Statistical model analysis of (n, α) reactions

Nualear Physics 51 (1964) 449---459; ( ~ North-Holland Publishing Co.. Amster, .~,, Not to be reproduced by photoprint or microfilm without written permission from the publishe

S T A T I S T I C A L M O D E L ANALYSIS OF (n, ~) R E A C T I O N S

(I). Excitation Functions and Energy Spectra of (n,~) Reactions in Light and Medium N uclei E. SAETTA-MENICHELLA, F. TONOLINI and L. TONOLINI-SEVERGNINI Laboratori C.LS.E., Milano

and lstituto di Fisica dell'Universitgl, Milano t

Received 17 April 1963

Abstract: The (n, ~)reactions referring to nuclei up to A ~ 70 are examined by means of the statistical model. The analysis of the energy spectra and excitation functions leads to results in agreement with the evaporation mechanism. In excitation functions computations it is necessary to introduce as a parameter an "effective threshold" for the (n, ~n) multiple reaction.

1. Introd,ction In the last few years many experimental results on (n, e) reactions have become available and allow the investigation of the reaction mechanism. The problem to be dealt with is that of evaluating the extent to which the evaporative and direct effects are present in the process. It should be pointed out, however, that while for the evaluation of the evaporative part the formulae of the statistical model may be applied, to far no attempt has yet been made to allow a quantitative evaluation of the contribusion of direct effects; furthermore, it may be stated that it is not yet clear which particular mechanism follows that part of the reaction that is not produced by evaporation from a compound nucleus in statistical equilibrium. The experimental results refer to the measurements of energy spectra, angular distributions and excitation functions, and they are rather numerous in the case of light and medium weight nuclei from Na to Br. In this range, an analysis by means of the statistical model is possible. In regard to heavy nuclei, on the other hand, the experimental data concerning the energy spectrum and the angular distribution of the emitted e-particle are very few, and the conclusions that may be drawn from the theoretical analysis often represent more a prediction or a suggestion than a model test. For light and medium weight nuclei, a statistical analysis has already been made for (n, p) and (p, e) reactions 1-3) and has led to conclusions rather in agreement with evaporation model predictions.

2. Experimental Data The energy spectra of the c~ particles in (n, c~) reactions induced on light and medium weight nuclei have been measured in a few cases by means of nuclear emult This work has been supported by the "Consiglio Nazionale delle Ricerche". 449

450

E. SAETTA-MENICHELLA et aL

sions , - 6 ) or, in particular, by means of an assembly using a crystal both as a target and a detector (e.g., scintillation or silicon detectors)7-10). In fig. 1 we have collected the energy spectra obtained by Bizzeti et al. and Patzak-Vonach for the reactions Na23(n, a)F 2°, A127(n, a)Na 24, CoS9(n, (x)Mn s6, Cu63( n, ~0 C06°. These spectra are not contaminated by ~-particles emitted in (n, ~n) and (n, n~) reactions; the Q(n, c0 and the Q(n, n~) values are collected in 23

20

Na(n,~}F

27 24 AI [n,o'] Na

800

400-

Z >.600 m

r] L

~300>. ,~ 2 0 0 -

"- 4 0 0 . <

".,3 ,,~ 100

200. • , 1 3 5 7 9 11 for MeV

0 5 7 9 11 Ecx MeV

63

60

Culn,~(} Co

59 56 Co{n,~}Mn 5 .;

800-

4

>, 600-

-0 2

"~ 4 0 0 < 200 -

1 •

5

_

0

4 6 8 10 12 14 16 18 /'o( NeV

7 9 11 13 15 17 E e MeV

Fig. 1. E n e r g y distribution o f x-particles e m i t t e d f r o m (n, c¢) reactions at 14~MeV. / mb/sr

27 24 At [n,(~)Na

mb/sr Co~tn,=}M~~

7

6

,

6 5

4

2 1

0o 3'oo 6'0° 9do ~oo 150o 1~oo

0

oo

~o o i 0 o 90 o 1~oo l g o ~ - T ~

Fig. 2. A n g u l a r d i s t r i b u t i o n o f c~-particles e m i t t e d from (n, c0 reactions at 14 MeV.

STATISTICAL MODEL ANALYSIS OF (n, or) REACTIONS (I)

451

table 1. On the other hand the spectrum of potassium obtained by Bormann s) is strongly contaminated by multiple reactions (n, n~) and (n, ~n) and therefore is not analysed here. Few measurements of the angular distribution of the emitted c~-particles in (n, ~) reactions are available. Some reliable results, which are shown in fig. 2 were obtained by Patzak and Vonach 6) for the A127(n, ~) and Co59(n, ~) reactions. Measurements of (n, ~) cross-sections as functions of the energy of incident neutrons have become available in recent years. Generally, they are obtained by activation methods, and therefore give the value of the (n, ~) cross-section not contaminated by multiple reactions. For nuclei from Na to Zr, about 10 such functions have been measured for neutron energy ranges 8-20 MeV or 12-20 MeV. We shall examine the results of Mani et al. 11) for A127, of Bayrust et al. 12) for Sc45, As 75, Zr 94 and Nb 9a and of Bormarm 3,1a) for p32, Co59 and Br 79.

3. Statistical Modal Analysis A recent analysis 1) has been made of the known measurements of nuclear resonances for slow neutrons and of energy spectra of nucleons emitted according to the evaporative mechanisms. This analysis has yielded a consistent set of the values of the parameter a occurring in the formula for the density of nuclear levels given by

e24aU p(U) = Cg~o(U+t)z.

(1)

Here p(U) represents the level density as a function of the "effective excitation energy" U = E + A, where E is the excitation energy and A the pairing energy; go is the average single particle density in the nucleus at the top of the particle distribution; a = ~(n2go), and t is given by U = a t 2 - t 14); c is a constant for each nucleus. We use the ~-energy spectra for 14 MeV neutrons to obtain the value of the parameter a, and calculate the cross-section as a function of the energy of the incident neutrons by applying the Weisskopf-Ewing formula 25) a(n, =)~ = a=.(E.)

g=

r

e=max

t°

P

=~,ro=(8Op(U=)~ (2)

~ gv f~Vm'p~a¢,(e,)p(U,)de ' according to which this cross section is determined by the competition of the =-emission with all possible emissions of v-particles with all possible energies. In this formula, a=,(E,) represents the cross section for compound nucleus formation; the quantities Pv, g,, e, and U, are the momentum, the multiplicity factor (2s, + 1) for particles of spin sv, the energy, of the outgoing particles, and the effective excitation energy of the corresponding residual nucleus from the different reactions; ac, represents the "inverse" cross section for emission of v-particles from the compound nucleus, and

452

p. SAETTA-MENICHELLA et aL

o(U~,) is the level density of the residual nucleus. The upper limits of the integrals are defined as follows: ....

=E,+Qn, v+A . . . .

where Qn, v is the Q value of the reaction, and A,,, the pairing energy of the residual nucleus. The lower limit eo for the integral of the numerator will be discussed later.

4. Inverse Cross-Sections The cross-sections ac~ used are those obtained by Huizenga and Igo 16) with the appropriate optical model potential. They are a little higher than those obtained by Shapiro 17) and Blatt and Weisskopf 18) with a square-weU potential and with a radius r o = 1.5 fm. These inverse cross-sections for ~-particles have recently been used with good success by many authors. It should be pointed out, however, that their values are particularly sensitive to the parameters of the optical model for energies below the classical barrier. As regards values of ac. not reported by Huizenga and Igo, some interpolations have been made on the curve which gives a¢~ as a function of Z for each ,-particle energy. Since the rise of the excitation functions for (n, , ) reactions depends critically upon the behaviour of a ~ with the ,-particle energy, the fit of this part of the curve, keeping the other parameters fixed, is a good test for the inverse cross-sections. For the calculation of the excitation functions we have used the inverse cross-section for neutrons acn obtained by Campbell, Feshbach, Porter and Weisskopf 19), with a Woods-Saxon optical potential. These authors report the total formation crosssections of the compound nucleus for a great number of nuclei and for various energy values. Values of inverse cross-sections for protons, acp obtained with an optical potential are not available at present. In fact, some calculations were made by Lindner 20) only for a few Z values, and therefore the interpolation that would be necessary would be extremely critical. For a few nuclei, such as Cu and Co, Hansen et al. 21) have reported the cross sections for the formation of a compound nucleus byprotons obtained from a Bjorklund-Feshbach optical model, and they have compared them with the a¢p calculated from the Dostronsky empirical formula with ro = 1.6 fm and ro = 1.5 fm. From these calculations we may conclude that the optical aCp are rather near to the values obtained from the empirical formula with ro = 1.6 fm. In our calculations we have applied the aop values reported by Shapiro 23) (from which Dostronsky deduced his formula) for a square-well of radius ro = 1.6 fm. Some excitation functions were calculated ¢¢ith aCp values obtained both with ro = 1.5 fm and with that appears only in one of the a(n, ~) value to the inverse cross-section for protons, ro = 1.6 fm. The sensitivity of the denominator terms of the Weisskopf-Ewing formula, is rather slight.

STATISTICAL

MODEL

ANALYSIS

( n , ¢x) R E A C T I O N S

OF

(I)

453

5. Parameter a of Level Density and Q Values Values of the parameter a are rather well defined for the nuclei from Na to Br. Fig. 3 shows the values of a for these nuclei, deduced from the spacing of the isolated resonances and from (n, p), (n, n') and (p, ~) spectra 1, 2). Interpolations on the curve of fig. 3 were necessary in many cases, since the residual nuclei of (n, ~) reactions rarely correspond to nuclei for which the values of the parameter a were reported. The Q values have been calculated from the Wapstra tables 22), while for the pairing energy the values reported by Cameron 23) were taken. a(MeV)

-

1

20

15

x x

x o o

o ox

I~ x x

x

x

x

x

o

o

10

x x

o

x x

x ~

~

x

o~ o

0

I 10

o

I 20

r 30

I 40

1 50

N

Fig. 3. The values of the level density parameter a from resonances for slow neutrons and from energy spectra plotted versus the neutron number Nfor 13 < N < 50. 6. Analysis of Spectra For the analysis of the energy spectra of the emitted ~-particles, we have applied the Weisskopf formula 24) for "reduced" spectra

n(O ~c p(uD. 60"c=

T h i s formula gives the level density of the residual nuclei

p(U,,)

as a function of the effective excitation energy. In fig. 4 in order to compare the value ofp(U=) obtained from the spectra with expression (1) we have plotted on the ordinate In

Fn(e) L~O'c=

(U~-t-t)21,

and on the abscissa the square root of the excitation energy of the residual nucleus. The resulting curves are straight lines. However, some deviations are present at the highest excitation energies; these may be interpreted as due to the presence of contaminations from multiple reactions.

454

E. SAETTA-MENICHELLA et a l . In [ ~

(Ua÷t~ x

L¢~cc~(~J 2I-

/

j

4Ji

x

3

/

2

! -I-

I >

0

-3~

-I

~3 (~) F2O -I

Na ( n

-4 i

/

//4

G

AI {n,~')Na

2

cl = 4.5MeV

a=5.5 MeV"~

-5-

/

6-

7" In F r~(~) (C~+t~ l L¢%a (c)~ J

5-

6 5.

4~ x

3"

0 -I -2

/

L.=.

o. //

,,u 1e3 Cu (n,a] a:9

eO

Co

MeV- I

5Q

-I"

58

Co I n,~) Mn a =I0

MeV -I

-3

vu~ iM,~;i Fig. 4. Values o f the parameter a f r o m energy spectra o f (n, ~t) reactions at 14 MeV. TABLE 1 a values o f residual nuclei f r o m energy spectra Reaction

Eo (MeV)

gel.

Q., = (MeV)

Qn, .~ (MeV)

a (MeV) -~

NaSa(n, ct)F 2°

14

~)

--3.86

-- 10.5

4.5

Al ~7(n, ct)Na 2.

14.6

e)

-- 3.13

-- 10.09

5.5

C o " ( n , ct)Mn"

14

6)

0.31

-- 6.96

10

Cuea(n, or)Co6°

14

4)

1.72

-- 5.8

9

STATISTICAL MODEL ANALYSIS OF (n, ~) REACTIONS (I)

455

The value of the parameter a can be deduced from the slope of the straight line. Table 1 gives the values of a obtained in this way from the four available spectra. These a values are in a good agreement with those reported by Erba et al. 1). It is interesting to recall that the position of the peaks in the spectra shifts from 5 to 8 MeV when one passes from the A127(n, 0t) to the Co59(n, 00 spectrum.

7. Excitation Function Computations Using formula (2), the excitation functions were computed for eight nuclei from A1 to Nb. The particles labelled v have been limited to neutrons, protons and ~ particles. The rise of the excitation functions is mostly due to the rise of the values of a~; for the highest energy values the (n, ~n) reaction will produce a decrease of the excitation function. In fact, after the ~ emission, when the residual nucleus has an excitation energy equal to or larger than the binding energy B, of its last neutron, the ~-emission is generally followed by the emission of a secondary neutron, producing a (n, ~n) reaction. The experimental method excludes (n, ~n) reactions, i.e., most of the ~-particles of the lowest energies. The contribution of these ~-particles should therefore be subtracted from the total ~ emission from the compound nucleus. A precise and quantitative evaluation of multiple emission should take into account the competition between neutron emission and ), emission as well as the transparence for the emission of the neutron. In our calculations the subtraction has been made in a simple way, assuming a lower limit in the integral of the numerator given by ~o = e (B, + 6). In this way the emission of secondary neutrons has been introduced through an "effective threshold" B. + ~. It appears that a 6 value of the order of 1 MeV gives satisfactory agreement with experiment. . . . .

--

TABLE 2 Values o f parameters used in excitation function computation Reaction

Ref.

Qnp

Qn~t

(MeV)

(MeV)

Bn

(MeV)

a=

an

ap

(MeV -~) (MeV -~) (MeV -I)

A137(n, ct)Na24

it)

-- 1.83

--3.14

6.96

4.9

4.9

4.9

pal(n ' Qc)AI2a

3)

--0.69

-- 1.94

7.72

4.6

5

5

Sc45(n, ~)K42

13)

0.53

--0.41

7.53

7

7

7

CoS~(n ' ct)MnS3

la)

--0.78

0.31

7.27

9

9

9

As~2(n, ct)Ga72

13)

--0.39

1.5

6.96

13

13

13

BrT,(n, ct)As~3

3)

0.62

1.86

7.29

13

12.6

12.6

Zra2(n, ct)Sr39

12)

--2.82

3.4

6.77

10.5

11.8

12.5

Nb33(n, ~)y~o

13)

0.72

4.97

6.85

10

11.8

12.5

E.

456

SAETTA-MENICHELLA

e t al.

> >-

% z

e" O

t.o

c~

J

Lo

o

"O

O

>

Lo

~s

O O

O o o

o~ ¢0 Lo

> O

T

c~ c~

o~

STATISTICAL MODEL ANALYSIS OF (n,

~)

REACTIONS (I)

457

Table 2 gives the values of the parameter a and of the Q values and binding energies used for the computation. In fig. 5 the experimental and the calculated excitation functions are plotted. The computation was carried out with the same value 8 = 1 MeV for all nuclei. The fit for light nuclei may be considered really good, while for the heaviest nuclei such as Zr 94 and Nb 93, the theoretical values begin to deviate from the experimental TABLE 3 The a values used for v a r i o u s nuclei in e x c i t a t i o n f u n c t i o n c a l c u l a t i o n s c o m p a r e d t o va l ue s deduced f r o m the spectra (aj) a n d resonance s for s l ow n e u t r o n s (aD) Residual nucleus

N

a (MeV) -1

N a 2~

13

4.9

A127

14

4.9

Mg ~

15

4.9

A12s

15

4.6

psi

16

5

Si 31

17

5

K 4~

23

7

Sc ~s

24

7

Ca ~

25

7

M n 56

31

9

Co 5~

32

9

Fe sD

33

9

G a TM

41

13

A s ~5

42

13

Ge ~5

43

13

A s ~e

43

13

Br ~°

44

12.6

Se ~9

45

12.6

Sr sp

51

10.5

yoo

51

10

Z r °2

52

11.8

N b ~a

52

11.8

yo~

53

12.5

Zr 9a

53

12.5

aD (MeV) -1

a. (MeV) -1 5.5

4.5

8.3

12.1

8.6-10

13 16.5

12.8

9.8

458

~. SAETTA-MENICHELLAet al.

ones. For instance, in the case of N b 93, for energies higher than 12 MeV the shape o f the curves is badly reproduced, and if we want to reproduce it better, it would be necessary to take a 6 value equal to 6 MeV. The chosen values of the parameter a are compared in table 3 with those previously obtained from the spectra and slow neutron resonances. In fig. 6 two curves calculated with 6 = 0 and 6 = 1 MeV for the Co 59 nucleus are reproduced to show the sensitivity to the value of the parameter 6. 6"1mDI

32

,.59 56 Co (n,od Mn

20 24

/

20 16 12

8 4

0

i

10

i

,

12

,

,

14

,

,

16

.

,

18

,

,

20

,

EnMeV

Fig. 6. Sensitivity of the Co5g(n, ~)Mn 5eexcitation function to d parameter value; represents the experimental points; • and the solid curve the calculations. 8. Conclusions The above analysis shows that the statistical computation of (n, ~) excitation functions and the statistical analysis of ~-particle spectra at 14 MeV energy give resuits in good agreement with the experimental data for the nuclei from A1 to Br. The values of the parameter a obtained from the spectra and those used for the calculation of the excitation functions are in a good agreement with values obtained previously by Erba et al. 1) for nuclei close to those considered here. The 6 parameter introduced in the excitation function calculations has been fixed equal to 1 MeV; for nuclei such as Zr 94 and N b 93 and incident energies higher than 12 MeV, strong components of direct effects m a y be present and for this reason the fit has not been enforced. We thank Professor U. Facchini for his kind interest and effective encouragement. We are grateful also to the I.B.M. Italia for their assistance in the use of the 1620 Electronic Computer. References 1) E. Erba, U. Facehini and E. Saetta-Menichella, N u o v o Cim. 22 (1961) 1237; D. L. Allan, Nuclear Physics 24 (1961) 274

STATISTICAL MODEL ANALYSIS OF (n, at) REACTIONS (I)

459

2) R. Sherr and F. B. Brady, Phys. Rev. 124 (1961) 1928; E. Erba, U. Facchini and E. Saetta-Menichella, Energ. Nucl. 9 (1962) 171 3) G. S. Mani and M. A. Melkanoff, private communication; M. Bormann, S. Cierjacks, E. Fretwurst, K. J. Giesecke, H. Neuert and H. Pollehn, private communication 4) B. Czapp and H. Vonach, Mitt. Inst. Radiumfors., Nr. 542 (1960) 5) M. Cevelani, S. Petralia, E. Righini, N. Valdr6 and G. Venturini, Nuovo Cim. 16 (1960) 950 6) W. Patzak and H. Vonach, Nuclear Physics 39 (1962) 263 7) P. G. Bizzeti, A. M. Bizzeti-Sona and M. I]occiolini, Nuclear Physics 36 (1961) 38 8) M. Bormann, Z. Naturf. 17a (1962) 479 9) M. G. Marcazzan, E. Saetta-Menichella and F. Tonolini, Nuovo Cim. 20 (1962) 903 10) W. M. Deuchars and G. P. Lawrence, Nature 191 (1961) 995; M. G. Marcazzan, F. Merzari and F. Tonolini, Phys. Lett. 1 (1962) 21 11) G. S. Mani, G. J. Callum and A. T. Fergnson, Nuclear Physics 19 (1960) 535 12) B. P. Bayrust and R. J. Prestwood, LA 2493 (1960) 13) M. Bormann, J. Phys. Rad. 22 (1961) 602 14) K. J. Le Couteur and D. W. Lang, Nuclear Physics 13 (1959) 32 15) D. H. Ewing and V. F. Weisskopf, Phys. Rev. 57 (1940) 472 16) J. R. Huizenga and G. J. Igo, A N L 6373 (1961) 17) M. M. Shapiro, Phys. Rev. 90 (1953) 171 18) J. M. Blatt and V. E. Weisskopf, Theoretical nuclear physics (Wiley, New York, 1952) 19) E. J. Campbell, H. Feshbach, C. E. Porter and V. F. Weisskopf, Technical Report N 73 (1960) 20) A. Lidner, private communication 21) L. F. Hansen and R. D. Albert, Phys. Rev. 128 (1962) 291