Inclusive particle spectra from light-ion fragmentation processes

Nuclear Physics ONorth-Holland

A442 (1985) 122-141 Publishing Company

INCLUSIVE PARTICLE SPECTRA FROM LIGHT-ION FRAGMENTATION PROCESSES A.G. SITENKO

and A.D. POLOZOV

Institute for Theoretical Physics, Academy of Sciences of Ukrainian SSR, Kiev, USSR and M.V. EVLANOV

and A.M. SOKOLOV

Institute for Nuclear Research, Academy of Sciences of Ukrainian SSR, Kiev, USSR Received 7 January

1985

Abstract: The theory of diffraction dissociation and stripping is extended to nuclei with low binding energy with respect to two-cluster break-up. The cross sections calculated using the formalism developed are compared with experimental data on the fragmentation of deuterons, 3He and 6Li ions in a wide range of incident beam energies and nuclear mass numbers. The calculations show that the diffraction approach provides a correct description of characteristic regularities in the experimental cross sections.

1. Introduction

In recent years, great interest has been shown in the experimental and theoretical studies of inclusive spectra of particles emerging from the interactions of both light and heavy ions with composite nuclei at energies from tens to hundreds of MeV [refs. ‘-13)]. The reason for these studies drawing researchers’ attention is that such investigations allow one to make definite conclusions about the mechanism of nuclear processes over a wide range of incident particle energies and nuclear mass numbers, and to obtain valuable information on the nuclear structure and various characteristics of nuclear reactions. Inclusive spectra were studied in refs. 5,8,9*14) by means of the distorted-wave Born approximation break-up theory for the cases of both elastic and inelastic break-up of incident nuclei. An analysis of inclusive reactions is given in refs. le4) on the basis of a simple stripping model introduced by Serber 15). For energies sufficiently high, the particle interaction is known to be diffractive. Diffraction phenomena occur if the wavelength associated with the relative motion of colliding particles is small compared to the characteristic dimensions of the interaction range; hence the criteria of the diffraction approach applicability are readily satisfied. Interactions between nuclei and composite particles are accompanied by various diffraction processes. Along with elastic scattering and total absorption, there occur stripping processes, such that only one of the particles initially 122

A.G. Siienko et al. / Inclusive particle spectra

123

contained in the incident system is absorbed by the nucleus, dissociation processes, when the scattered particle disintegrates into two or more fragments due to the nuclear field, and other direct nuclear reactions. The detailed theory of diffraction deuteron-nucleus interaction was developed in ref. 16). The diffraction dissociation of deuterons was first predicted by Akhiezer and one of the present authors “) and, independently, by Glauber ‘*) and Feinberg 19). It was later investigated experimentally and theoretically20-23). The theory of deuteron diffraction stripping was developed in ref. “). The influence of nuclear edge diffuseness and Coulomb forces on the stripping cross section was studied in refs. 25*26). It was pointed out in ref.27) that diffraction phenomena (elastic scattering, dissociation, and stripping) can also occur during the collisions of weakly-bound light nuclei with heavier ones. Elastic scattering and dissociation of weakly-bound light ions having cluster structure was studied in ref. 28) in the framework of the diffraction approach with neglect of the nuclear boundary diffuseness. The consideration of nuclear boundary diffuseness is of considerable interest owing to the high sensitivity of the diffraction cross sections to the peripheral range of the nucleus. As was shown in ref. 29), the integral cross section of deuteron diffraction dissociation greatly decreases as the nuclear boundary diffuseness increases. The nonmonotonic dependence of the cross sections of deuteron-nucleus processes on the target-nucleus mass number was then investigated both experimentally and theoretically30-32). As follows from calculations carried out in this work, the conclusion about the sensitivity of the cross sections to the nuclear boundary is also correct in the case of an interaction between weakly-bound two-cluster ions and nuclei. The purpose of the present paper is to show and illustrate by typical examples that, within the applicability range of the diffraction model, stripping and dissociation reactions provide the main contribution to the inclusive spectra. To do this, we extend the diffraction dissociation and stripping theory to the case of nuclei having low binding energy with respect to two-particle fragmentation. The nuclear boundary diffuseness and the finite range of nuclear forces between clusters are taken into account. The incident particle size is not assumed to be small compared to the target nucleus. The formalism thus developed is applied to numerical computations. The latter are compared to experimental data on the reactions (d,pX) for the deuteron energy 56 MeV [ref. 3)], (3He, dX) for beam energies 70, 90, and 110 MeV [ref. 4)], and (6Li, dX) and (6Li, CWX)for the beam energy 156 MeV [refs. 7,8)], over a wide range of target mass numbers. 2. Diffraction dissociation of light ions The differential cross section of the two-body nuclear dissociation may be written as 28)

(1)

124

A.G. Sitenko et al. / Inclusiveparticle spectra

where k is the c.m. momentum of the incident nucleus (k = k, + k,, k, and k, are the momenta of corresponding clusters), q = k - k’ is the momentum change in the scattering (q = 2ksinf@, 13 is the scattering angle), and f= &k, - &k, is the momentum of the fragments relative motion after the collision (pi = mj/( m i + mj), mj is the mass of the j-cluster). The two-body diffraction dissociation amplitude may be presented in the form

(4 where r = r, - r2 is the radius vector of the relative motion of the clusters, the total profile function w, being expressed in terms of the individual profile functions for the clusters according to Wrl(b) = q(U

+%(bz)

- %(b,)%(b*);

(3)

b = /3,b, + &b2 is the projection of the c.m. radius vector of the incident nucleus in the plane perpendicular to k, and b, and b, are the projections of the cluster radii vectors in the same plane. Let the wave functions of the cluster relative motion before and after the collision be of the form cpO(r) = (2y2/n)3’4exp( G,(r) = exp(if*r)

-y2r2),

- J8exp( -f2/4-r2

(4) - y2r2),

(5)

where y is a parameter associated with the beam particle dimensions. The functions ‘pa and ‘pr are orthogonal. The cluster profile functions c+(b), taking into account the diffuseness A of the nuclear boundary, are of the following form:

o,(b)q,,)(b)=

I

R-+Acb

1,

(R+;;12-b2,

R-+A
(6)

where R is the nuclear radius. In order to simplify the calculations, we assume the decrease of the function w(b) to be parabolic. For the black-nucleus model, A = 0 and the profile function (6) reduces to q,(b): b
125

A, G. Sitenko et al. / Inclusive particle spectra

Substituting expression (8) into (2), we can write the diffraction plitude in the form

dissociation am-

(9) where

F(q, f ) = /dvj?r)eiq%dr)

= (2n/y2)3'4(

exp[ - ( f-

q)*/4y2]

- exp[ - (f2/4v2

+ q2/32y2)]

) . 00)

With w(b) of the form (6), the amplitude f(q) f(4)

= Mq)

is determined by the expression

= ik[go(q)

(11)

+~ddl~

where (12)

hdd = -_&

[p2J2(q3)]

(S=R+A’2 _ RJl(qR) . 4

3-R-A/2

(13)

Expression (13) depends on the nuclear edge diffuseness and vanishes if w(b) is taken in the form (7). Thus, similarly to ref. 28), the diffraction dissociation amplitude is calculated in the impulse approximation, i.e. the last term in expression (9) is disregarded. This approximation is quite justified for comparatively small scattering angles, where the contribution of double scattering is insignificant. To obtain the energy distribution of the second fragment, it is necessary to pass in (1) from the variable f to k,, and q2 = - ( f, + p2q), where q2 is the component of the cluster wave vector perpendicular to k. Integrating (1) over the variable q and the directions of the two-dimensional vector q2, we obtain the energy distribution of the second fragment in the solid angle da,: d2ud (kR)2&? dL?,dE, = 2Y2Ja exp X

where x=qR,

x2 = q2R,

(F2-Gz)2

mdxxg2(x) i 4(x, i-

/0

p = fiyR,

-- x2’

L

P2 I

4,

(14)

1

L=ii2y2/m2,

E is the incident

particle

A.G. Sitenko et al. / Inclusive particle spectra

126

energy, and the functions Hi(x, x2) are of the form H,(x, ~2) = 1, H,(x,x,)=

H,(x,x,)

[exp( -$x2)

= -2[exp(

+exp(

- 5x2)

- “~Pf”x2)~Io(

+enp(

Fx2x),

- 2’i;“‘x2)]Io(

$x2x),

H,( x, x2) = 2 exp

H,(x,x,)=

-2 [exp ( - %x2)

H,( x, x2) = exp ( - -x ;

Integrating fragment:

+exp( - 2”~~~+2x2)]10(

+/x2x),

2)Io( ;x2x).

(14) over energy, we obtain the angular distribution

(kRJ2P,2 exp

d% -= dQ2

of the outgoing

6

dxxg2(x)

Y2

C Hi(x>x2)*

05)

i=l

3. Diffraction stripping of light ions

Let us consider now the stripping reaction. The differential cross section of the stripping of the second cluster, resulting from the absorption of the first cluster by the nucleus, is the following: d2aS dGdk,z

=

06)

~J[1ao(k2,r,)12-1a(k2,r,)12]dB,,

where a(k,, ri) is the amplitude of the probability for the outgoing cluster to possess the wave vector k, and the absorbed cluster to be at the point r,, a(k,,q)=glexp(-ik,

.r2)(I - ~d)~O(r)cxp[ik*(P,rr

+PA)]

dr2,

(17)

and a,(/~,, ri) is the same amplitude neglecting the interaction between the absorbed cluster and the nucleus. Using the evident form (4) of the wave function q,(r) and expression (6) for the profile functions, we obtain the following energy distribution of the cluster for a fixed scattering angle: d2uS dQ,dE,

= 2( kR3) 2-$$xp[

- k(&

- m)Z]p(+).

(18)

A.G. Sitenko et al. / Inclusive particle spectra

127

The function F(x*) is expressed in terms of a rapidly convergent series,

5 (/l-adslSlew( -p23f)IGm(x2, L) I2

F(x2) =

0

pJ-_m

G,(x,,S,) =[TodS212ew( - b23;)L( p2S1S2)Jm(x2Yd

+ /1+adS2S2W2)exp( - 3p25z)L(~2L52)Jm(&) (20) T

l-a

with the sign conventions as follows: Q(l)

=

r2- 0 -

4’ ’

4a

Q(3)= 1 - ti2(3),

A

a=2R’

(21)

Integrating (18) over energy, we obtain the angular distribution of particles for the stripping process:

2 =4(kR3)2j3;y2F(x2).

(22)

dfJ2

Since both dissociation and stripping contribute to the inclusive spectra of light-ion fragmentation, the energy and angular characteristics of the fragmentation processes are determined by the sum of (14) and (18), as well as of (15) and (22): d2ai z-+--sd2ad d2a dGdE’ dQdE dDdE dq da, do, dS2 -d0+dQ*

(23) (24)

Integrating expressions (15), (22) and (24) over the solid angle, one can obtain the integral cross sections of the corresponding processes. 4. Comparison with experimental data As was mentioned in the introduction, a great number of experiments are devoted to the measurement of inclusive spectra during the interaction between light ions and nuclei in a wide range of incident particle energies and nuclear mass numbers. Now we shall apply the formalism developed in the preceding sections to explain the measurements of proton inclusive spectra from the (d,pX) reaction for an incident deuteron energy 56 MeV [ref. 3)], deuteron inclusive spectra from the (3He,dX) reaction for incident ion energies 70, 90, and 110 MeV [ref. 4)], and inclusive spectra of particles from the (6Li,dX) and (6Li, ax) reactions for an incident energy 156 MeV [refs. 7,8)]. These experiments have revealed some general characteristic features in the behavior of the cross sections: (i) wide maxima are observed in the measured spectra

128

A.G. Sitenko et al. / Inclusive particle spectra

Fig. 1. Inclusive spectra of protons from deuteron fragmentation and deuterons from 3He break-up for various targets; the proton emission angle i$ = 9.5” for deuteron energy E = 56 MeV (a); the deuteron emission angle 0, = 13”, for energies of incident )He ions equal to 70 MeV (b) and 90 MeV (c). Experimental data are taken from refs. 3,4).

A.G. Sitenko et al. / Inclusive particle spectra

129

for all target nuclei; (ii) the maximum peaks correspond to the energies E' = (m'/m)E, where E',E, m',m are the energies and masses of outgoing and incident particles, respectively; (iii) the cross sections reach their highest values at small angles, rapidly decrease at greater angles, and increase as the target mass number increases. Let us begin with the analysis of the (d, pX) and (3He, dX) reactions. To interpret their results, the authors of refs3y4) used a method, based on the Serber model ref. 15), that assumed the incident particle dimensions to be small in comparison with the target nucleus. Moreover, only the mechanism of elastic dissociation of deuterons and 3He ions was taken into account, while inelastic processes, in which the fragment initially included in the beam particle is transferred to the target nucleus, were ignored. Therefore, the authors failed to achieve quantitative agreement between the calculations and the experimental data and thus had to fit the absolute values of calculated cross sections for each target nucleus. Making use of (23) and (24) enables one to carry out the calculations irrespective of the normalization procedure; the theoretical cross sections in all figures are shown in absolute units (solid curves). The radii of the target nuclei are assumed to be R = r,,A'/3, where r, = 1.25 fm, and the diffuseness parameter A = OAR. For the deuteron, the magnitude of the parameter y that was used in the calculations (0.19 fm-‘) is close to the value corresponding to the rms radius obtained for the Hulthen

Fig. 1. (continued).

130

A.G. Sitenko et al. / Inclusive

particle spectra

Fig. 1. @ontimed).

and Gartenhaus wave functions with correct asymptotics 34*35).For the 3He nucleus, the value y = 0.39 fm-’ is close to the result corresponding to the rms radius equal to 1.88 fm that was obtained from experiments on electron-nucleus scattering36s37). The inclusive spectra of protons produced by the fragmentation of deuterons and deuterons from jHe ion break-up reactions are shown in fig. 1. Evidently, the theory describes fairly well the shapes, widths, and absolute values of the cross sections, and also the experimental fact that the position of the cross section peak corresponds to the energy E’ = (m’/m)E. Fig. 1 shows also that the absolute values of the spectral peaks are correct for light nuclei and underestimated for heavy nuclei. Apparently, this is due to the considerable Coulomb interaction for nuclei with a large charge. The analysis of this effect is a separate problem.

131

A. G. Sitenko et al. f Inclusive particte spectra

Fig. 2 gives the inclusive spectra of protons resulting from the 27A1(d,pX) reaction and deuterons from the WZr(3He,dX) reaction, It shows that our calculations provide a satisfactory description of the experimental dependence of the cross sections on the scattering angle: the magnitudes of the cross sections rapidly decrease with increasing angle of fragment emission.

15’

I

I

I

-

g”Zrf3He,d)

0

10

30

w Ey

I

I

I

I

(b):

l13”

40

50

60

70

(MeVJ

Fig. 2. Inclusive spectra of protons from 27Al(d,p) reaction for the deuteron energy E = 56 MeV (a); deuterons from w Z$ He, d) reaction for incident 3 He ion energies 70 MeV (b) and from g2Zr(3 He, d) for energy 110 MeV (c) for various angles of fragment emission. Experimental data are taken from refs. 3*4).

132

g2Zr(%, d) E<

IfQMeV

40

60

80

100

EdbbfMeVl Fig. 2. (continued).

Fig, 3. Expefimenta angular distributions of protons from the break-up of deuterons with energy 56 MeV for the nuclei *‘Al, 58Ni,40Zr [ref. 3)] (a); deuterans from ‘He fragmentation for the nuclei CIXWS we =‘A&9QZr, 2wBi [ref.4)] for incident beam enfqies 70 MeV (b) and 90 McV (c). ‘lharetical calculated for the reactions 27Al(d,p) and w Zr(3 He, d) at corresponding energies.

A.G. Sitenko ef al. / Inclusive particle spectra

Fig. 3. (continued).

133

134

A.G. Sitenko et al. / Inclusive particle spectra

Fig. 4. Target-mass-number dependence of the peaks of deuteron inclusive spectra from the A(3He,d) reaction for a deuteron emission angle 0, = 13” and incident beam energy 90 MeV. Experimental data are taken from ref. 4).

Fig. 5. Calculated integral cross sections of the diffraction dissociation c,, , stripping cS, and their sum ei as functions of the nuclear edge diffuseness from the 92Zr(3 He, d) reaction for an incident beam energy 110 MeV.

Fig. 3 shows theoretical angular distributions of protons resulting from the 27A1(d pX) reaction, deuterons from the 90Zr(3He, dX) reaction, and corresponding expedmental data for a number of nuclei. The figure demonstrates a distinguishable maximum in the fragmentation cross sections at small angles, which is peculiar to the direct processes. Fig. 4 shows the peaks in the inclusive spectra of deuterons produced by 3He fragmentation as functions of the mass numbers of target nuclei. AS follows from

A.C. Sitenko et al. / Inclusive particle spectra

135

figure, the theory correctly reproduces the experimental observation that the increase of the cross sections is proportional to the nuclear radius. It shows the peripheral character of the nuclear interaction mechanism. For the 92Zr(3He,dX) reaction, fig. 5 illustrates the influence of the nuclear boundary diffuseness on the calculated integral cross sections of diffraction dissociation q,, stripping us, and their sum ui. It shows that the stripping cross section us increases as the diffuseness increases, while the dissociation cross section ud decreases. Moreover, the decrease of ua is rather steeper than the increase of u,. Since the inclusive cross section ui is the sum of these, it is nearly independent of the nuclear boundary diffuseness. Fig. 6 shows the dependence of the peaks of proton inclusive spectra in the 27A1(dpX) process on the parameter y for various angles of proton emission. As can be seen from the figure, the peak value considerably depends on y for small angles of fragment emission. At the same time, the shape of the spectrum remains unchanged, which is in accordance with our calculations. Therefore, precise experimental studies of inclusive processes for small angles of fragment emission make it possible to draw important conclusions about the relative motion of fragments in the incident nucleus. Let us consider now the inclusive spectra of o-particles and deuterons produced due to 6Li fragmentation in the field of the nuclei. It has been experimentally established that the probability of 6Li fragmentation on an a-particle and a deuteron is considerably higher than the probab~ties of other break-up channels of this

this

Fig. 6. The dependence of the peaks of proton inclusive spectra on the parameter y (z7Al(d,p) reaction) for various emission angles and an incident beam energy 56 MeV.

(a )

(b)

41CaP

Li,dX)

40CLl(6Li,~X)

E LAB Fig. 7. Inclusive spectra of deuterons (a) and a-particles (b) from 156 MeV 6Li dissociation on a 40Ca target for various fragment emission angles. Experimental curves are taken from ref. *). 136

A.G. Sitenko et al. / Inclusive particle spectra

nucleus7*8). This is a direct

131

consequence of the cluster structure and low binding energy of ‘Li (E = 1.47 MeV). In ref. 8), inclusive spectra of particles, induced by the interaction of 6Li beam with @Ca and ‘08Pb targets at the energy 156 MeV, are analyzed in the DWBA break-up theory including both elastic and inelastic break-up of the incident particle 38). In order to simplify involved numerical calculations the authors of ref. 8, applied the approximation of zero-range interaction. Hence, in the authors’ opinion, the theory could not reproduce the angular dependence correctly. Investigation of the above-mentioned reactions by means of the diffraction model allows one to avoid using this approximation. In our calculations y is taken equal to 0.20 fm-‘, which is close to the value extracted from the experimental magnitude of the 6Li radius 39). Fig. 7 shows the inclusive spectra of deuterons and o-particles produced by 6Li fragmentation on a %Za target. The theory is seen to describe correctly the magnitude and the positions of the cross-section maxima. The peaks of the cross sections correspond to Ed = $E for deuterons and E, = $E for o-particles associated with the momenta of these fragments in the incident nucleus. Moreover, theoretical calculations correctly reproduce maximum experimental values of the cross sections at small angles and their rapid decrease as the angle of fragment emission grows. The discrepancy between our theory, based on the direct-reaction mechanism, and the experimental data is due to Coulomb effects (especially significant for o-particles) and to preequilibrium and equilibrium emission of particles of the composite nucleus that give higher contributions to the inclusive spectra for greater angles of particle emission. The angular distributions of fragments for ‘jLi-nucleus collisions are shown in fig. 8. The theory is seen to predict, according to the experimental data, the exponential decrease of the cross sections with an increase of the angle of fragment emission. An example of the theoretical calculation of deuteron inclusive spectra from the 12’Sn(‘jLi dX) reaction is shown in fig. 9a. The figure demonstrates the relative contributions of diffraction dissociation and stripping to the inclusive spectrum. The cross sections of these processes are seen to have the same form. However, the contribution of inelastic break-up of the incident ion, when one of the fragments is absorbed by the target, is higher than that of elastic break-up, when both fragments become free after the collision. The fragmentation of 3He ions confirms this conclusion too. A similar dependence of particle angular and energy distributions was established for the diffraction dissociation and stripping of deuterons in ref. 40). Fig. 9b shows the dependence of the integral cross sections of diffraction dissociation a,, stripping us, and their sum ui, on the mass number of the target nucleus for the A(6Li, dX) reaction. As follows from the figure, the theory predicts the increase of the cross sections to be proportional to the nuclear radius. Such a feature is peculiar to peripheral nuclear processes. The estimation of the integral cross sections of 156 MeV 6Li fragmentation carried out in ref.7), and the A-dependence of the

ELi

‘= 156 MeV

103

I

I

I

I

I

t

I

2osPb l6 Li ,dX)

i

!-

2o”

IO0

3o”

e,,

Fig. 8. Fragment angular distributions from the reactions 12CeLi, nX) (a) and ZosPb(%i,dX) incident beam energy 156 MeV. Experimental data are taken from ref. 7). 138

(b) for an

A.G. Sitenko et al. / Inclusive

particle spectra

139

angular distributions, obtained in this paper, show that, in accordance with the predictions of the diffraction model, the cross sections are proportional to AlI3 for deuterons, protons, tritons, and 3He nuclei.

5. Conclusion

As follows from the above calculations, the developed diffraction theory of inclusive processes correctly describes the main experimental regularities of the fragmentation of 56 MeV deuterons, 3He ions with the energies 70, 90, and 110 MeV, and 156 MeV 6Li nuclei for a wide range of targets. The agreement of the theoretical results with the experimental data is direct evidence for the diffraction mechanism of light-ion fragmentation. Owing to comparatively weak dependence of the inclusive spectra on the nuclear edge diffuseness, one has to carry out kinematitally complete experiments on various isotopes in order to study in detail the isotopic effect that is manifested through the nonmonotonic dependence of the dissociation cross section on the mass number of the target nucleus. Such coincidence

I

i20Sn (%,

dX)

(a )

E = 156 MeV e, = 12”

20 -

0

I

20

I

60

loo

I

C

Efp6( MeV’

Fig. 9. Relative contributions of diffraction dissociation (curve 1) and stripping (curve 2) to the deuteron inclusive spectrum (curve 3) from the “‘Sn(6Li, dX) reaction for an emissi& angle 0, = 12O and incident beam energy 156 MeV (a); calculated integral cross sections of the diffraction dissociation o,+, stripping 4, and their sum q as a function of the target mass number A from the A(6Li, dX) reaction (b).

140

A.G. Sirenko et al. / Inclwivepariiele

spectra

CTCb) (b)

48

0,s

0,4

92

0

L

f

I

s

3

I

5

‘

-

A%

Fig. 9. (continued).

experiments will enable one to separate stripping and dissociation processes which are sensitive to the peripheral nuclear diffuseness. The approach developed to explain light-ion inclusive spectra is useful for the simple theoretical interpretation of experimental results. It contains a few clear physical parameters, and does not lead to complicated calculations, which cannot be avoided in DWBA or optical-model analysis. At the same time, as follows from our calculations, the diffraction description of the inclusive spectra allows one to explain correctly many characteristic regularities in the experimental cross sections, However, in order to obtain better agreement between theory and experiment, it is necessary to improve the model; in particular, to take into account the Coulomb interaction.

A.G. Sitenko et al. / Inclusive particle spectra

141

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