Heavy-ion reactions induced by 16O, 18O and 19F ions

I 2.A.1

[

Nuclear Physics A140 (1970) 548--570; (~) North-HollandPublishing Co., Amsterdam Not to be reproduced by photoprint or microfilmwithout written permissionfrom the publisher

HEAVY-ION REACTIONS I N D U C E D BY 160, t S o A N D 19F I O N S W. R. FALK University o]"Manitoba, Winnipeg, Canada and A. HUBER, U. MATTER, R. W. BENJAMIN t and P. MARMIER Eidg. Technische Hochschule, Ziirich Received 27 October 1969 Abstract: Excitation curves have been determined from y-ray yield measurements for heavy-ion reactions induced by Elaa=12-30 MeV leO, l s o and 19F ions incident upon thick targets of 9Be, I°B, liB, 12C and 2aNa. The yields of radioactive decay products with half-lives greater than one second were measured; hence a large number of the outgoing reaction channels could be observed. The preponderance of heavy reaction products suggests compound-nucleus formation as the dominant reaction mechanism. Statistical-model calculations with a spin-dependent level density have been performed, in which the nuclear moment of inertia was treated as a parameter. Many of the results can be explained satisfactorily with a nuclear moment of inertia 0.55 to 0.7 of the rigid body value. EI

I

NUCLEAR REACTIONS 9Be, l°B, x2C(160, x), 9Be, 10.11B, 23Na(lSO, x), 9Be(19F, x), E = 12-30 MeV; measured o'(E; Er); or(E).

I I

1. Introduction A survey has been m a d e o f the exit channels leading to r a d i o a c t i v e nuclei for h e a v y ion reactions i n d u c e d by 160, 1SO a n d 19F ions at l a b energies o f 12 to 30 M e V inc i d e n t u p o n t h i c k targets o f 9Be, 10B, 11B, 12C a n d 23Na. The e x p e r i m e n t a l a r r a n g e m e n t p e r m i t t e d m e a s u r e m e n t o f half-lives in the range f r o m seconds to hours a n d hence a large n u m b e r o f the exit channels were a m e n a b l e to investigation. T a b l e 1 lists the various projectile-target c o m b i n a t i o n s t h a t have been investigated t o g e t h e r with the reaction p r o d u c t s identified. T o t a l cross-section m e a s u r e m e n t s as a function o f b o m b a r d i n g energy have been o b t a i n e d for the reactions listed. The results o f these m e a s u r e m e n t s m a y be s e p a r a t e d into two g r o u p s : (i) one- o r t w o - n e u t r o n transfer r e a c t i o n s a n d (ii) r e a c t i o n s which p r o c e e d t h r o u g h the f o r m a t i o n o f a c o m p o u n d nucleus which subsequently decays b y the emission o f nucleons or light-nucleon clusters. Results for the transfer r e a c t i o n s have been r e p o r t e d in an earlier p a p e r 1); the p r e s e n t p a p e r is d e v o t e d to a n investigation o f the latter g r o u p o f reactions a n d their interp r e t a t i o n in terms o f the statistical model. t Present address: Savannah River Laboratory, Aiken, South Carolina. 548

lsO

1sO 19F

2aNa

l°B 9Be

2SA1

+IK

29A1

27Mg

26A1

2asi

25Mg

Compound system

2ct+Z°F pct+ZaNe 2n+ZeAl

3n+3aK

9Be(19F, Z°F)aBe

23Na(lSO, xTO)2'*Na 2aNa(laO, 160)2SNa

1.63 0.439 0.511

0.511 2.752 0.392

0.345 0.47 1.368 0.584 0.842 0.511

0.198 2.752 0.392

2~-]-190 p2n +24Na np+2SNa 2otW21F p~-t-Z4Ne n~-k~4Na ~+2SNa np+27Mg 3n+Z6Al

0.511 0.511 2.752

2u+lSF n~+ZtNa 2p +24Na

2.752 0.511

E~, (MeV)

0.511

9Be(tsO, t90)aBe

Transfer reactions ")

n+27Si

p+24Na 2n+2aMg

Reaction products

a) Yields and cross sections for these reactions have been published in ref. t). b) Ref. 29). 0 Ref. so). a) Ref. 31). e) Ref. z2). f) Ref. an).

280

160

l°B

ttB

260

12C

laO

160

9Be

9Be

Projectile

Target

TAnLE 1 Reactions and analysed 7-rays

s min h s min s

10.3 s 37.6 s 6.38 s

7.68 min 14.98 s 60 s

4.35 3.38 14.98 60 9.46 6.38

26.8 s 14.98 h 60 s

1.83 h 22.8 s 14.98 h

4.17 s

14.98 h 12.0 s

T/t

f) b) b)

b) b) ~)

0 b) b) b) b) b)

b)

b)

a)

~) b) b)

b)

b) b)

Ref.

43 27 27

38 35 25

26 41 26 +100, --50 22 31

35 28 30

29 24 31

40

23 25

Maximum error in or(%)

o

o

550

W . R . FALK et al.

Evidence for compound-nucleus formation in heavy-ion collisions is manifest in the preponderance of heavy reaction products, that is, products whose mass differs from the combined target and projectile mass only by a few nucleon masses. Much of the early information regarding such reactions was obtained by workers at Oak Ridge 2) using 14N ions. Other investigations have been made of the light-particle spectra and angular distributions 3), results of which also indicated a compound-nucleus mechanism. Several unique features characterize compound-nucleus reactions initiated by heavy ions, chief among which is the high angular momentum with which the compound system can be formed. In addition, high excitation energies are achieved at relatively low bombarding energy, thus minimizing the probability for spallation-like reactiGns. The effect of high spin in the decay of the compound system has been investigated experimentally 3,4) and also theoretically 5-7). The primary conclusion from these studies pertinent to the present discussion is that alpha-particle emission is enhanced with respect to nucleon emission. This is due to the paucity of high-spin states in the residual nucleus which inhibits nucleon emission. Consequently, any theoretical description of the decay of the compound system within the framework of the statistical model must take into account the angular-momentum dependence of the particle decay widths. This can be accomplished by using a spin-dependent level-density formula as discussed in the next section. The theoretical calculations performed and presented in this paper represent an attempt to see to what degree the statistical model is able to describe the large number of data available from these measurements. Two pairs of reactions, l°B+X60 and 12Cd-14N proceeding through the compound nucleus 26A1, and l ° B + l s O and 9Beat" 19F proceeding through the compound nucleus 28A1, have been examined in some detail to see whether differences in the cross sections within each pair are explicable in terms of angular-momentum effects. Because of the large number of parameters available in the statistical-model calculations any effort to "fit" the experimental results by variation of all the available parameters would provide little new insight; the approach taken for these calculations was to make reasonable and consistent estimates of the statistical-model parameters using only one free parameter, the quantity I/Irigid , where I is the nuclear moment of inertia and /rigid the calculated rigid-body moment of inertia of the nucleus. Calculations for different values of this ratio are compared with the experimental results. The above approach should have considerable validity in the present case since the calculations are applied to the relativelynarrow mass region A = 21-27, and the excitation energies in the compound systems are sufficiently high (25-38 MeV) that shell effects and transitions to low-lying levels are not too important. Because of the high excitation energies involved, emission of up to three particles was energetically allowed in many cases. A complete calculation taking into account second- and third-particle emission from the compound nucleus is prohibitively costly in computing time if a spin-dependent level density is used. Consequently, the calculations performed were carried only as far as the emission of the first particle.

551

HEAVY-ION REACTIONS

2. Theoretical considerations

2.1. STATISTICAL MODEL The general features of the statistical model require no elaboration at this point; numerous review articles are available to the reader 8, a). The following outline of the model, summarizing the pertinent formulae, parallels that of Butler and Santry 10). According to the formula of Le Couteur and Lang 11), the density of states for a Fermi gas at an energy U summed over all angular momenta is ÷ og(U) = 1A-~(61 (U+t)-5/4 exp [E(~rc2gU)½], (2.1)

\gI where g is the density of single-particle states at the Fermi energy, and t the thermodynamic temperature related to U by the expression

U = ~2gt2--t. The usual level-density parameter ~ is defined in terms of the density of single-particle states by the relation ~ = ~rc2g. Because of pairing effects in real nuclei the energy U of the Fermi gas is not simply the excitation energy in the nucleus. Following Newton's prescription 12) this equivalent energy U of the Fermi gas is obtained by subtracting from the nuclear excitation energy the nuclear pairing energy. Values for this pairing energy have been tabulated by Cameron 13) and are given in table 2 for the nuclei of interest to this discussion. The spin-dependent level density is obtained under the assumption of a random coupling of the angular momenta of the individual nucleons. This yields the result [refs. 10,14)] for the number of levels of spin J at the equivalent excitation energy U f2(U, J ) - (2J+ 1) {exp [-- (J+ ½)2/2ct]} ~(U), n~(2ct) ~

(2.2)

where ch 2 can be shown to be equal to the nuclear moment of inertia. The quantity ct in the exponential term is frequently represented by tr2, referred to as the spin cut-off parameter. Combining the foregoing equations one obtains

12(U, J) - [~/(8c3)]*(2J+l)exp[-(J+½)Z/2ct]exp[E(~U)½]. 12

(2.3)

(U+t) 2

As explained in the introduction, the ratio of the nuclear moment of inertia to the rigid-body value (defined by G = I/Irisid) was taken as the on12¢free parameter in the model. A nuclear radius parameter of 1.25 fm was used to calculate/rigid" The leveldensity parameter ~ is normally taken to be proportional to the nuclear mass A which, when combined with the result/rigid OC A s/n, leads to the formula

Q(U, J) oc ( 2 ( + 1) exp [ - (j+½)2 h2/(2tGirigid)] exp[2(eU) ½] G~A (U+ t) 2

(2.4)

2tNe

3.41

2.25

Nucleus

Level density parameter,

Pairing energy (MeV) 0.0

3.52

22Na

0.0

3.73

24Na

2.17

3.41

2SNa

4.58

3.73

2*Mg

2.10

3.83

2SMg

Statistical-model parameters

TABLE 2

2.10

3.14

ZTMg

3.84

2.98

2aMg

2.48

3.83

2SAl

0.17

3.59

27A1

0.0

3.22

28A1

2.13

3.59

27Si

HEAVY-ION REACTIONS

553

Although a more complicated expression, not linear in A, was used to determine a as discussed below, the above form for the spin-dependent level density was retained. Following the prescription given by Newton 12) and a more recent analysis by Lang [ref. is)i, the level-density parameter is calculated from = 0.0748 (Jp+Jn+ 1)At, where jp and jn are, respectively, the mean values of the proton and neutron angular momenta close to the Fermi level. Values of ~ calculated from this expression using Newton's tabulation are given in table 2. The expression (2.4) for the level density predicts a finite value for arbitrarily high values of J. This is clearly unphysical, however, since the energy associated with the rotational motion of the nucleus cannot exceed U. Thus, at an excitation energy U the maximum permissible value of the angular momentum, Jm, has been restricted to the value given by the formula 6)

(Jm+½)~h~/2I = U.

(2.5)

The rate of emission of a particle with channel energy E from a compound nucleus with excitation energy Uc and spin J to a final state of a residual nucleus with spin j and excitation energy between Uf and U f + d U f can be expressed as 16)

R( Uf, j )d Uf

I f2(Uf, j_))

•

h a(Uo, J)

S'

~+~

s+s

S', Tt(E)dUf,

(2.6)

where f2 is the level density as previously defined, s is the spin of the emitted particle and Tz(E) is the transmission coefficient for the lth partial wave at the channel energy E. The probability for the compound nucleus decaying to the final state of angular momentum j and excitation energy between Uf and Ue+ d Uf is then readily obtained by taking into account all the particle exit channels and averaging over the spin distribution in the compound nucleus. A computer program to carry our the above computation was written for the University of Manitoba 360/65 IBM computer with provision for angular momenta in the compound and residual nuclei of up to 20h. Only neutrons, protons and alphaparticles were considered in the outgoing channels. Deuterons are not considered to play a significant role at these excitation energies since the density of states available for deuteron emission is generally considerably lower than for nucleon or alpha-particle emission. The essential results of the computer calculation are the relative particle-decay widths and the distribution in spin and energy of the levels populated in the product nuclei through the emission of neutrons, protons and alpha-particles from the compound nucleus. A knowledge of the compound-nucleus formation cross section then permits calculation of the cross sections for single- and multiple-particle emission. The levels populated below the lowest particle-emission threshold of the product nucleus contribute to the cross section for that product; all the levels populated above this particle threshold are assumed to decay by further particle emission and hence contribute to the cross section for multiple-particle emission.

554

w.R. FALK et aL

2.2. SPIN DISTRIBUTION IN T I - ~ C O M P O U N D N U C L E U S

The absorption cross section for the lth partial wave of two interacting nuclei trt = n~2(2l+ 1)Tt,

(2.7)

in the absence of spin-orbit interactions, leads to the following expression 16) for the formation cross section of the compound nucleus with spin J o(J) = ~,~2

(2J+ I) J,+J, ~+s E E T,. (2J, + I)(2J2 + I ) s= I , , - , , I ,=l~-sl

(2.8)

Here J1 and J 2 a r e the spins of the interacting nuclei and ~ is the reduced de Broglie wavelength. The manner of calculating the transmission coefficients for heavy-ion interactions is discussed below.

2.3. TRANSMISSION COEFFICIENTS FOR HEAVY IONS: PARABOLIC POTENTIAL The optical model has been applied with considerable success in the calculation of total absorption cross sections for interacting heavy ions 17,18). Unfortunately, elastic scattering data for determining the optical-model parameters were not available for many of the projectile-target combinations required in these measurements. Therefore, a method employed by Huizenga and Igo 19) for calculating reaction cross sections for alpha particles, and adapted by Thomas 2o) to heavy-ion collisions, was used. The total potential for the interacting charged ions is assumed to have the form

_ z, 2e2 + r

[2/~r2

0.574

)3

The approximation is made that this total potential can be represented by a parabola matched in position, height and curvature at the maximum of the potential. An analytical calculation can then be performed 2 l) yielding for the transmission coefficients

Tt(E) = {l + exp [2rc(Bt-E)/(hco,)]}- 1.

(2.10)

Here B t is the height of the barrier, E the channel energy, and oh the vibrational frequency of the harmonic oscillator having a potential-energy function given by the negative of the potential-energy function describing the barrier. To check the transmission coefficients calculated with this formula several com. parisons were made with experimental total absorption cross-section measurements and optical-model calculations. For the systems 12C-]-12C and 160-3t-12C the calculated total cross sections agreed with the experimental values 17,22) to within 20 over the energy range from about 1 MeV below the Coulomb barrier to 12 MeV (c.m.). Comparison with recent optical-model analyses of 160-~12C by Gadioli et aL is) revealed agreement to within 5 ~ over the same energy range. Similarly, the transmission coefficients calculated using formula (2.10) were in good agreement with the

HEAVY-ION REACTIONS

555

optical-model results for the 12C--[-12C system 17) and the 1 4 N + 2 7 A 1 system 3) except for the highest contributing/-values where the values of the transmission coefficients became less than approximately 0.3. One can conclude from these comparisons that the parabolic potential can be used with some confidence in the region from the Coulomb barrier and above. 2.4. TRANSMISSION COEFFICIENTS FOR LIGHT PARTICLES Calculations for the statistical model, eq. (2.6), require the transmission coefficients for all the outgoing channels, i.e. in the present case for neutrons, protons and alphaparticles. The optical-model parameters given by Butler and Santry 1o) were used with a few modifications. For the imaginary part of the potentials they employed a Gaussian surface term. The code used for these calculations was a modified version of Smith's program 23) and employed a Woods-Saxon derivative-surface term. Table 3 TABLE 3

Optical-model parameters used in the statistical theory calculations

System

Vo (MeV)

nq-ZSA1 p+2SMg ct+22Na n+27Al pq-27Mg ~+24Na

54.2 56 45 58.1 56 45

Cx --0.71 --0.50 --0.98 --0.50

Wo (MeV) 2.4 5.0 10.0 0.3 5.0 10.0

C2

r, r' (fin)

a (fro)

a' (fro)

0.67 0.50

1.35 1.35 1.35 1.35 1.35 1.35

0.5 0.5 0.5 0.5 0.5 0.5

0.45 0.45 0.45 0.45 0.45 0.45

0.80 0.50

gives the optical-model parameters used for the two cases of particle emission from the compound nuclei 26A1 and 2SA1. The quantities r and r' are the real and imaginary nuclear radius parameters and a and a' the real and imaginary diffuseness parameters, respectively. The energy dependence of the real and imaginary parts of the potential is expressed in terms of the constants CI and C2, respectively, in the form V = Vo + C1E and W -- W 0 + C2E. In order to obtain the same neutron-absorption cross sections at low energies as reported by Butler and Santry for 2 4 M g + n and 27Al+n a peculiar energy dependence of the potentials had to be introduced as shown in the table. With neutrons at low energies the difference between the Gaussian and WoodsSaxon derivative form of the potential may be responsible for this anomaly; for charged particles the Coulomb potential likely outweighs this difference and hence the values of the parameters C1 and C2 given by Butler and Santry were adopted. 3. Experimental procedure and results

The techniques of activation analysis were chosen for these measurements because this approach is ideal for a rapid survey of the many exit channels present in such reactions. A detailed description of the experimental techniques and data analysis

556

w . R . tALK et aL

16s ICY6 10-7

•2 4 N ° ~ C

, . o - ~ 9Be+lSO

10-$

]0"5 "~ =.

Io"7

J

i

10-4 J o ZOF

=

t o 27Si

/

ic~

10-7

J0B.160

10.'

,

,

• 2sAI

/

104

lO"a

/ 10.9

io~

o

IO'eI 10.9t_

9Be + 19F

l 30

~

Incidenl Energy (MeV)

~ 2'0

~'5

30.

Fig. l(a). Yield-per-incident ion as a function of bombarding energy for 1~O ions on 9Be, X°B and 12C and age ions on 9Be. procedure has been presented in an earlier paper ~). Following is a brief description of the experimental approach. Measurements were made using the HVEC-EN tandem Van de Graaff of the Swiss Federal Institute of Technology in Ztirich. The targets used were pure metallic Be discs, reactor-grade graphite discs, and discs compacted at high pressure from powdered ~°B, z~B or NaBr. The compacted discs presented a smooth, hard target surface, and beam currents were held low enough so that there was no visible target deterioration after irradiation. All targets were thicker than the range of the incident ions. Measurements of the ?-ray yields of the radioactive product nuclei were converted to yield-per-incident-ion and these yield curves are shown in figs. 1a and 1b. Corrections for analyser dead-time and sum effects in the detector were made which required a detailed knowledge of the decay scheme for each radioactive isotope. The sources for these decay schemes are listed in table 1, along with the reactions investigated and the ?-rays analysed in each case.

HEAVY-ION REACTIONS

IO'S

i

i

557

10-r

i

o 38K

o

i(~e o

o

j07 9Be +is0

I(~8

,]

,£

2'0

o

20F

• ,= • "="

~AI ONe 24N0 27Mg

2'5

3O

IO's .~ =/...,r ~ ~ _/¢ ~(" / ~ / .....

IO"6

29 i o 27M O • 24NQ ZIF ~ 'SNa o 24Ne

i

i

~

-

/

, / .o,..t ~=

i(~7

i0-r

y f/

I0-

,.,'~

=°BJ80

#~/g ,'5

2'0

2'5

1,0-' J ~o

,B.Iso

,/ ,;

2'o

2'5

~o

Incident Energy ( MeV )

Fig. l(b). Yield-per-incident ion as a function of bombarding energy for 1sO ions incident on 9Be, 23Na, I°B and alB. Yield curves for the 1sO induced transfer reactions listed in table 1 have been published in ref. 1). The yields of 2*Na and 27Mg from the a°B+lsO reaction are attributable to the 11B impurity in the a°B target. The yield for the reaction 11B(~80, 2~)21F represents a lower limit only, since the assumption was made that 2XF decays 100 ~ through the 0.345 MeV level of 21Ne.

The total cross sections were calculated from the yield curves using the expression

where the first bracketed term is the differential energy loss of the incident heavy ion at energy E t a n d the second bracketed term the derivative of the yield curve for the given reaction at energy E t . The total cross sections d e t e r m i n e d in this m a n n e r are shown in figs. 2a a n d 2b a n d error estimates for the cross sections are listed in table 1. These estimates are based u p o n reasonable attempts to fit different curves to the experimental yield points a n d their associated error bars. I n a d d i t i o n to uncertainties c o n t r i b u t e d b y c o u n t i n g statistics, y-ray p h o t o p e a k stripping a n d analyser dead-time corrections, the errors also

558

W.R. i

i

FALK

i

i

et al. L

i

i

i 18F

I0

J

2aMg-

i

i

i

IOC

i

i

i

i

_=rsi

o

Po F K)O

f

-

=C+'%

2'5

I

30

I

,5

2o

2'5

i.

Incident

Energy

(MeV)

Fig. 2(a). Cross sections as a function of bombarding energy for radioactive reaction products from leO ions incident on 9Be, l°B, 12C and 19F ions incident on 9Be.

include uncertainties associated with detector efficiencies, branching ratios, energyloss values and, in the case of 1oB and 11B, uncertainties in the target composition. The errors quoted represent the maximum error in the cross section over the investigated energy range. The enriched 10B and 11B powders furnished by Oak Ridge National Laboratory contained, respectively, 4 % 11B and 1.4 % 1°B. These relatively small amounts of impurities caused considerable difficulties in at least one case. Bombardment of both these targets with an 180 beam resulted in the reaction products 2aNa, 27Mg and 26A1. The yields of 24Na and 27Mg are much greater with a 11B target than with a 10B target and a numerical calculation revealed that essentially all the yield for these products in the case of the l°B target could be explained by the lZB impurity. This conclusion is supported by a statistical-model calculation which yields cross sections for these products from 10B considerably smaller than those measured. A comprehensive decay scheme for 21F showing the branching ratios was not avail-

HEAVY-IONREACTIONS

559

100 ~

'

-

ZSNaJ

10

I0

+l

~

// f

115 I

15

i

I

I

4

I

I

25

30

I

30 100

25

ioB.+o 20

I

I

~0

23N1~

24 NO

i

~

M

II

g

I

j.-No

°" / / zh

zh

/

I

1

28 15 Incident Energy(MeV)

"a+% I

20

I

25

50

Fig. 2(b). Cross sections as a function of bombarding energy for radioactive reaction products from 1sO ions incident on 9Be, X°B, 11B and Z3Na. The cross section for the reaction 11B(laO, 2~)2aF represents a lower limit only, since the assumption was made that 2XF decays 100 ~o through the 0.345 MeV level of 21Ne. In the calculation of the cross section for 26A1 from the reaction 1~B(180, 3n)26Al, the contribution from X°B impurity in the target was subtracted.

able. The formation cross sections for this activity were calculated, therefore, assuming that 21F decays 100 ~ through the 0.345 MeV level of 21Ne. The results for the cross sections calculated on this basis thus represent lower limits only. Two reaction products, 26A1 and 38K, have isomeric states. For 26A1 only the decay of the isomeric 0 + level was observed since the ground state (5 +) has a half-life of 7.4x 105 y. In 38K, however, which has a 3 + ground state and a 0 ÷ isomeric state, only the 7.68 min ground state decay was observed. Hence it should be borne in mind that cross sections for reactions involving these nuclei as reaction products represent only some undetermined fraction of the total cross section for that product. Supporting the interpretation of these results as compound-nucleus reactions is the increasing prominence of multiple-particle emission with increased bombarding energy. In the absence of effects due to high angular momentum in the compound nucleus

560 I0

w . R . FALK e t al.

IOC

CROSS SECTION RATIOS =IAI ,-- lOB(leO,2p)24 NO -- t°B(ISO, n',~)it No =°B( leO,2~)is F --

J~c(B4N,2p)a'INo IZc(14N,n~')zlNo / f~C(b:lN,2~) m F ~ f .

Mg=s (p,2p)2*No

/

j

---

CT(2"<)/U(2p)

_

_

CROSS SECTION RATIOS

SeAl

--

9 Be(ISF. 2n)=eAi e Be (eF, p~)'zh'qe

t° B(leO,2n)=eAI to B(=eO ip,)lSNe

__

= Be(mF,2=<)=°F

=oB(mO,2*<)=OF

--

Mg zs (p,2") mF /

-

ulO I-.

F-t~ z

./

o1.0 Ic~

/

/.1, . o l J

'--

_ I.-

_

O-(n,~)/'0.~p )

leo 0~

--

1.0

./..I"

0" ( n=<)//0-(2p)

/-/

--

//°

/

le F.~..~.-/" o/" /./"

.~./"I

_

t"

/

I~.i"10"

( P~<)/O" 12n)

(2 p)

OIL

I

I

26

27

28

I

I

I

I

:29

"'50

31

32

EXCITATION

0.11 33

I

3I

ENERGY

IN

I

I

i

I

I

I

l

32

33

34

35

36

37

38

COMPOUND NUCLEUS

Cross-section ratios for reactions induced by protons, ~4N ions, and 160 ions proceeding through the compound nucleus 26A1, and zsO- and 19F-induced reactions proceeding through the compound-nucleus 2aAI. Data for the reactions 12C-}-14N and ZSMg+p were taken from ref. 24) and ref. 25), respectively. Fig. 3.

one would expect the ratio of any two cross sections proceeding through the same entrance channel to be independent of the mode of formation of the compound nucleus. That angular-momentum effects do, indeed, play a significant role even at these moderate bombarding energies can be gathered from an examination of the two sets of reactions 160+t°B ~ / 24Na+2p t4N+12C • 26A1. > 2tNa+n+ct p+25Mg / ~ ~aF+2~ t9F+gBe 1So+1O B

~ ~

~ 2aAI*

26Al+2n > 23Ne+p+~ 2OF+2~

Fig. 3 shows cross-section ratios a(2a)/a(2p) and a(na)/a(2p) for 2 6A1 formed by the different incoming channels, and ratios a(2a)/a(2n) and a(pa)/a(2n) for the 2SAl compound nucleus. The data for the reaction 12C+ t a N were taken from the measure-

HEAVY-ION REACTIONS

561

I00

I0

IO

\ b

I0

"0 2 J-SPIN

4 6 B IN C O M P O U N D

I0 12 NUCLEUS

14 26AI

16

Fig. 4. Partial compound-nucleus-formation cross sections as a function of the spin in the compound system 26A1. Distributions are shown for three incident channels for excitation energies in 26A1 o f 26, 29 and 32 M~V. A parabolic potential was used for the heavy-ion calculations as explained in the text.

562

FALK et al.

w.R. I

I

t

I

I

I

I

I

I

I

I

I

I

I0

=aAI

9

drms

....

dQv

8

.//./-

~ /

7i

/

/ , o ~>.>."

..i"

~ t " 4

~ . / ' / ' / ./" i

I

26

e=M(j+ p I

I

28

.° ./.. I

I

30

I

. I

32

1

34

.

"

"

I

I

36

I

I

38

EXCITATIONENERGYINCOMP(~UNDNUCLEUS Fig. 5. T h e average a n d r m s spin, d e n o t e d by Jay a n d Jrms, respectively, i n the c o m p o u n d s y s t e m 26A1 a n d 2SAI as a f u n c t i o n o f the e x c i t a t i o n energy.

ments of Gaedke et al. 24) and those for the 2SMg+p reaction from Cohen et aL 25). The ratios (r(2~)/tr(2p) and ~r(n~)/tr(2p) are significantly greater for the 14N induced reaction than for the 160 induced reaction. The spin distribution in the compound nucleus 26A1 formed by the different incoming channels is shown in fig. 4 from a calculation using a parabolic potential for the heavy ions as outlined in sect. 2. Fig. 5 further summarizes these spin distributions for the compound nuclei 26A1 and 2SA1 by showing the average and rms spin, denoted by Jay and J,=s, respectively, in the compound nuclei. A qualitative interpretation of the experimentally determined crosssection ratios is that a nucleus with high spin preferentially emits alpha-particles since nucleons are unable to remove the angular momentum required for their emission. The apparent anomaly in the tT(20t)/tT(2n) ratio for 19F ions incident on 9Be is due to a large contribution from the one-neutron transfer reaction 9Be(19F, 2°F)SBe, results of which have been reported earlier 1). The statistical-model calculations given in the following section substantiate the interpretation that the observed cross-section ratios can be understood in terms of angular-momentum effects. It should be noted at this point that with activation analysis one cannot differentiate between one-neutron transfer and two-alpha evaporation in the case of a 9Be target. However, in the case of one-neutron transfer from 9Be, the two alpha particles resulting from the decay of the residual SBe nucleus are emitted with very small angular

HEAVY-ION REACTIONS

563

separation due to the small decay energy of aBe (only 95 keV). It is possible, therefore, to identify the neutron-transfer through a coincidence measurement if one detector is large enough to intercept the two alpha-particles together. This technique has been applied recently at the ETH tandem laboratory to study a similar reaction, 9Be(160, 170)SBe, and a large cross section was observed 26). This, together with arguments presented earlier ~), would appear to justify the assumption that the measured 9Be (19F, 2°F)aBe cross section is largely due to a direct neutron-transfer. In the same manner the reactions 1°B(160, 2~)~8F and l°B(~sO, ct)2°F could, perhaps, proceed partially by deuteron transfer. The former of these reactions has been examined with the method described in the preceding paragraph. This measurement have no indication of a transfer reaction with aBe as a reaction product; therefore, this reaction must proceed primarily by compound-nucleus formation and subsequent two-alpha evaporation. 4. Statistical-model calculations 4.1. SINGLE-PARTICLE EMISSION

Since the computer calculations were carried out only to,the stage of the emission of the first particle from the compound nucleus, absolute cross-section values could be obtained only for single-particle emission. Hence the first calculations discussed are those for single-particle emission, namely for the reactions 9Be(160, p)2*Na, 12C(160, n)27Si and llB(~80, ~)25Na. In the calculations it was assumed that only the levels populated below the particle-emission threshold of the product nucleus contributed to the observed cross section for that product. The effect of competition between ~-ray emission and particle emission in systems of high spin has been investigated by Sperber 27) and Grover 28), but was not taken into account in these calculations. The values of the level-density parameter ~ and the pairing energies are given in table 2. Transmission coefficients calculated from a parabolic potential were used to calculate the spin distributions in the compound nuclei and the total absorption cross sections. Then, using the expressions given by eqs. (2.4) and (2.6) cross sections were calculated for various values of the parameter G(= I/frigid). The results are shown in fig. 6 for the three reactions mentioned earlier. For the reaction 9Be (160, p)2,Na a value of G between 0.55 and 0.7 provides the best fit to the experimental results. A simple, qualitative interpretation of the dependence of the cross section on the value of G is difficult to make because of the complex interplay of competition between the various exit channels. Examination of the calculated proton spectra for this reaction reveals that for G = 0.7 about 20 % more protons are emitted than for G = 0.4, most of them at energies below 6 MeV. This result is understandable since the higher value of G implies a greater abundance of high-spin states available to the proton at high excitation thus favoring its emission. The enhanced emission to states of high excitation in the product nucleus depletes the number of protons emitted to states below the particle-emission threshold and hence the cross section decreases

564

W . R. t A L K e t al.

4°t

°Be

('SO,p) "~No

/

/';°t.I e-0.4

/

/

20 /

~ . ~ ....................... .~....~...-- ~

~

~

,

~

.

.~o - o. 4 s

~

~,........... ..

I0

\ I

IO 14

i 18

, I 22

I

I01

I 26 [k~" /

8 L-

at_ 'i*_ ~¢1.. U Up'
6--

.---

...

4--

'

G*..~" ~ / ~ / 7 ~

G I~0

/

.,'Y

_o

/- /-J i6

20

24

28

2

I

.8

,/,d" .....

-6

I

16

]

I

20

I

I

24 EIob (MeV)

I

I

.

= c ['°o~)*' sJ

SO

I

28

Fig. 6. Statistical-model calculations for single-particle emission from compound-nuclear reactions. Results are s h o w n for various values of the parameter G = I/I, jsia, the ratio of the nuclear moment of inertia to the rigid-body value. Values of the other statistical-model parameters are those given in the text. No normalization has been applied to the calculations except in the case of the reaction 12C(~60, n)2~Si where the curves for G = 0.4, 0.55 and 0.7 have been multiplied by a factor of 2.5 and the curve for G = 100 by 1.5. with increasing G as shown i n fig. 6. The curves for G = 100, essentially a n infinite m o m e n t o f inertia, represent the situation where n o spin effects would occur. Physically, of course, G c a n n o t be larger t h a n one. A l t h o u g h the absolute error in the cross section for the reaction 11B(180, c~)2 SNa was very large, a value for G between 0.4 a n d 0.7 is compatible with the results. I n contrast to all the other cases studied, the m a x i m u m decay energy for n e u t r o n s from the c o m p o u n d nucleus 2aSi is only 6-11 MeV for lab b o m b a r d i n g energies of 16-27 MeV. It has been d e m o n s t r a t e d 17) that in such cases b r a n c h i n g to a few lowlying levels can d o m i n a t e the decay. Accordingly, a n u n d e r e s t i m a t e of the level den-

HEAVY-ION REACTIONS

565

I000

.a 100 E

13

I0

16

20 24 EIo b (MiiV)

28

Fig. 7. Statistical-model calculations for multiple-particle emission from 29A1 formed by 1sO ions incident on XlB. The curve crc=~¢ is the total cross section for multiple-particle emission calculated using the value G = 0.55. The sum of all the measured cross sections corresponding to multipleparticle emission is shown by the curve (rexp. The large difference between these curves is attributable to two-neutron emission that was not detected experimentally. A lower limit on the experimentally observable multiple-particle emission cross section is given by the curve Crmlnwhich has been calculated by considering all the decay channels that could follow the emission of a first proton or alpha particle. sity could seriously affect the calculated cross section. This is believed to be the reason that n o r m a l i z a t i o n factors had to be applied in this case before reasonable agreement was o b t a i n e d with experiment. The calculated curves for G = 0.4, 0.55 a n d 0.7 have been multiplied by a factor of 2.5 a n d the one for G = 100 by 1.5 before plotting. N o n o r m a l i z a t i o n has been applied to the calculations shown for the two other reactions. Table 1 shows that 180 ions incident o n 11B yield four reactions c o r r e s p o n d i n g to two-particle emission f r o m the c o m p o u n d nucleus, Emission of two n e u t r o n s results in the stable nucleus 27A1 and, a l t h o u g h a mass value for 27Na was n o t available, it is unlikely t h a t t w o - p r o t o n emission is energetically possible. Therefore, with the exception of t w o - n e u t r o n emission a n d the possible three-particle emissions, d n a n d p2n, most of the p r o m i n e n t reaction channels for multiple-particle emission could be detected. Fig. 7 shows the calculated a n d experimental total cross sections for multi-

w.R. FALKet aL

566

ple-particle emission. The much larger value of the calculated cross section indicates the importance of two-neutron emission that was not detected experimentally. A lower limit on the experimentally observable total cross section for multipleparticle emission can be calculated by considering all the particle-decay channels that could follow the emission of a first proton or alpha-particle from the compound nucleus 29A1. Such a calculation is shown by the curve marked trmin in fig. 7 using a value for G of 0.55. 4.2. COMPARISON OF CROSS-SECTION RATIOS The exit channels for light-particle emission together with the range of excitation in the compound nucleus 26A1 are shown in fig. 8. It is clear that three-particle emission is of importance and must be considered in a complete analysis; however, as explained earlier, the computation time for such an analysis was prohibitive and hence calculations limited to one-particle emission only were carried out. Nevertheless, under certain simplifying assumptions, some gross predictions of the effects of high angular momentum in the compound nucleus can be made. These assumptions are that differences in the mean spin of the compound nucleus affect only the relative total-emission probabilities for the various particles, and that the distributions in spin and energy of the levels populated in the product nuclei resulting from firstparticle emission are the same. Although these assumptions will be shown to be invalid, the result is only to underestimate the effects due to high angular momentum. With these assumptions, the further decay of the intermediate nuclei subsequent to the first-particle emission from the compound nucleus is independent of the mode of formation of the compound nucleus. Thus one is able to predict the ratio of the crosssection ratios shown in fig. 3 since this simply requires a knowledge of the distribution in spin and energy of levels populated in the product nuclei resulting from first particle emission. Here, as in the earlier calculations, the various particle emission thresholds must be specified, in this case for the nuclei 25A1, 25Mg and 22Na. In order to take into account Coulomb effects, charged particle thresholds were raised above their real values by about one MeV for protons and three MeV for alpha particles, with the exception that the lowest particle threshold was assigned its real value. We define the ratio

ca, where x and y specify outgoing reaction channels (e.g. na, 2p, etc.) and Ca and C~ different incoming channels. For example, the ratio of the '4N-induced to ' 6 0 induced cross-section ratios, a(2a)/tr(2p) is given by R(2~, 2p, '4N, ' 6 0 ) = [tr(2°0~

/[cr(2ct)~

.

\ a - ~ / , , N / \g(Zp)/16o This ratio and the ratio R(n~, 2p, ' 4N, ' 60) have been calculated for various values

HEAVY-ION REACTIONS

567

~iii!iiii 3O

iiiiiiiiiii 2n

+m4Af

p.2n +2:~Mg 2p,=< ,2o F

3p*Z~Ne

n, 2:'< *l't F

/ ii

20

p+2~ +170

I

I/ / l

+

p+~< , + n +2°Ne 3 ~ 4 N

rl +:.< +21NcI

....,d.- I ~oB

n*2p +:Z~.No

,6 0

f

I

2~+=SF 2p+ 24No

I

t I I +

2=

p*~* Ne

14N

> p,n+24Mg

11 + 2~1

10

p+~Mg

C

'%

F i g . 8. Exit channels for light-particle emission from the compound system 26A1. The range of excitation energies in 26A1 in these experiments was from 26-32 M e V .

of G and are shown in fig. 9 as a function of the excitation energy in 26A1. Also shown are the experimental curves calculated from the information given in fig. 3. Only broad limits can be defined for the ratio R(n~, 2p, 14N, 160) since both the n~ and ~n reactions contribute to the observed yield and their relative contributions are unknown. While these results are not intended as a fit to the experimental data, they do indicate a dependence of the calculated value of the ratio R on the parameter G which, for a value of G of about 0.7, is in qualitative agreement with the experimental result. As expected, putting G = 100 yields the result R--- 1. A value of G = 0 . 4 predicts a decreasing R with increasing excitation, in contradiction to the observed result.

568

w.R.

tALK

t

20 1 "'+'.%. I. 8

et aL

*',....

l

I

I

I

I

I

t

1.6

w --o

.9.:~

.

'.+,

",.

..,...+..,.-"""*"

1.2

....."~ - a.o

1.0 "'I ....

26

I

I

28

I

I

.30

I

I

32

I

26

28

~0

32

EXCITATION ENERGY JN 26AI (MeV)

Fig. 9. Calculated values of the ratio R(2u, 2p, X+N, 160) and R(nu, 2p, 14N, 160), as defined in the text, for the compound system 26AI. Only broad limits can be defined for the latter ratio since both nu and 0on reactions contribute to the observed yields. The experimental values of the ratio R are included for semi-quantitative comparison only.

Evidence showing that the assumptions made earlier are invalid is readily found by examining the particle-emission spectra and the spin distributions in the residual nuclei. The spectra differ not only in intensity but also in their shape and the position of their maxima. The average value of the spins populated in the residual nuclei are higher if the compound nucleus has higher spin. These results clearly indicate that effects due to high spin have not dissipated after the emission of the first particle, and that conceivably a complete calculation with a value of G in the vicinity of 0.7 would predict the results observed. Similar calculations for proton-induced reactions on 2SMg, when compared to °B + 160, exhibited very little spin dependence. This is not too surprising in view of the relatively small difference in the calculated average spirt of 26A1 when formed by these two incoming channels as shown in fig. 5. A possible reason for the low experimental value of a(2~)/a(2p) for proton-induced reactions is the presence of direct reactions which increases ~r(2p). The small differences in the calculated average spin in the compound nucleus 2SAl formed in the reactions 9 B e + 19F and I°B+ 1so necessarily led to very small spin

H E A V Y - I O N REACTIONS

569

effects. M o r e a c c u r a t e d e t e r m i n a t i o n s o f the c o m p o u n d - n u c l e u s spin d i s t r i b u t i o n f r o m o p t i c a l - m o d e l calculations a r e r e q u i r e d before it is p r o f i t a b l e to extend the inv e s t i g a t i o n s for these reactions. I n s u m m a r y , the h e a v y - i o n i n d u c e d r e a c t i o n s o b s e r v e d with a n u m b e r o f targets a n d projectiles have been s h o w n to be o f a p r e d o m i n a n t l y c o m p o u n d - n u c l e a r nature. Spin effects in the c o m p o u n d nucleus have a m a r k e d effect on the relative emission p r o b a b i l i t i e s , o f n e u t r o n s , p r o t o n s a n d a l p h a particles. A value o f G between 0.55 a n d 0.7 has been f o u n d to describe the fits to the experim e n t a l d a t a m o s t a d e q u a t e l y . This value represents an effective average over a r a n g e o f excitation in the residual nuclei o f u p to a p p r o x i m a t e l y 20 MeV. E x p e r i m e n t a l l y d e d u c e d values for this p a r a m e t e r in o t h e r m e a s u r e m e n t s in this m a s s r e g i o n 3,17) show t h a t the n u c l e a r m o m e n t o f i n e r t i a is essentially the r i g i d - b o d y value at high excitation (say, 15 M e V a n d higher). A t lower excitations ( o f the o r d e r of, o r less than, the n e u t r o n b i n d i n g energy) the m o m e n t o f inertia m a y be c o n s i d e r a b l y less t h a n the r i g i d - b o d y value, as is evidenced b y the spacing o f the levels in r o t a t i o n a l b a n d s o f nuclei. The a u t h o r s w o u l d like to t h a n k the a c c e l e r a t o r o p e r a t o r s o f the L a b o r a t o r i u m fiir K e r n p h y s i k for their c o n t i n u i n g efforts to p r o d u c e a n d m a i n t a i n stable h e a v y - i o n b e a m s . One o f us ( W . R . F . ) w o u l d like to express his a p p r e c i a t i o n t o the N a t i o n a l R e s e a r c h C o u n c i l o f C a n a d a for financial s u p p o r t in the f o r m o f a P o s t - D o c t o r a t e Overseas Fellowship. This w o r k was s u p p o r t e d b y the Swiss Science F o u n d a t i o n and, in p a r t , by the A t o m i c E n e r g y C o n t r o l B o a r d o f C a n a d a .

References I) 2) 3) 4) 5) 6) 7) 8) 9) I0) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21)

W. R. Falk, U. Matter, A. Huber, R. W. Benjamin and P. Marmier, Nucl. Phys. A l l 7 (1968) 353 J'./. Pinajian and M. L. Halbert, Phys. Rev. 113 (1959) 589 F. E. Durham and M. L. Halbert, Phys. Rev. 137 (1965) B850 M. L. Halbert and A. Zucker, Phys. Rev. 114 (1959) 132 T. Ericson, Nucl. Phys. 17 (1960) 250 D, C. Williams and T. D. Thomas, Nucl. Phys. A107 (1968) 552 S. J~igare, Nucl. Phys. A95 (1967) 491 D. Bodansky, Ann. Rev. Nucl. Sci. 12 (1962) 79 T. Ericson, Adv. in Phys. 9 (1960) 425 .I.P. Butler and D. C. Santry, Phys. Rev. 152 (1966) 1034 K. J. LeCouteur and D. W. Lang, Nucl. Phys. 13 (1959) 32 T. D. Newton, Can. J. Phys. 34 (1956) 804 A. G. W. Cameron, Can. J. Phys. 36 (1958) 1040 D. C. Williams and T. D. Thomas, Nucl. Phys. A92 (1967) I D. W. Lang, Nucl. Phys. 26 (1961) 434 T. D. Thomas, Nucl. Phys. 53 (1964) 558 E. W. Vogt, D. McPherson, J. Kiihner and E. Almqvist, Phys. Rev. 136 (1965) B99 E. Gadioli, T. Tori and L. Zetta, Phys. Rev. 174 (1968) 1140 J. R. Huizenga and G. Igo, Nucl. Phys, 29 (1962) 462 T. D. Thomas, Phys. Rev. 116 (1959) 703 D. L. Hill and J. A. Wheeler, Phys. Rev. 89 (1953) 1102

570 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32) 33)

w.R. FALK et aL J. A. Kiihner and E. Almqvist, Phys. Rev. 134 (1964) B1229 C. J. Kost, Ph.D. Thesis, University of Manitoba 1968, unpublished R. M. Gaedke, K. S. Toth and I. R. Williams, Phys. Rev. 140 (1965) B296 B. L. Cohen, I-L L. Reynolds and A. Zucker, Phys. Rev. 96 (1954) 1617 H. Knoth, A. Gobbi, A. Huber, U. Matter, J. L. Perrenoud and P. Marmier, Helv. Phys. Acta 41 (1968) 1278 D. Sperber, Phys. Rev. 141 (1966) 927 J. R. Grover and J. Gilat, Phys. Rev. 157 (1967) 814 P. M. Endt and C. van der Letm, Nucl. Phys. A105 (1967) 1 T. G. Ebrey and P. R. Gray, Nucl. Phys. 61 (1965) 479 J. W. Olness and D. H. Wilkinson, Phys. Rev. 141 (1966) 966 J. L. C. Ford Jr., J. K. Bain, C. M. Jones and H. B. Willard, Nucl. Phys. 63 (1965) 588 H. Yule, Nucl. Phys. A94 (1967) 442