In the opposite direction (/3, + 180“) (anti-quasi-elastic kinematics) the constructive coherence vanishes, and the quasi-elastic background is strongly suppressed [see ref. 13) for an analogous effect observed in atomic physics]. In the (e, e’p) and (e, e’n) reactions, the shape of the coincidence excitation spectra is much more complicated, because it involves both isoscalar and isovector transitions. The main features of the coincidence spectra between 25 and 35 MeV are determined by the interference of El and E2 transitions.

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V. V. Balashov et al. / Coincidence experiments 3.5. ON THE

DISTORTION

EFFECTS

IN REACTIONS

WITH

HIGH-ENERGY

PROTONS

The distortion of the wave functions of the passing proton in the (p, 2p) and (p, p’n) reactions affects the absolute value of the cross section (decreasing it, in our case, by a factor of about 2) as well as the shape of the angular distribution of the ejected nucleons. In the plane-wave approximation (for the passing particle) the angular distribution of the ejected nucleon, in the (p, p’N) reaction, is characterized, as in the (e, e’N) reaction, by an axial symmetry with respect to the momentum transfer vector q (see subsect. 2.2). The strong interaction of the incident hadron with the nucleus disturbs this symmetry by introducing a competing factor of the axial symmetry relative to the incident beam direction. In order to illustrate the violation of the q-axis symmetry the angular distributions of ejected particles in two particular cases are presented: (a) in the scattering plane, and (b) in the plane perpendicular to the momentum transfer vector q. In fig. 9 the angular correlation function for protons ejected in the scattering plane is given for the excitation energy E = 30 MeV which corresponds to the energy of the giant quadrupole resonance. It shows a large number of oscillations due to the considerable

contributions

of higher

multipolarities.

The distortion

deforms

the

shape of this correlation function, so one can hardly find any new symmetry axis for it. The strongest deformation of the angular correlation function is expected in the directions perpendicular to the transferred momentum q. q = 1.0 frri’

E ~30 bleV

q = 1.5 li-ri’

E-SOMeV

q= 1.5 fti’

E=60MeV

Fig. 9. Polar diagrams of the angular correlation function dR,C’lS (E, 4, k^+J/dkN (in arbitrary units) for the reaction ?(p, 2p)“B as a function of the polar angle 0~ calculated in the scattering plane with (solid curves) and without (dashed curves) taking into account the distortion of the wave function of the high-energy proton. The arrow indicates the direction of the momentum transfer 4. On all the polar diagrams the incoming particle momentum lies in the left half-plane.

In order to show the peculiarities of the (p, p’N) reactions in non-coplanar geometry we present in fig. 10 the azimuthal distribution of the protons ejected in the plane perpendicular to the vector q. The violation of the axial symmetry turns out to

be rather strong. It is interesting to note that the distortion effects lead to a preferable ejection of nucleons in the directions (PN= *f?r.

V. V. Balashov et al. / Coincidence experiments

382

270°

-.

yz o”-

@

-WI”

90°

Fig. 10. Polar diagram of the azimuthal dependence of the angular correlation function dRF’“(E, q, k^,)/dk, (in arbitrary units) calculated for the reaction ‘*C(p, 2p)“B in the plane perpendicular to the vector q(& = 90”). The circle indicates the correlation function obtained without taking into account the distortion effect.

3.6.

ON THE NUCLEAR

RELATION

BETWEEN

THE

PROTON

AND

NEUTRON

CHANNELS

OF

DISINTEGRATION

All characteristics of the proton and neutron channels of nuclear disintegration by protons are, if they are taken at the same nuclear excitation energy, on the whole close to one another. This is a manifestation of the isoscalar character of the nuclear excitation in the high-energy (p, p’) reaction and of an almost rigorous protonneutron symmetry of the structure of the r2C nucleus in the ground and excited states. Naturally, at the lowest excitation energies, where the difference between the proton and neutron thresholds and the Coulomb interaction of the ejected proton with the residual nucleus are more important, there is a substantial difference between the proton and neutron channels. It would seem that the role of these factors should decrease gradually as the nuclear excitation energy increases. The calculation shows, however, that a different situation takes place. For instance, the ratio of the yield of the proton channel to that of the neutron channel in the (p, p’N) reaction varies non-monotonically as a function of the excitation energy. In the region of the giant quadrupole resonance, one observes a maximum of this ratio, the proton yield being about 1.5 times higher than the neutron one (table 1, fig. 11). Such violations of

to the Fig. 11. Proton Rz”(E, q) (solid curve) and neutron Rz” (E, q) (dashed curve) contributions at generalized isoscalar response function for the laC(p, p’) reaction with 1 GeV protons calculated q=l.Ofm.‘.

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the proton-neutron symmetry in the (p, p’N) reaction at high excitation energies may present considerable interest from the viewpoint of elucidation of the nuclear disintegration mechanism. For the nucleus “C being considered the cause of the effect is, that in the region of the giant quadrupole resonance, the role of the transitions O- + 2+ and 0’ + 4’, which correspond to the f-wave of the ejected proton or neutron, is especially great. The centrifugal barrier in the f-wave is very high and a relatively small difference in the kinetic energies of the proton and neutron, which is associated with the threshold difference, leads to a large difference between the penetrabilities of the barrier in the proton and neutron channels. Thus, in the case under consideration, the preponderance of the proton channels of the nuclear disintegration in the (p, p’N) reactions near E = 30 MeV serves as a peculiar indicator of the contribution from the single-particle f-wave shape resonance to the collective excitation of the multipolarities J = 2 and J = 4. In the (e, e’N) reactions, the characteristics of the proton and neutron channels are, as a rule, very different from one another. The very yield of neutrons in the (e, e’N) reaction in the condition for which our calculations have been performed is entirely due to the charge-exchange process p + n in the final state, because we neglect the direct electron-neutron interaction. The effect of this process is maximum in the resonance states characterized by a definite value of the isospin. However, the coupling between the proton and neutron channels, which gives rise to the chargeexchange, remains quite appreciable to energies much higher than those of the giant resonances (fig. 2, upper row). In this region, the angular distribution of neutrons in the (e, e’n) reactions is very close to that of protons in the (e, e’p) reaction and is quite consistent with the conception of the direct quasi-elastic knock-out (figs. 4 and 9). As was already stressed in ref. “), an experimental study of the (e, e’n) reaction in the conditions enabling one to exclude the contribution from the transverse electron form factors is highly important for getting information about the role of such “post knock-out” charge-exchange effects in disintegration processes induced by highenergy particles. 4. Conclusions In the present paper, the main features of nuclear disintegration by high-energy electrons and protons over a wide region of the nuclear excitation energy from the threshold up to several tens of MeV have been considered from a unified point of view. We do not have experimental information about the (e, e’N) reactions or, with high-energy protons, about the (p, p’N) reactions investigated in the giant resonance region to relate it to our theoretical predictions. At the same time we would like to mention that the ‘*C nucleus, for which all the calculations have been made in this paper, may be less suitable for the application of the lp-lh approach than, say, the doubly magic nuclei I60 or 4nCa. Yet we started from such a nucleus to emphasize the methodological aims of our study. It is important for us to have a qualitatively true

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picture of the nuclear disintegration by electrons and protons, but we do not think that the calculations made will permit a quantitative analysis of the future experiment just on the 12C nucleus and particularly will lead to a close correspondence between the calculated and observed resonances. Our main purpose was to forecast the new general problems, unrelated merely to the chosen nucleus as well as to the chosen nuclear model, to be encountered by the experimentalist when applying the coincidence technique in high-energy nuclear physics as a method for investigating the structure of the nuclear continuum. The main result of the analysis performed lies, to our opinion, in the demonstration of the various differences between the nuclear disintegration process by electrons (e, e’N) and the process (p, p’N). Indeed, these two types of experiments could give much information supplementing each other about the nuclear structure in the continuum. A part of the differences between the (e, e’N) and (p, p’N) processes is caused simply by the strong AT = 0 selectivity of the high-energy proton inelastic scattering. But we have also shown the non-trivial role of the distortion effects in the (p, p’N) reaction. One aspect of this problem -the transformation of the symmetry properties of the angular correlation function when coming from the (e, e’N) to the (p, p’N) reaction-seems to be especially important for the search of an optimal combination of coplanar and non-coplanar (p, p’N) experiments in future giant resonance studies. Due to strong absorption effects, the (p, p’N) reactions are of a more peripherical character than the corresponding (e, e’N) reactions. In principle, this difference can be reproduced quite well by the theory suggested. However, the 12C nucleus chosen for the calculation is too light to consider this problem here. A similar calculation for the 40Ca nucleus, where this difference is much more important, will be published elsewhere. The authors are grateful to N.N. Titarenko for helpful discussions.

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