Reactions of 14N and 16O ions with p and s-d shell nuclei at Elab = 16 to 75 MeV

]

2,---"~'-~ Nuclear Physics A246 (1975) 515--532; (~) North-Holland Publishing Co., Amsterdam

|

~

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

R E A C T I O N S O F 14N A N D 1~O I O N S W I T H p A N D s-d S H E L L N U C L E I A T Elab ~ 16 T O 75 M e V R. HOLUB t and A. F. ZELLER tt Department of Chemistry **¢

and H. S. PLENDL and R. J. DE MEIJER

Department of Physics ~, The Florida State University, Tallahassee, Florida 32306 Received 13 November 1974 (Revised 10 April 1975) Abstract: Yield curves were measured for formation of 7Be and of residual nuclei formed by u-particle and nucleon emission following the reactions of 16 to 75 MeV 14N and xeO ions with thick, natural C, Mg, A1 and Si targets. The results are discussed in terms of statistical model calculations. Yield curves calculated with computer codes based on the statistical model are compared with the experimental results.

E

NUCLEAR REACTIONS C, Si(14N, X), C, Mg, 27A1(teO, X), E = 16-75 MeV; measured yields of activation products 7Be, z2'24Na, 27Mg, 3aCl, 3s'a2K. Natural targets. Ge(Li) detector.

1. Introduction M o s t e x p e r i m e n t a l studies o f h e a v y - i o n r e a c t i o n s with light nuclei h a v e been m a d e a t i n c i d e n t energies e i t h e r n e a r the C o u l o m b b a r r i e r o r in excess o f 100 M e V . T h e p r i m a r y c o n c e r n s o f these studies have, a c c o r d i n g l y , been e i t h e r scattering a n d r e s o n a n c e p r o c e s s e s o r r e a c t i o n s o f a d i r e c t n a t u r e . I n t h e less e x p l o r e d i n t e r m e d i a t e e n e r g y r e g i o n , f o r m a t i o n o f a c o m p o u n d s y s t e m a p p e a r s to b e t h e d o m i n a n t process. This p o s s i b i l i t y was s u g g e s t e d b y s o m e e a r l i e r e x p e r i m e n t a l results [see, e.g., t h e w o r k r e v i e w e d i n refs. ~' 2)] a n d was f u r t h e r i n v e s t i g a t e d in m o r e r e c e n t w o r k at several l a b o r a t o r i e s . O n e o f t h e s e m o r e r e c e n t studies is a d e t e r m i n a t i o n o f highr e s o l u t i o n e x c i t a t i o n f u n c t i o n s f o r t h e r e a c t i o n ~2C(~ 6 0 , c024Mg f r o m 19 t o 25 M e V t Present address: US Department of the Interior, Bureau of Mines, Bldg. 20, Denver Federal Center, Denver, Colorado 80225. tt Present address: Department of Physics, Florida State University, Tallahassee, Florida 32306. ttt Research supported in part by the US Atomic Energy Commission through Contract At (40.1) 1797. Exchange visitor from the Netherlands under the Senior Fulbright-Hays program. Present address: Kernfysisch Versneller Instituut, University of Groningen, Groningen, The Netherlands. $~ Research supported in part by the National Science Foundation, Grants no. Gu-2612, GP25974, and GJ-367. 515

516

R. HOLUB et al.

at the Argonne National Laboratory and a successful explanation of these and earlier results on reactions of this type in terms of Hauser-Feshbach (HF) calculations 3). In a recent study at the Oak Ridge National Laboratory, cross sections from (160, xn yp z~) reactions on ss, 6ONi at 38, 42 and 46 MeV were measured, and cross sections calculated from a statistical evaporation (SE) code were found to be in good agreement with the data 4). Relative yields for similar reactions on Fe, Ni, Ge and Zr in the 3'0-60 MeV region measured at Florida State University were fairly well reproduced by an SE calculation including angular momentum effects 5). In earlier work by the present authors, yields of residual nuclei formed in C + 14N, Si+14N, and 27A1-~160 reactions were measured from 16 to 58 MeV; the results were compared with both SE and HF calculations, and the possibility of transition state effects was considered 6). HF calculations with the computer code STATIS 7) by Hanson e t al. s) were found to be in excellent agreement with the excitation curves of ref. 6) for C + 14N and with angular distributions of other reactions in the same mass and energy range s). In the present study, the work of ref. 6) has been extended to several additional reactions, to higher incident energies, and to a comparison with calculations that take additional effects into consideration.

2. Experimental procedure OH- and NH~ ions were produced in a Heinicke direct extraction source 9) from an Nz'-O2 mixture and NH3, respectively, and accelerated in the Florida State University super FN tandem Van de Graaff accelerator. Oxygen ions with charge states up to 7 + and nitrogen ions with charge states up to 6 + were produced by foil or gas stripping, depending on the terminal voltage. Thick, natural C, Mg, A1, and Si targets were prepared from materials of high chemical purity. The surfaces were thoroughly cleaned before they were exposed to the beam. Cold traps were used to minimize the amount of carbon deposit on the target surfaces during the runs. The targets were exposed in a simple target holder that also served as Faraday cup. The beam currents used ranged from 30 to 2000 nA, depending on the beam energy. Fluctuations in beam current were recorded and taken into account in the yield determinations, affecting mainly the shorter-lived activities. The total charge collected per irradiation was between 0.5 and 10 mC, depending on the expected yield. Following the bombardment, the residual ~,-ray activities were counted off-line with a 20 cm 3 Ge(Li) detector having a resolution of 2.3 keV at Er = 1.33 MeV. The activities were identified by their energies, half-lives, and relative intensities. The counting time was gradually increased from 1 min shortly after irradiation to two days several months later. After the first day, the counting was done inside lead shielding to suppress room background. The spectra were stored on magnetic tape and computer-analyZed. Residual ~-ray spectra recorded at three different times following the reaction

teO

14N AND

IONS

517

2 7AI + 160 are shown in fig. 1. Relevant decay data of the product nuclides for this and other reactions are summarized in table 1. While the first spectrum in fig. 1 is dominated by short-lived activities, these activities had almost died out when the second spectrum was taken. The 6 6Ga and 6 9Ge lines are the results of Fe impurities that were introduced when the targets were cut. The longer-lived activities in the

it~5~l

,,~

HI

,o* z"~]¢I('~o~p':)

r

5oor

Ilk.,_..

"l

' 27AI +I60'ACTIIVATION Y-RAY SPECTRA 1

~c,('~'~ 3oKi~.)

I

R

~'" ":" ~

otJLJ 0.5

,

1,0

1.5

1

I

"-

•

2D

•

"

'

.J

~4

2.5

t

3.0

'q

I

TSTART =

.57 a~

0.59

1

,,o

'J~;:~,'.~,/,~:,,-.,K~,-~'~,'~

TSTOP

3Omin j

•30hr.30rain

j

GS? 1.37

I! I~"=~ ~ I It ~ I

Ill;~':'~)

,.,,

2'INo (~9"}

%.~')

I

~o,, ,

I oi'

LO i

1000~- 0.5~ 0.67

I

2~1

"B;c/y)

I

x~

13

2.0

2.5

I

I

I

3.0 I

TSTART=I6d 15h ]'STOP =17d 23h

1.46

40K(8-) ~oo~- , ~,~,.L,I

2175

;L~TI(~ ") 24N{~{~")

261"

L':t t;Tl

1.76 )

2

1

4

,

2,61 Ti ~

•

1.5

2.0

2.5

I

3.0 E~(MeV)

Fig. 1. Activation 7-ray spectra from the reaction Z7AI--F z60 at EL,b = 56 MeV. The total accumulated charge was 2.14 mC. The times elapsed between end o f bombardment and starting and stopping o f counting are given.

second spectrum have, in turn, decreased sufficiently when the third spectrum was taken so that the 53.5 d, 7-ray activity following the decay of 7Be begins to be noticeable. The radon daughter products resulted from the decay of uranium present in the walls of the building. A spectrum from the same activity one month after irradiation is shown in fig. 2.

518

R. HOLUB e t al. t

I

I

0

27AI + Js0 : ACTIVATION ) , - R A Y SPECTRUM

3 0

I

I

0.51MeV (mo~)

E,s° = 56 MeV I = 2.1 mC TCOUNT = 1800 rn]n

0.48MeV

Z

o 7Be (EC)

.Jo o

I

I

450

!

500

550 Ey(keY)

Fig. 2. Detail of activation 7-ray spectrum from the reaction 27A1-~-160at E~=b= 56 MeV, taken one month after bombardment, showing the 0.48 MeV 7,-ray following the decay of 7Be.

TABLE 1 Decay characteristics of observed reaction products Reaction product

T~

Decay

Observed E~, (MeV)

Branching ratio (~) or relative intensity

Ref.

7Be 22Na

53.5 d 2.62 y

EC t+

15.0 h

t-

2"/Mg

9.46 rain 32.3 rain 7.6752 min 12.36 h

flt+ fl+ t-

10.3 ~o 180 90 99 99 71 ~ 40.6 ~o 99.8 ~ 18.3 ~O

2t) 22) 22)

24Na

0.477 0.511 1.275 1.369 2.754 0.844 2.127 2.167 1.524

3*mc1 3aK aZK

22)

22) 22) 22) 22)

22)

3. Methods of calculation 3.1. BLANN STATISTICAL EVAPORATION CALCULATIONS F o r c o m p a r i s o n w i t h t h e e x p e r i m e n t a l y i e l d curves, r e a c t i o n cross sections were c a l c u l a t e d with a statistical e v a p o r a t i o n ( S E ) c o d e d e v e l o p e d b y B l a n n a n d Plasil [refs. lo,11)]. T h i s c o d e is b a s e d o n W e i s s k o p f - E w i n g t h e o r y 12) a n d i t calculates cross sections f o r p r o d u c t i o n o f r e s i d u a l nuclei b y successive n e u t r o n , p r o t o n , a n d ~-particle e v a p o r a t i o n f o l l o w i n g f o r m a t i o n o f a c o m p o u n d nucleus. T o t h e basic

t*N AND ttO IONS

519

code was added a standard optical model program, JIB 13) to calculate formation and inverse reaction cross sections, and an internal subroutine based on the parabolic barrier method t 4) to calculate transmission coefficients and cross sections. Optical model parameter sets were taken from the compilation by Perey and Percy 15). Experimental masses 16) with corrections for pairing energy 17) were used when available; otherwise, calculated masses la) were used. While this code does not take y-ray decay competition explicitly into account, it does correct for it by subtracting from the excitation energy the energy tied up in rotation, i.e. the yrast energy J ( d + 1)h2/2I; I is the moment of inertia. This procedure reduces the particle emission width, because near the yrast level only fission-like processes and y-ray emission are likely 19); in the energy region below the fission barrier, it results in a y-ray cascade down the yrast level. Because the rotational energy depends strongly upon the moment of inertia,/, the effect of changes in I was studied in some detail. The code also allows for fission mode competition by calculating the fission width from transition state theory 2 0). 3.2. HAUSER-FESHBACH STATISTICAL MODEL (STATIS) CALCULATIONS Reaction cross sections for t 2 C + t 4 N and 27A1(160, 7Be)36CI were calculated v~ith the Hauser-Feshbach (HF) statistical model code STATIS 7). Optical model parameters were the same as those used in the Blann code. Average pairing energies were used iT) and the level density parameter was taken to be {A, the same value used in the Blann code. Discrete levels for all exit channels were taken fromrefs. 2t, 22). The exit channels consisted of n, p, ~t, 7Be, and, to provide a competing heavy particle exit channel for high angular momentum processes, aBe. 4. Results

Since the incident ions were stopped in the targets, they contributed to the measured reaction cross sections over an energy range from the incident energy down to the reaction threshold energy. The experimental yields represent, therefore, the value of the cross section integrated over this effective energy interval. These yields, expressed in units of cross section times effective target thickness, are shown as a function of bombarding energy in figs. 3 to 7. The error bars indicate statistical uncertainties only. The reactions producing the observed nuclides are listed in table 2 (first column). In figs. 3a, 4, 5, 6, and 7a, the solid lines are drawn through the data points to indicate the shape of the excitation function, and the broken lines represent the results of Blann statistical evaporation calculations (dashed lines: I = Iris; dotted lines: I = 0.5 I~is). In fig. 3b, the solid line is the result o f a HF statistical model calculation by Hanson et aL a); the dashed lines were obtained by changing the radius parameter in the level density equation (upper dashed line: R = 1.2 fm, lower dashed line R = 1.5 fm). In fig. 7b, the solid line is the result o f a HF statistical model calculation;

520

R. H O L U B e t al.

i

12

i

14

[

1"'"

....... .T..................... /<.7............... f--

C," N

.d

..i--

i

I

I

,,.,.~ io ~

~-,o ~

rqo

// ~,,o-7~

'"" /I ~~ / / /.,k.,.-%

~

#/,.# i i,' Wf-" i i."

IOI

Na

[~,0/

I

I

22No" 0 00_=

/ ' [ ....... 2 2 .

/ .,"

~.~"_ I N ~ - ...... .~O"~......... _

i

12C ~14N

% ,o~ E

x ...0 lOI

__1 .LLI

I

lo°

~-id I

~-Id'

~o ~'o

4;

I° z

~o 6'o

20'

' 30

~ E (MeV) Fig. 3a. Energy-integrated yield curves (solid lines) for 1 2 C + 1 4 N reactions compared with Blann calculations using I ---- I, js(dashed lines) and I = 0.5 Iris (dotted lines).

4 'o

~'

50'

6'0

E (MeV)

Fig. 3b. Energy-integrated yields compared with STATIS calculations by Hanson et al. s) (solid lines). The upper dashed line is a calculation for 7Be using a radius parameter in the spin cut-off of 1.2 fro; in the lower dashed curve 1.5 fln is used.

i

102

I03~-

E .O .o10~ E>< to

l

.// / / ~ . - . . . 7 ~ , / ~ ~-Z~

U - ,~ZK ~ ' - - (XVS)

C + ~60

< ,o'

,,..'5~

,,/

~o

,j,,5~

xD

I, ~ ~ 3 8 K yg.a

E3 D _J W Jd~ I

l*"~Se

"

>--

i(j2

2O

Cb idl

~o 46 ~'o ~ E (MeV)

Fig. 4. Energy-integrated yield curves (solid lines) for Si-FI4N reactions compared with Blann calculations using I = l,,s (dashed lines).

20

P"7 50

40

50

60

70

>E(MeV) Fig. 5. Energy-integrated yield curves (solid lines) for C+1~O reactions compared with Blann calculations using I = Irls (dashedlines). iTMg yields have been multiplied by I/1oo.

I')N A N D 160 IONS

E

l03

521

/

F

1/'~.,~

:x: 'o2

_o

,o, d3 / Ld

,oO

/

104 I

20'

3'0'4'0'

,

50

,

i

i

60

i

70

) E(MeV) Fig. 6. Energy-integrated yield curves (solid lines) for M g + l ~ O reactions compared with Blann calculations using / = / , I s (dashed lines).

I0 ~

. . . .

~AI J.~-'.:::.::....

1~ E x

'~' ~°°° tl

tO

E

//" 38K 00'

,

,

102

(D

/

0

,

2~1,~60

E

..... o / / :

"~10 ~

i

/ .... /..I:

" -

E lo X

..J LLI

"

~_ Io"~

ic~3

20

I

50

40

50

60

> E(MeV) Fig. 7a. Energy-integrated yield curves (solid lines) for 27Al+lZO reactions compared with Blann calculations for I = 1,is (dashed lines) and for I = 0.5 ;,is (dotted lines).

2O

30

I

40

I

t

50

60

E (MeV) Fig. 7b. Energy-integrated yields for ~A1(160, 7Be)a6C1 compared with STATIS calculation (solid line) and with transition state calculations using I = 0.8 I,j s (dottedline) and 1 = l,j, (dashed line) for the saddle point.

522

R. HOLUB et al.

the broken lines represent the results of calculations in which transition state effects were considered (dashed line: I = Iris, dotted line: I = 0.8/rig). The calculated yield curves shown in the figures were obtained by summing the calculated differential cross sections times the calculated effective range in the target material 23). For the reactions 14N+Si, 160+C, and 160+Mg, the effect of all target isotopes was considered, and the yields were calculated by A

/3

Ytotal = E E i=1 j f E ' r

o',./Rij

ai,

where the summation extends over all isotopes (A) and energies (E); alj, Rij, and at are, respectively, the differential cross section, the effective range, and the isotopic abundance; E T and E are the threshold and incident energies, respectively. For the reaction ~4N+C, the presence of a3C was not taken into account because of its relatively low abundance (1 ~o) and its high threshold relative to 12C for the observed reactions. I

3C

.=

I

o % 8or,!er

/0'

2C

o h=

I0

I

!

I0

20

--Z

Fig. 8. Coulomb barrier heights for 14N and ~60 ions as a function o f Z of the target nucleus (see text).

To obtain reaction cross sections from the experimental yield curves shown in figs. 3 to 7, the threshold energies have to be known. The entrance channel threshold energy is equal to the Coulomb barrier height, and the exit channel threshold energy equals the Coulomb barrier height plus the reaction Q-value. Coulomb barrier heights for the entrance channels were extracted from the optical model analyses of elastic scattering data using the method of Obst et aL 24). The results are shown in fig. 8. To calculate the exit channel Coulomb barrier heights, the standard expression, ZIZ2e2/(A~+A~)Ro, was used. The subscripts indicate emitted particle and residual nucleus, respectively; the radius constant R o was chosen to be 1.6 fm for emitted particles up to Z = 2 and 1.2 fm for heavier fragments to take into consideration deformations from spherical shape is). The experimental results (figs. 3 to 7) are

+ 3"C1

7

6

5

4

3

Fig.

3

1 2 1, 92 Yo (2 or 3, 8 %) 1, 92 ~o (2 or 3, 8 ~ ) 2, 92 ~. (3 or 4, 8 %) 2 4, 99 ~. (5, 1 ~ ) 3, 99 ~. (4, 1 ~o) 1, 99 ~0 (2, 1 ~o) 2, 79 70 (3, 10 ~o) 3, 79 ~. (4, 10 ~o) 1, 79 ~o (2, 10 ~o) 2 1

1

2

No. of particles removed ")

1

0.1 to 0.2

0.1 to 0.2 1

~ 0.2 ,~, 0.2 ~ 1 1

~ 0.3 ~ 0.02

~ 0.1

1

0.01

Agreement of calculations with data Hczp (Tezp O'Blgnll O'$TATIS

0.5 IriB

0.5 lrj,

]rill

/'ri.

/'a.

/'rl.

/rig Iris

Agreement determined by no. of levels at saddle point

Agreement determined by no. of levels in final state 0.5/~,. 0.5 I~,.

Comments concerning STATIS calculations

Moment of inertia used in Blann calculations

Observed reaction products are underlined. Low-abundance target isotopes and corresponding products are given in parentheses, if they play a significant role in the reaction. a) If ~-emission is possible, it is counted as one particle; see text.

2~+n

-t- 19F q- 't2K q- aSK -I- 3sCl -I- 34mc! q- 27Mg q- 2'tNa + 22Na q- 21Ne q- aSK q- 34C1 + ssS --]- 3SK + 3eCl

-I- ZZNa

¢z

7Be a°Siq-14N ~ 2p 2s(zg. ao)si_k14N ~ g(-l-n; 2n) 7Be(+n; 2n) 2~(+n; 2n) t3C-l-160 --)-2p 12(13)C"~-160 ~ 3p-kn(-kn) ctWp+n(Wn) 7Be(+n) z4(ZS)Mg-k 160 ~ p-l-n(q-n) c(-[-p-kn(-kn) 7Be(+n) ZTAlq-l°O --* 0~q-n 7Be

q- 24Na

12Cq-14N ~ 2p

Reaction

TABLE 2 Reaction products and comparison of data and calculations

7,

Po

0

>.

524

R. HOLUB et

aL

shown as yield curves rather than as cross section curves, so that the calculated results can be compared with purely experimental results. 5. Comparison of experimental and calculated results

The extent to which the experimental and calculated excitation curves agree is summarized in table 2. The Blann program in its present form does not consider emission of particles heavier than alphas and lighter than symmetric fission products. Hence the 7Be excitation c m , e could not be calculated. Extension of the Blann code to what amounts to asymmetric fission is in progress 25). From table 2 it is evident that, for a given particle-projectile system, the agreement with the data is generally better the more particles are evaporated from the system. The agreement is seen to get worse as more states with higher J and higher excitation energy contribute to a particular process. A large discrepancy between the experimental yield curves for 38K and those calculated with the Blann program is evident in the figures and in table 2. This disagreement is due, in addition to the already mentioned effect of high angular momentum, to an isomeric state in 3SK (J~ = 0 +, T = 1, T÷ = 0.59 see) that does not decay to the 3SK (J= = 3 +, T = 0) ground state 22) but to the 3SAr ground state by B +. The experimental yield curves, which are determined from the 2 + --, 0 ÷ transition in 3SAr following the ~+ decay of the 3SK ground state, do not include the contributions from the decay of excited states through the isomeric state and represent, therefore, only a fraction of all 3s K produced. Since no detailed ?-cascade scheme for 3SK is known 22), a reliable calculation of that fraction is not possible. The experimental yields of 2 7Mg were difficult to determine in some reactions, since it was formed from only the x3C content of natural carbon, so that the photopeaks from this nuclide were often buried under high background. Since the STATIS program requires considerably more computer time than the Blann program, fewer calculations were made with it. F r o m the results summarized in table 2, it is noted that, without normalization of the curves, there is excellent agreement for the 12C+ l'tN system, but poor agreement for the 2 7AI + 160 system. The agreement for 2 7AI + x60 improved very little even when principal parameters used in the calculations (radius parameter in the level density, Coulomb radius parameter, and number of discrete levels used) were adjusted rather drastically. 6. Discussion 6.1. GENERAL CONSIDERATIONS CONCERNING STATISTICAL EVAPORATION

CALCULATIONS The assumptions and approximations in both SE calculations that have been made in this work stem from angular momentum restrictions and from the existence of a transition state as a rate-determining step (bottleneck). These factors may explain the discrepancies between experiment and theory.

1aN AND 160 IONS

525

In reactions of the type heavy ion in-particle(s) out, the basic assumption is that they can be described by the statistical model a, 5,s.26). Except for the transition state approach, SE calculations are based on either reciprocity (i.e. spins reversed under time reversal) or detailed balance (i.e. spins averaged over) between the initial (compound system) state and the final (particle(s) plus residual nucleus) state. If the compound system with angular momentum J decays into a particle and a residual nucleus the angular momenta of which, Jp and $R, cannot readily have any value s), i.e. if the s-wave approximation does not hold, restrictions are placed on the system. Assuming that Jp ,~ 0 and JR -- J, the equations corresponding to detailed balance are given in refs 27, 2s). Using transmission coefficients, T~sz, for entrance and exit channels, one obtains the average cross section expression on which the STATIS code is based, 7z ~ 2J+l k~ ~', (2j°+ 1)(2jb + 1)

_

Torsi T ~ , s , l,

E T~.S.I,, o~"S"I"

(1)

where S and l are the spin and orbital angular momentum in the entrance channel, ~t; the primed quantities refer to the exit channel and the doubly primed quantities to all channels. The dilemma in calculating cross sections using either Ericson's treatment 27) or the Weisskopf-Ewing (WE) formula 12) is one of accuracy versus computational feasibility. The equation for the cross section given by Ericson is oo

a(n, E,) = ~-- ( 2JTfdJ #'p(ER' O) k2 J ° Eft, dE,

f"

v

dO

x

2o2 ] Jo \ o21

p(ER,O)f:2tT? exp( j2 +12~-jJo ~ (i Jl~)dl

, (2)

where o(n, E,) is equivalent to a~, p(ER, 0) is the level density of the residual nucleus for state j = 0, o R is the spin cut-off parameter, jo(Jl/oR 2) the zero-order Bessel function, and W.ts(n, E,) the angular distribution part. The WE formula, which is a gross approximation of the Ericson equation (especially since it contains the s-wave approximation), was used in the Blann code, since it satisfied computer core limitations. Eq. (1) is much simpler than eq. (2) and takes the angular momentum restrictions into account in a natural way. In eq. (2), the exponential term with the Bessel functions and Wjt takes care of what in the H F code is calculated by means of Clebsch-Gordan geometry. When all states in the residual nuclei are known, eq. (I) is exact while eq. (2) is not. If the states are unknown, then both eqs. (I) and (2) utilize some approxi-

R. HOLUB et aL

526

mate level density formula. The formula used here is

p(U, J) = (2J+1) exp (-J(J+ 1)/2tr2)p(U) (2~2)~

,

(3)

where tr is again the spin cut-off parameter and p(U) is the number of levels in the equidistant spacing model for excitation energy U = E.-b&; tS, in this case, is the pairing energy, and b is 0, 1, or 2 depending on whether the nucleus is doubly odd, odd-even, or doubly even. When eq. (3) is applicable, the total level density depends strongly on tr, a collective feature that relates to an essentially single-particle phenomenon (internal excitation). Thus, the uncertainties in tr (and consequently in the level densities), in the optical model parameters, and in the correct application of angular momentum restrictions cause a large uncertainty in the yrast energy and ultimately in the calculated cross section. 6.2. BLANN CALCULATIONS In order to explain the trend observed in the comparison of experimental and Blann calculation results (table 2), the s-wave approximation used in that code must be examined. It assumes an infinite moment of inertia and zero width for the angular momentum distribution of the emitted particles. The kinetic energy carded away by the emitted particle is underestimated in the code by a term depending on the rotational energy, so that subsequent decay channels, particularly high-J channels, are not depopulated. If these were taken into account, the calculated cross sections would be in better agreement with the data. Another factor contributing to the observed discrepancy is the disposition of angular momentum in the process, which is shown schematically in fig. 9. The areas marked 1 and 2 (separated by the binding energy of the neutron) determine the ratio of particle to y-ray cross sections weighted by the formation cross sections. Fig. 9 also shows the effects of changing I on the yrast energy and on the radiative capture cross section (shaded areas). The s-wave approximation is seen to mean that the excitation energy is disposed of by successive evaporations without any change in angular momentum. In the Blann code, rotational energy is removed only by y-ray emission. From momentum and energy conservation laws, it follows that heavier particles carry away more angular momentum. Gilat et al. 29) have shown that competition between ~- and y-emission at high angular momentum results in oremission becoming the dominant mode of decay. Therefore, the larger the angular momentum that must be disposed of by heavier particles, the fewer particles are emitted (i.e. high-J channels), and the lower the cross section for formation of a particular residual nucleus. A third contributing factor is the effect of the moment of inertia. If it is less than I,i~, then the amount of rotational energy present is correspondingly greater, resulting in a shift of the excitation functions to higher energies. As seen in figs. 3a to 7a,

1aN A N D 160 IONS

527

E'J

b. / I

/

/

/

/

/P' w Z

2

/.

i_.

2 // t

/

/

/

/

~e.

J--

,.1

E.:

I

2

C.

,

//

I I I

I

\

u.I

d'--

Fig. 9. Disposition o f angular momentum in heavy-ion compound nucleus formation. The dotted lines in all figures are the optical model cross sections for formation o f a compound nucleus with excitation energy E * and angular momentum Y. The lower solid line represents the yrast energy, and the upper solid line the binding energy o f the neutron. Regions 1 and 2 are discussed in the text. In (a) the Eyrast line is calculated by J2[21, where I = Iris; the shaded region corresponds to the radiative capture cross section. In (b), I = oo. In (c) I < Iri, (irrotational), and the shaded zone is again the radiative capture cross section.

using I = 0.5 lab generally gives a better fit to the data. Obst et aL s) also obtained better agreement with their data using I = 0.5 Iris. A fourth contributing factor is the use of the level density formula for low excitation energies. A more realistic approach would be to use all known levels for the entire evaporation grid and then use the level density equation for the continuum. This approach appears to reduce cross sections for products close to the compound nucleus [ref. 3o)]. Finally, entrance channel effects may also contribute to the observed discrepancies, i.e. some high-J states may not be formed. This may result from both single-particle damping 3) and from collective (liquid drop) dynamics. In fig. 9c, where I < Iris, some higher-J states are not formed, i.e. an entrance channel restriction appears to be effective. It is important to note that there is no direct evidence that I = Iri s at high excitation energies. The approach of Vandenbosch et al. 3~, 32) is to calculate the effective moment of inertia, [eft, given by left

-

I-I- -- III

1±+III

-

Ko2 h2 T

,

(4)

528

R. HOLUB et aL

where 1± and III are the perpendicular and parallel moments of inertia with respect to the z-axis (fission), Ko2 the z-component of the total angular momentum, and T the temperature. Their use of 1± = I±ris, III = IIIrig and certain level density parameters leads to the conclusion that Iaf = I~i, at high excitation energies. However, identical results may be obtained when both Iz and III are smaller than the rigid body values or if a different temperature is used. The problem of entrance channel effects has been discussed in a more quantitative way by Robson 33) who introduced J©rlt, above which no compound nucleus may be formed. Plasil 34) and Galin et aL 35) have also investigated the problem. The net result is that if at.,. < trR, the cross section is, of course, reduced. 6.3. STATIS CALCULATIONS; CONSIDERATION OF TRANSITION STATE EFFECTS

The STATIS calculations were seen to be in excellent agreement with the experimental yield curves for the 12C + 14N reactions, while there was considerable disagreement for the 2~AI(~60, 7Be)36C1 reaction. In both the STATIS and the Blann calculations, only the rotational collective degree of freedom has been taken into account, and changes in the level density due to rotation 36) have been neglected. Another collective degree of freedom that needs to be considered, a shape change leading to fission, cannot be described by reciprocity or detailed balance, and hence the statistical model is no longer applicable. We require only that the final fission products be identical to the evaporation residues (ER) calculated in the Blann and STATIS codes. If the evaporated particle is small (i.e. n, p, ~), then collectivity is not applicable because of lack of coherence and the inability of the Coulomb forces to drastically affect the residual nucleus. Also, the shape of the corresponding asymmetric saddle shape requires so much surface energy that the barrier becomes too high. Because n, p, a evaporation happens, we know that the initial and final states determine the cross sections and that this is certainly a single-particle phenomenon. On the other hand, if the evaporated particle and the ER have a mass that is larger than some critical value, then a collective description is natural. As a consequence, an intermediate or "transition state" (TS) is formed whose properties are determined, to a first approximation, by the Coulomb and surface energies 37). Whether we describe a particular reaction channel as either a single-particle (SP) or a collective phenomenon, the initial states are, of course, the same. The number of open channels for a given ejectile, i.e. 7Be, as a function of J is, in the HF notation, I- U~ Gsp ( J ) : E s,l,

t~U..,~x T~s,v+

-I

I)T~s,vd

,

(5)

where the sum from U = 0 to Ue is for discrete levels and the integral from Uc to Urea, for continuum states given in eq. (3); the primed quantities refer to the exit

14N AND leO IONS

529

channel. In Ericson's semi-classical approach (eq. (2)), the corresponding expression is

Gse(J, E,) = 2g, ~ 2 exp (-JZ/2a~)p(UR, O)f o °exp (-12/2o'~) sin (Jl/a~)T¢')(E,)dl, (6) where P(UR, 0) =

P(UR) =

p(UR)[(2a~)~,

1 .. . . . .

exp 2 x / a - U R .

12alt/'}(U+ TR)~/'~

The subscript R refers to the residual nucleus, and a, a 2 and T are the level density parameters (-~A), spin cutoffs, and temperatures, respectively. The excitation energy, U, contains the pairing term 6 given by ref. i), 6 = 16.2 MeV/A°'Sll; g, -- 2A,+ l is the statistical weighting factor, where A,. is the spin (½ and ½ for the ground and first excited states of VBe). We have previously made use of the TS approach for the saddle point corresponding to the equal mass split 6). However, since in fission the mass asymmetry degree of freedom is considered frozen with respect to other collective degrees of freedom as), we now assume that each asymmetric deformation at the point of no return constitutes a new saddle point. Each of these saddle points has a different potential energy (barrier height Br) as well as a different lowering of this barrier due to rotation, ABr, but is, in all other respects, analogous to the saddle point for the equal mass split. A more detailed description of this approach will be given elsewhere 2s). The number of levels at the saddle point in the TS case is given by G~=(J, e,) = (2J + 1) exp (-j2/2a2) exp 2x/as(Us + ABf) 12a~/'(Us + rs)S/4(20"2)a/2

(7)

The subscript S refers to the saddle point. The total excitation energy of the initial state, E, is related to other energies in eq. (6) through

E = ER(J)+Qout+C+E,,+Erot(J), where eR(J) =

edO)-e o (J),

and through E = Es(J ) + Bt(J) + E, + Erot(J). Here, Qo.t is the Q-value for formation of 7Be and 36C1 from the 43Sc compound nucleus, C is the Coulomb barrier. The fission barrier used here is Bf(J) = Qout + C -

ABf(J),

where ABf(J) represents the lowering of the fission barrier calculated according to refs. 1I. 25).

530

R. HOLUB et al.

In the TS case, the number of collective degrees of freedom available at the saddle point can affect the shapes of the kinetic energy distributions 37) and angular distributions 3s), but not the number of levels, which depends on the barrier Br(J) only. This, of couxse, leads to the notion of the bottleneck. In order to determine whether the disagreement between the STATIS-calculated yield curve (solid line in fig. 7b) and the experimental yields for 27Al(t60, 7Be)36C1 is due to collective effects, the ratio (Gsp/GTs)TBc of the single-particle, H F expression and the collective, TS expression for G (eqs. (5) and (7)) was integrated over all appropriate values of J, l, and E~ and evaluated over the energy range of the experimental points. The integrations were performed with a program based on Simpson's rule ,0). Values of I = 0.8 I~is and I = I~is for the moments of inertia of the saddle point shapes were used. No other parameters were adjusted. The yield curves, obtained by folding the ratio into the experimental curve, are plotted in fig. 7b (dotted line: I = 0.8 Iris; dashed line: I = Iris). These curves are seen to follow closely the STATIS-calculated yield curve; their sharper initial rise can be attributed to the neglect of fission-barrier penetration in the calculation. That these curves disagree with the experimental yield curve to the same extent as the curve calculated by STATIS is taken as an indication that the 27Al(t60, 7Be)36C1 reaction is of a collective (fission-like) type. On the other hand, the agreement observed for the experimental 12C+ X*N yield curves and the curves calculated by STATIS is interpreted to mean that these reactions are essentially of a single-particle type (i.e. simple evaporation from the compound system). A value for f27Be -,sp was also calculated using Ericson's semi-classical approach (eq. (6)). Since that value is a factor of 10 s lower than the corresponding H F value obtained from eq. (5), the yield curve in this case would be well below the range of fig. 7. One problem in using eq. (6) for this calculation is that the J-dependence appears to approach zero too quickly. More detailed analysis is needed to determine the source of this discrepancy. 6.4. SUMMARY The Blann calculations are seen to give generally good agreement with the data, expeciaUy in cases where several particles are evaporated and when I = 0.5 IriB is used. Changing I was found to be justified in view of the angular momentum approximations made in the program. With that adjustment alone, the shape of the experimental excitation functions was well reproduced. No attempt was made to reproduce the absolute magnitude by adjusting optical model, level density, or pairing parameters or by using a J-cutoff in these calculations. STATIS calculations, which take angular momentum restrictions into account, were made for a few cases. Excellent agreement with the data was found for t 2C + 1,N. The discrepancy between the experimental yield curve for 27A1(160, 7Be)36C1 and the curves calculated by STATIS and on the basis of transition state considerations is taken as an indication that this reaction is a fission-like process. To test this tentative

I*N A N D 160 IONS

531

conclusion, more reactions of this type need to be studied, and existing computer programs must be modified so that such reactions can be treated more realistically. The authors wish to thank G. R. Choppin for his continued interest and support and J. D. Fox for his participation in the early stages of this work. We also wish to thank M. Blann, F. Plasil and R. Stokstad for making copies of their programs available. Valuable discussions with A. W. Obst, D. Robson, and F. Plasil and assistance in data taking by R. Eakcr are gratefully acknowledged. We also acknowledge the efforts of Deborah Raines in typing the manuscript. References 1) T. D. Thomas, Ann. Rev. Nucl. Sci. 18 (1968) 343 2) T. D. Thomas, Prof. heavy-ion summer study, Oak Ridge National Laboratory (1972), ed. S. T. Thornton, Report no. CONF-720669 (1972) p. 148; M. Blann, ibid, p. 269 3) L. R. Greenwood, K. Katori, R. E. Malmin, T. H. Braid, J. C. Stoltzfus and R. H. Siemssen, Phys. Rev. C6 (1972) 2112 4) R. L. Robinson, H. J. Kim and J. L. C. Ford, Phys. Rev. C9 (1974) 1402 5) A. W. Obst, D. L. McShan, M. B. Greenfield, R. Holub and R. H. Davis, Phys. Rex,. C8 (1973) 1379 6) R. Holub, A. F. Zeller, G. R. Choppin, R. J. de Meijer and H. S. Plendl, Phys. Lett. 43B (1973) 375; R. Holub, R. J. de Meijer, H. S. Plendl and A. F. Teller, Bull. Am. Phys. Soc. 17 (1972) 921; 18 (1973) 599; R. Holub, A. F. Teller, H. S. Plendl and R. J. de Meijer, Prof. Int. Conf. on reactions between complex nuclei, Nashville, vol. 1, ed. R. L. Robinson, F. K. McGowan, J. B. Ball and J. H. Hamilton (North-Holland, Amsterdam, 1974) p. 52 7) R. G. Stokstad, Yale University Wright Nuclear Structure Laboratory Internal Report no. 52 (unpublished) 8) D. L. Hanson, R. G. Stokstad, K. A. Erb, C. Olmer and D. A. Bromley, Phys. Rev. C9 (1974) 929 9) E. Heinicke, K. Bethge and H. Bauman, Nucl. Instr. 58 (1968) 125 10) M. Blann, Nucl. Phys. 80 (1965) 223 11) M. Blann and F. Plasil, Phys. Rev. Lett. 29 (1972) 303 S. Cohen, F. Plasil and W. J. Swiatefki, Ann. of Phys. 82 (1974) 557 12) V. F. Weisskopf and H. Ewing, Phys. Rev. 57 (1940) 472 13) F. G. Perey, Phys. Rev. 131 (1963) 745; A. W. Obst, Florida State University Report (1973) unpublished 14) T. D. Thomas, Phys. Rev. 116 (1959) 703 15) C. M. Perey and F. G. Perey, Nucl. Data Tables 10 (1972) 539; R. E. Malmin, Ph.D. thesis, Indiana University (1972) 16) A. H. Wapstra and N. B. Gore, Nucl. Data Tables 9 (1971) 267 17) A. Gilbert and A. G. W. Cameron, Can. J. Phys. 43 (1965) 1446 18) W. D. Myers and W. J. Swiatecki, Report no. UCRL-11980 (May 1965) 19) J. R. Grover and J. Gilat, Phys. Rev. 157 (1967) 802 20) J. R. Huizenga and R. Vandenbosch, in Nuclear reactions, vol. 2, ed. P. M. Endt and P. B. Smith (North-Holland, Amsterdam, 1962) p. 42 21) T. Lauritsen and F. Ajzenberg-Selove, Nucl. Phys..4227 (1974) 1; F. Ajzenberg-Selove and T. Lauritsen, Nucl. Phys. Al14 (1968) 1; F. Ajzenberg-Selove, Nufl. Phys. A152 (1970) 1; A166 (1971) 1; A190 (1972) 1 22) P. M. Endt and C. van der Leun, Nucl. Phys. A214 (1973) 1

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