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Photodisintegration of light and medium-weight nuclei at intermediate energies—VI
J. inorg, nucl. Chem. Vol. 43. No. I1, pp, 2589-2592, 1981 Printed in Great Britain
0022-1902/81/112589-04502.00/0 Pergamon Press Ltd.
PHOTODISINTEGRATION OF LIGHT AND MEDIUM-WEIGHT NUCLEI AT INTERMEDIATE ENERGIES--VI YIELDS OF ~r-MESONSFROM (y, ~'x) REACTIONS ON NUCLEIt E. S. de ALMEIDA~:and M. FOSHINA Centro Brasileirode Pesquisas Fisicas-CNPq, 22290 Rio de Janeiro, Brasil and V. di NAPOLI* and M. L. TERRANOVA Istituto di Chimica Generale ed Inorganicadell'Universitr, 00185 Roma, Italy
(Received 12 January 1981) Abstract--The mass-yielddependence of ¢r-mesonphotoproductionfrom complex nuclei has been investigatedin the energy range from threshold up to 1GeV. From both the available experimental data and the results of the present calculation, evidence for a volume production of photopions has been found. Agreement with recent theories is quite satisfactory. INTRODUCTION
A previous paper[l] was almost entirely dedicated to a critical review of data regarding the (y, n) and (3, p) true direct processes in complex nuclei, from 7Li to 23sU. In that paper some reference was made, although in a qualitative manner only, to r-meson photoproduction. The present paper will be devoted to an analysis of processes of the type A(y, 7r°x)B,
(1)
A(y, zr+x)C,
(2)
A(y, 7r x)D,
(3)
with x standing for a variable number of nucleons or nucleon clusters which may escape from the affected nucleus during the cascade and/or evaporation steps (see Ref. [l] for details). After the first experiments on nuclear photoproduction made in the early fifties, two hypotheses were proposed to interpret the dependence of the yield of pions on the mass number A. The first assumed that pion photoproduction were dominated by surface interactions of photons (mainly bremsstrahlung), leading thus to a A 2/3dependence. The second considered volume interactions which would have led to At-dependences. More careful analyses of experimental data gave exponents higher than 2/3 and this strongly supported the idea of a volume
tBased on part of the Doctoral Thesis presented by E. Santos de Almeidaat the Centro Brasileirode Pesquisas Fqsicas,Rio de Janeiro, December t979. ~:Permanentaddress: Instituto de Fisica, UniversidadeFederal Fluminense, 24000 Niter6i-RJ,Brasil. *Author to whom correspondencemay be addressed. §Yields available in the literature are in general differential cross sections per equivalent quantum, d2o/(dllx dp,, x Q); consequently, in the course of this paper we shall compare our calculated integrated yields with those obtained at different 0 and p~ (or T~).
production model with some hindrance factor due to reabsorption of pions mostly in the bulk of the nucleus. This question has been the subject of a number of papers that we shall not review here, kindly referring the reader to [2]. In more recent years, however, exponents between 0.72 and 0.86 were found/3-7], depending on the pion momentum p~ or kinetic energy T, and emission angle 0, in the laboratory system§, and the photopion yield turned out to be proportional to A raised to a power considerably higher than 2/3. Data furnished by Weise [8], Heimlich et al.[9] and by Ziegler[10] also confirmed such a trend, making the volume interaction model the most appropriate in describing these phenomena. It was the purpose of the present work to add further evidence for a volume-dependent yield of processes (1)(3) by means of a Monte Carlo calculation on cascades initiated by photons in nuclei. THE MONTE CARLO CALCULATION The essentials of the calculation, as well as the nuclear model chosen, cutoff energies, and lower and upper limits of excitation energies E* of the post-cascade nuclei, were already reported in[l]. A deeper discussion about the E* values may be found in [11]. As a basic point, we assumed in the calculation that the mean-free-paths of a photon, with an energy k ranging from the threshold up to 1 GeV, were large in relation to the nuclear sizes, also for heavy nuclei, and pions were thus assumed as uniformly produced within the entire nuclear volume. The calculation was carried out on a total of 12 target nuclei, from 7Li to 2°9Bi. For each nucleus, 15 different photon energies were chosen from 0.2 GeV up to 1 GeV (nota bene, photon beams were considered as monochromatic in the course of the calculation). A cascade was assumed as "favourable" whenever it led to the emission of one pion only, regardless of whether the number of nucleons (or nucleon clusters) ejected was different from zero or not (i.e. x -> 0). A total number of 104- 4 × 104 cascades was followed at each k,
2589
E.S. DE ALMEIDA et aL
2590
according to the likelihood of the process under study, and for each target nucleus. Although the calculation was programmed in such a way that (a) it would separate cascade events leaving the residual nucleus in a truly de-excited state (E*-<2 MeV) from others leading to residual nuclei with E* > 2 MeV, and (b) also for E* -< 2 MeV, it would distinguish between cascades resulting in the escape of one pion alone (x = 0) and cascades which led to the escape of one pion plus other nucleons (x > 0)[1,11], we preferred to confine ourselves to the study of all those events with E*-> 0, that is we focused our analysis towards reactions (1)-(3), as both experiment and theory do not give unambiguous information on (1, ~r) "cold" reactions for target nuclei more massive than a few light ones[12-15] and, very often, for energies around the A(1232) resonance oily. The calculation gave as results the probabilities @~(A,k) that a particular reaction i occurred during either the fast cascade step or the de-excitation of the post-cascade nucleus. In the present case, the index i refers to reactions (1)-(3). The cross sections per photon, ~rk.~, were then obtained by simply multiplying @i by the total inelastic cross section of y-nucleus interaction, which accounted for the quasi-deuteron and photomesonic elementary processes, as indicated in eqn (12) of Ref. [1]. Finally, by averaging the trk./S over the whole energy range explored, the mean cross section per photon #k.~ was deduced for each nucleus.
Owing to the somewhat large error in @, we thought it preferable to analyse only mean cross sections and compare them with the results of experiment. RESULTS AND DISCUSSION
As, from our point of view, any distinction between cold and hot events would be meaningless, we report in Table 1 the calculated mean cross sections of reactions (1)-(3). A least-squares analysis made with the data listed in Table 1 gave, in any case, an A-dependence of the type
#k.i(A)
=
aA b,
a being expressed in #b. For the sake of simplicity we shall eliminate the index k and read eqn (4) as
#i(A)
=
aA b.
Target
7L1
ok: {pb) :: (x
>
I*
I ° i-
D}
1
11
43
35
55
12C
74
70
89
160
93
92
112
27A1
150
140
153
51V
269
249
345
55Mn
266
263
358
75As
367
324
469
88Sr
414
358
556
127I
591
474
777
138Be
612
487
868
197Au
840
625
1190
209Bi
872
654
1238
No data r e j e c t i o n t e c h n i q u e was employed i n the f i t t i n g procedure used to o b t a i n mean cross s e c t i o n s from r e a c t i o n p r o b a b i l i t i e s (see Ref. I ). fStatistical
uncertainty
~Statlstical
u n c e r t a i n t y m 20~.
(5)
Further, in what follows we shall change the index i with the proper symbol (rr°, 7r+, It-, and the sums zr+ + It- and 1r° + Ir+ + It-) of the outgoing pion(s), with reference to eqns (1)-(3). A number of fits were made in order to achieve ALdependences for the different reactions. The results are summarized in Table 2. To compare more easily the present calculation data with the experimental ones, it will be more useful to write explicitly the most significant relationships of Table
Table 1. Calculated mean cross sections per photon of ~--mesonproduction from nuclei in the energy range 0.2-1 GeV
Nucleus
(4)
20 to 25~.
¶Statistical
uncertainty
~ 25~.
Photodisintegration of light and medium-weightnuclei at intermediate energies--VI Table 2. Least-squares values of the coefficient a and exponent b of eqn (5)~'
if one gives due account to the fact that 7r+-mesons arise mostly from the primary interaction
Outgolog
a
pion(s)
2591
y+p-~r++n
(9)
7 + n-~ ~" +p,
(10)
b
(pb)
and 7r -mesons from 'n°
B ± 2
0.68
± 0.10
9 ± 3
0.83
± 0.09
8 ± I
0.93
* 0.08
17 ± 3
0.88
+ 0.06
25 +- 5
0.88
± 0.05
+
n
+
,
o
-
+ ~
+ ~
*
l
* ~
- ¶
# E r r o r s q u o t e d in the T a b l e a r i s e f r o m a f i t t i n g p r o c e d u r e b a s e d on d a t a r e j e c t i o n a n d p r o p e r s t a t i s t i c a l w e i g h t s f o r e a c h set of c r o s s s e c t i o n s . ¶Yields
*
o
relative ~+
+ ~-
to the sum of ~+ and ~- e v e n t s a n d
(x > 0),
events
2, that is d~+(A) = (9 -+ 3)A (°'s3-+°'°9),
(6)
d,~-(A) = (8 -+ 1)A (°'93-+°'°s),
(7)
#o,++,-~(A) = (17 ___3)A (°'ss-+°°6).
(8)
and
The eqns (6)-(8) lead to some interesting conclusions. If one considers the calculated ratio d~-/#,,÷ as a function of the neutron to proton, N/Z, ratio of the target nucleus, the graph shown in Fig. 1 is obtained. The observed trend agrees very well with the experimental data on differential cross sections given by Grishaev et al.[3]. Complete agreement is as well found with the data of [3] as far as the cross sections per nucleon of charged pion photoproduction are concerned. As a matter of fact, 3.5[_ [ l
•
T -
] - ........
I
I
both the calculated and experimentally determined cross sections[3] must be divided by Z and N, respectively. These cross sections per nucleon are shown in Fig. 2 as a function of A. We shall now discuss the most important problem we studied, i.e. the mass number dependence of pion photoproduction. Apart from obvious differences in the coefficient a (one must remember, indeed, that our comparison is actually being made with differential cross sections), the exponent 0.88 in eqn (8) gives an A-dependence for charged pion photoproduction, whose trend almost overlaps the experimental ones found by Shramenko et a/.[5], which are 0.81_0.03 (at 0~ab= 120°) and 0.83-+0.05 (at 0~ab= 60°) at a fixed kinetic energy (T,~ = 40 MeV) of the outgoing pions and for a bremsstrahlung maximum energy Eo = 1.2 GeV (see Fig. 3). The set of experimental data reported in Refs. [3-7] compare quite favourably with the calculated data of the present work. Moreover, all these data regard pions either charged or neutral[6], and cover rather large ranges of masses (from C to Pb), bremsstrahlung energies (from 0.4 to 4.68 GeV), and pion kinetic energies and emission angles (40 MeV-2.26 GeV, and 16°-1200 in the laboratory system). They give exponents b ranging from 0.72 up to 0.86. All these experiments together with others reported in [8-10] do support a volume interaction mechanism for pion photoproduction. For "cold" (1, ~r) (i.e. x = 0 and E* <- 2 MeV) events, our calculated exponents were 0.67-+ 0.12(zr°), ~.2 r
l-
I
' I ' l''q
I
li
I 1
5 Z ~ 2
0.8
i
I
i¸ u.l 0,6
L I
•
I
o.41 0.6 ....
I
_
I
I
I
I
I
10
I 20
I
1 40
,
I ,_L,I 60 MASS
Fig. 1. Ratio of (r(Tr-)/o'(~"+) as a function of the neutron to proton ratio N/Z. Each set of data has been normalized to the ratio for nC (x). @, Ref. [3] at 0w,= 60. and T, = 40MeV. Iq, Ref. [3] at 0ab = 120" and T,, = 40 MeV. A, Ref. [3] at 0~,b= 1200
and T~ = 60 MeV. O, present work (integrated yields). - - , has been drawn by eye. JINC Vo}. ,43, No. 11--B
........ I 100 NUMBER
200 A
Fig. 2. Yield per nucleon as a function of A. Each set of data has been normalized to nC (cross). A, Ref. [3] at 0lab= 120° and T,,+ =40MeV. ¥, Ref. [3] at 0tab= 1200 and T,-=40MeV. A, present work for rr+ (integrated yields). ~7, present work for zr(integrated yields). O, present work for the cumulative yield (integrated) of charged pions (in this case divided by A). - has been drawn by eye.
2592
E.S. DE ALMEIDA et al. 10 4
!
I
r~ ill 10 3
Acknowledgments--The authors wish to express their gratitude to Professors H. G. de Carvalho and F. Salvetti for their interest and criticism. The financial support of the Italian Consiglio Nazionale delle Ricerche and Brazilian Comisfio Nacional de Energia Nuclear is also gratefully acknowledged. One of us (E.S.A.) wishes to express her deepest thanks for the hospitality she received at the Centro Brasileiro de Pesquisas Fisicas during the development of the Doctoral Thesis.
rF
!--
_z a~ O~
~:,10
uJ~ 6N
REFERENCES
121 _1 UA >
nuclei at energies above the A(1232) resonance are clearly called for to clarify the situation. This is, however, beyond the purpose of this paper, which was aimed at the study of a more general and frequent mode of nuclear de-excitation.
10
"r n
I
1 1
I
io MASS
ioo N U M B E R ~I
Fig. 3. Trends of aA b. The straight lines are marked with the exponent b. b = 1 represents the trend arising from a volume model without any hindrance factor, b = 0.88 is the result of the present calculation for the cumulative yields of ~r+ and ~rphotoproduction (x>0). b =0.81 from Ref. [5]. b =2/3 from surface model, b = 0.67 is the result of the present calculation for the cumulative yields of ~-0, ~r+ and ~-- from (y, ~-) processes (x = 0, E*-<2 MeV); the slope found seems to suggest a surface interaction model for these latter reactions.
0.60_+0.12(¢r+), and 0.71-+ 0.10(~r-). Some recent experiments[12-15] seem to confirm this trend and the conclusion should be drawn that such very peculiar reactions would be dominated by surface production. In Fig. 3 the trends are shown for charged pion photoproduction (x > 0, E* > 0) and for the cumulative yield of (3, ~r) cold processes. In concluding this note, we may say that for x > 0 the volume production model is without doubt effective in yielding reactions (1)-(3). On the contrary, it is still an open question whether a surface model alone be responsible for (y, ~r) events or not. Other experimental data for medium-weight and heavy
1. J. B. Martins, E. S. de Almeida, V. di Napoli, M. Foshina, O. A. P. Tavares and M. L Terranova, J. Inorg NucL Chem. 43, 1115 (1981). 2. V. di Napoli, Lett. Nuovo Cim. 12, 609 (1975); and references cited therein. 3. I. A. Grishaev, A. N. Krinitsyn, N. I. Lapin, V. I. Nikiforov, G. D. Pugach~v and B. I. Shramenko, Soy. J. Nucl. Phys. 14, 20 (1972). 4. N. V. Goncharov, A. I. Derebchinskii, O. G. Konovalov, S. G. Tonapetyan and V. M. Khvorostyan, Soy. J. Nucl. Phys. 17, 124 (1973). 5. B. I. Shramenko, I. A. Grishaev, V. I. Nikiforov and G. D. Pugach~v, Kharkov Physico-Technical Institute Report No. KhFTI 73-31, p. 74 (in Russian) (1973). 6. G. N. Dudkin, V. N. Eponeshnikov, Yu. F. Krechetov and V. A. Tryasuchev, Soy. J. Nucl. Phys. 19, 153 (1974). 7. L. O. Abramyan, A. O. Agan'yants, F. D. Adamyan, G. A. Vartapetyan, N. A. Dem~khina, A. N. Lebedev, Zh. V. Manukyan, E. G. Muradyan, S. E. Piliposyan, A. M. Sirunyan and A. G. Khudaverdyan, Soy. J. Nucl. Phys. 23, 394 (1976). 8. W. Weise, Phys. Rev. Lett. 31,773 (1973). 9. F. H. Heimlich, G. Huber, E. R6ssle, P. David, H. Mommsen and D. Wegener, Nucl. Phys. A267, 493 (1976). 10. B. Ziegler, Int. Conf. Nuclear Physics with Electromagnetic Interactions, Mainz, June 5-9, 1979, published in Lecture Notes in Physics (Edited by H. Arenh6vel and D. Drechsel), Vol. 108, p. 148 (1979). 11. V. di Napoli, E. S. de Almeida and J. L. Vieira, Lett. Nuovo Cim. 29, 231 (1980). 12. I. Blomqvist, P. Jane6ek, G. G. Jonsson, H. Dinter, K. Tesch, N. Freed and P. Ostrander, Phys. Rev. C15, 988 (1977); and references cited therein. 13. J. M. Laget, Nucl. Phys. A335, 267 (1979). 14. V. De Carlo, N. Freed, W. Rhodes, B. Billow, G. G. Jonsson, K. Lindgren and R. Pettersson, Phys. Rev. C2I, 1460 (1980). 15. M. Nilsson et al., Z. Phys. A294, 253 (1980).
0022-1902/81/112589-04502.00/0 Pergamon Press Ltd.
PHOTODISINTEGRATION OF LIGHT AND MEDIUM-WEIGHT NUCLEI AT INTERMEDIATE ENERGIES--VI YIELDS OF ~r-MESONSFROM (y, ~'x) REACTIONS ON NUCLEIt E. S. de ALMEIDA~:and M. FOSHINA Centro Brasileirode Pesquisas Fisicas-CNPq, 22290 Rio de Janeiro, Brasil and V. di NAPOLI* and M. L. TERRANOVA Istituto di Chimica Generale ed Inorganicadell'Universitr, 00185 Roma, Italy
(Received 12 January 1981) Abstract--The mass-yielddependence of ¢r-mesonphotoproductionfrom complex nuclei has been investigatedin the energy range from threshold up to 1GeV. From both the available experimental data and the results of the present calculation, evidence for a volume production of photopions has been found. Agreement with recent theories is quite satisfactory. INTRODUCTION
A previous paper[l] was almost entirely dedicated to a critical review of data regarding the (y, n) and (3, p) true direct processes in complex nuclei, from 7Li to 23sU. In that paper some reference was made, although in a qualitative manner only, to r-meson photoproduction. The present paper will be devoted to an analysis of processes of the type A(y, 7r°x)B,
(1)
A(y, zr+x)C,
(2)
A(y, 7r x)D,
(3)
with x standing for a variable number of nucleons or nucleon clusters which may escape from the affected nucleus during the cascade and/or evaporation steps (see Ref. [l] for details). After the first experiments on nuclear photoproduction made in the early fifties, two hypotheses were proposed to interpret the dependence of the yield of pions on the mass number A. The first assumed that pion photoproduction were dominated by surface interactions of photons (mainly bremsstrahlung), leading thus to a A 2/3dependence. The second considered volume interactions which would have led to At-dependences. More careful analyses of experimental data gave exponents higher than 2/3 and this strongly supported the idea of a volume
tBased on part of the Doctoral Thesis presented by E. Santos de Almeidaat the Centro Brasileirode Pesquisas Fqsicas,Rio de Janeiro, December t979. ~:Permanentaddress: Instituto de Fisica, UniversidadeFederal Fluminense, 24000 Niter6i-RJ,Brasil. *Author to whom correspondencemay be addressed. §Yields available in the literature are in general differential cross sections per equivalent quantum, d2o/(dllx dp,, x Q); consequently, in the course of this paper we shall compare our calculated integrated yields with those obtained at different 0 and p~ (or T~).
production model with some hindrance factor due to reabsorption of pions mostly in the bulk of the nucleus. This question has been the subject of a number of papers that we shall not review here, kindly referring the reader to [2]. In more recent years, however, exponents between 0.72 and 0.86 were found/3-7], depending on the pion momentum p~ or kinetic energy T, and emission angle 0, in the laboratory system§, and the photopion yield turned out to be proportional to A raised to a power considerably higher than 2/3. Data furnished by Weise [8], Heimlich et al.[9] and by Ziegler[10] also confirmed such a trend, making the volume interaction model the most appropriate in describing these phenomena. It was the purpose of the present work to add further evidence for a volume-dependent yield of processes (1)(3) by means of a Monte Carlo calculation on cascades initiated by photons in nuclei. THE MONTE CARLO CALCULATION The essentials of the calculation, as well as the nuclear model chosen, cutoff energies, and lower and upper limits of excitation energies E* of the post-cascade nuclei, were already reported in[l]. A deeper discussion about the E* values may be found in [11]. As a basic point, we assumed in the calculation that the mean-free-paths of a photon, with an energy k ranging from the threshold up to 1 GeV, were large in relation to the nuclear sizes, also for heavy nuclei, and pions were thus assumed as uniformly produced within the entire nuclear volume. The calculation was carried out on a total of 12 target nuclei, from 7Li to 2°9Bi. For each nucleus, 15 different photon energies were chosen from 0.2 GeV up to 1 GeV (nota bene, photon beams were considered as monochromatic in the course of the calculation). A cascade was assumed as "favourable" whenever it led to the emission of one pion only, regardless of whether the number of nucleons (or nucleon clusters) ejected was different from zero or not (i.e. x -> 0). A total number of 104- 4 × 104 cascades was followed at each k,
2589
E.S. DE ALMEIDA et aL
2590
according to the likelihood of the process under study, and for each target nucleus. Although the calculation was programmed in such a way that (a) it would separate cascade events leaving the residual nucleus in a truly de-excited state (E*-<2 MeV) from others leading to residual nuclei with E* > 2 MeV, and (b) also for E* -< 2 MeV, it would distinguish between cascades resulting in the escape of one pion alone (x = 0) and cascades which led to the escape of one pion plus other nucleons (x > 0)[1,11], we preferred to confine ourselves to the study of all those events with E*-> 0, that is we focused our analysis towards reactions (1)-(3), as both experiment and theory do not give unambiguous information on (1, ~r) "cold" reactions for target nuclei more massive than a few light ones[12-15] and, very often, for energies around the A(1232) resonance oily. The calculation gave as results the probabilities @~(A,k) that a particular reaction i occurred during either the fast cascade step or the de-excitation of the post-cascade nucleus. In the present case, the index i refers to reactions (1)-(3). The cross sections per photon, ~rk.~, were then obtained by simply multiplying @i by the total inelastic cross section of y-nucleus interaction, which accounted for the quasi-deuteron and photomesonic elementary processes, as indicated in eqn (12) of Ref. [1]. Finally, by averaging the trk./S over the whole energy range explored, the mean cross section per photon #k.~ was deduced for each nucleus.
Owing to the somewhat large error in @, we thought it preferable to analyse only mean cross sections and compare them with the results of experiment. RESULTS AND DISCUSSION
As, from our point of view, any distinction between cold and hot events would be meaningless, we report in Table 1 the calculated mean cross sections of reactions (1)-(3). A least-squares analysis made with the data listed in Table 1 gave, in any case, an A-dependence of the type
#k.i(A)
=
aA b,
a being expressed in #b. For the sake of simplicity we shall eliminate the index k and read eqn (4) as
#i(A)
=
aA b.
Target
7L1
ok: {pb) :: (x
>
I*
I ° i-
D}
1
11
43
35
55
12C
74
70
89
160
93
92
112
27A1
150
140
153
51V
269
249
345
55Mn
266
263
358
75As
367
324
469
88Sr
414
358
556
127I
591
474
777
138Be
612
487
868
197Au
840
625
1190
209Bi
872
654
1238
No data r e j e c t i o n t e c h n i q u e was employed i n the f i t t i n g procedure used to o b t a i n mean cross s e c t i o n s from r e a c t i o n p r o b a b i l i t i e s (see Ref. I ). fStatistical
uncertainty
~Statlstical
u n c e r t a i n t y m 20~.
(5)
Further, in what follows we shall change the index i with the proper symbol (rr°, 7r+, It-, and the sums zr+ + It- and 1r° + Ir+ + It-) of the outgoing pion(s), with reference to eqns (1)-(3). A number of fits were made in order to achieve ALdependences for the different reactions. The results are summarized in Table 2. To compare more easily the present calculation data with the experimental ones, it will be more useful to write explicitly the most significant relationships of Table
Table 1. Calculated mean cross sections per photon of ~--mesonproduction from nuclei in the energy range 0.2-1 GeV
Nucleus
(4)
20 to 25~.
¶Statistical
uncertainty
~ 25~.
Photodisintegration of light and medium-weightnuclei at intermediate energies--VI Table 2. Least-squares values of the coefficient a and exponent b of eqn (5)~'
if one gives due account to the fact that 7r+-mesons arise mostly from the primary interaction
Outgolog
a
pion(s)
2591
y+p-~r++n
(9)
7 + n-~ ~" +p,
(10)
b
(pb)
and 7r -mesons from 'n°
B ± 2
0.68
± 0.10
9 ± 3
0.83
± 0.09
8 ± I
0.93
* 0.08
17 ± 3
0.88
+ 0.06
25 +- 5
0.88
± 0.05
+
n
+
,
o
-
+ ~
+ ~
*
l
* ~
- ¶
# E r r o r s q u o t e d in the T a b l e a r i s e f r o m a f i t t i n g p r o c e d u r e b a s e d on d a t a r e j e c t i o n a n d p r o p e r s t a t i s t i c a l w e i g h t s f o r e a c h set of c r o s s s e c t i o n s . ¶Yields
*
o
relative ~+
+ ~-
to the sum of ~+ and ~- e v e n t s a n d
(x > 0),
events
2, that is d~+(A) = (9 -+ 3)A (°'s3-+°'°9),
(6)
d,~-(A) = (8 -+ 1)A (°'93-+°'°s),
(7)
#o,++,-~(A) = (17 ___3)A (°'ss-+°°6).
(8)
and
The eqns (6)-(8) lead to some interesting conclusions. If one considers the calculated ratio d~-/#,,÷ as a function of the neutron to proton, N/Z, ratio of the target nucleus, the graph shown in Fig. 1 is obtained. The observed trend agrees very well with the experimental data on differential cross sections given by Grishaev et al.[3]. Complete agreement is as well found with the data of [3] as far as the cross sections per nucleon of charged pion photoproduction are concerned. As a matter of fact, 3.5[_ [ l
•
T -
] - ........
I
I
both the calculated and experimentally determined cross sections[3] must be divided by Z and N, respectively. These cross sections per nucleon are shown in Fig. 2 as a function of A. We shall now discuss the most important problem we studied, i.e. the mass number dependence of pion photoproduction. Apart from obvious differences in the coefficient a (one must remember, indeed, that our comparison is actually being made with differential cross sections), the exponent 0.88 in eqn (8) gives an A-dependence for charged pion photoproduction, whose trend almost overlaps the experimental ones found by Shramenko et a/.[5], which are 0.81_0.03 (at 0~ab= 120°) and 0.83-+0.05 (at 0~ab= 60°) at a fixed kinetic energy (T,~ = 40 MeV) of the outgoing pions and for a bremsstrahlung maximum energy Eo = 1.2 GeV (see Fig. 3). The set of experimental data reported in Refs. [3-7] compare quite favourably with the calculated data of the present work. Moreover, all these data regard pions either charged or neutral[6], and cover rather large ranges of masses (from C to Pb), bremsstrahlung energies (from 0.4 to 4.68 GeV), and pion kinetic energies and emission angles (40 MeV-2.26 GeV, and 16°-1200 in the laboratory system). They give exponents b ranging from 0.72 up to 0.86. All these experiments together with others reported in [8-10] do support a volume interaction mechanism for pion photoproduction. For "cold" (1, ~r) (i.e. x = 0 and E* <- 2 MeV) events, our calculated exponents were 0.67-+ 0.12(zr°), ~.2 r
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Fig. 1. Ratio of (r(Tr-)/o'(~"+) as a function of the neutron to proton ratio N/Z. Each set of data has been normalized to the ratio for nC (x). @, Ref. [3] at 0w,= 60. and T, = 40MeV. Iq, Ref. [3] at 0ab = 120" and T,, = 40 MeV. A, Ref. [3] at 0~,b= 1200
and T~ = 60 MeV. O, present work (integrated yields). - - , has been drawn by eye. JINC Vo}. ,43, No. 11--B
........ I 100 NUMBER
200 A
Fig. 2. Yield per nucleon as a function of A. Each set of data has been normalized to nC (cross). A, Ref. [3] at 0lab= 120° and T,,+ =40MeV. ¥, Ref. [3] at 0tab= 1200 and T,-=40MeV. A, present work for rr+ (integrated yields). ~7, present work for zr(integrated yields). O, present work for the cumulative yield (integrated) of charged pions (in this case divided by A). - has been drawn by eye.
2592
E.S. DE ALMEIDA et al. 10 4
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Acknowledgments--The authors wish to express their gratitude to Professors H. G. de Carvalho and F. Salvetti for their interest and criticism. The financial support of the Italian Consiglio Nazionale delle Ricerche and Brazilian Comisfio Nacional de Energia Nuclear is also gratefully acknowledged. One of us (E.S.A.) wishes to express her deepest thanks for the hospitality she received at the Centro Brasileiro de Pesquisas Fisicas during the development of the Doctoral Thesis.
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REFERENCES
121 _1 UA >
nuclei at energies above the A(1232) resonance are clearly called for to clarify the situation. This is, however, beyond the purpose of this paper, which was aimed at the study of a more general and frequent mode of nuclear de-excitation.
10
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I
1 1
I
io MASS
ioo N U M B E R ~I
Fig. 3. Trends of aA b. The straight lines are marked with the exponent b. b = 1 represents the trend arising from a volume model without any hindrance factor, b = 0.88 is the result of the present calculation for the cumulative yields of ~r+ and ~rphotoproduction (x>0). b =0.81 from Ref. [5]. b =2/3 from surface model, b = 0.67 is the result of the present calculation for the cumulative yields of ~-0, ~r+ and ~-- from (y, ~-) processes (x = 0, E*-<2 MeV); the slope found seems to suggest a surface interaction model for these latter reactions.
0.60_+0.12(¢r+), and 0.71-+ 0.10(~r-). Some recent experiments[12-15] seem to confirm this trend and the conclusion should be drawn that such very peculiar reactions would be dominated by surface production. In Fig. 3 the trends are shown for charged pion photoproduction (x > 0, E* > 0) and for the cumulative yield of (3, ~r) cold processes. In concluding this note, we may say that for x > 0 the volume production model is without doubt effective in yielding reactions (1)-(3). On the contrary, it is still an open question whether a surface model alone be responsible for (y, ~r) events or not. Other experimental data for medium-weight and heavy
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