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Breakdown of the thermal moving-source description of fast neutron production in (α, xn) and (3He, xn) reactions
Volume 114B, number 2,3
PHYSICS LETTERS
22 July 1982
BREAKDOWN OF THE THERMAL MOVING-SOURCE DESCRIPTION OF FAST NEUTRON PRODUCTION IN (ez, xn)AND (3He, xn) REACTIONS C.A. FIELDS, F.W.N. DE BOER 1, R.A. RISTINEN and E. SUGARBAKER 2 Nuclear Physics Laboratory a, Department of Physics, University of Colorado, Boulder, CO 80309, USA Received 4 March 1982 Revised manuscript received 29 April 1982
The production of fast neutrons in (a, xn) and (aHe, xn) reactions is analyzed in terms of a thermal moving-sourcemodel. Surprising discrepancies are found between the behavior of the data and the predictions of the model.
Semi-classical moving-source models provide a simple method for calculating the energy and angular distributions of energetic particles emitted in nuclear reactions at sufficiently high energy. The simplest such model is the quasi-free projectile breakup model of Serber [1 ], which was originally proposed to explain the continuum neutron spectrum from the 190 MeV (d, n) reaction. This model has more recently been applied with considerable success to the analysis of composite-particle emission in various heavy-ion (notably 12C and 160) induced reactions (see, e.g. refs. [ 2 - 4 ] ) , and to the production of fast protons (as well as d, t, and 3He) in 80 and 160 MeV a-induced reactions [5]. This breakup picture can easily be extended to the case in which the projectile is absorbed by the target: here the moving source is the excited "hot spot" formed in the collision. The production ot~fast neutrons in a-induced reactions at various energies [6-10] has been successfully parameterized with the use of such a model. The central feature of such models is that fast-particle emission can be regarded as a thermal process. While various more realistic approaches to continuum scattering have been developed in recent years (e.g. refs. [ 11-13]), the thermal moving-source approach remains 1 Present address: Physics Department, University of Fribourg, c/o SIN, CH-5234 Villigen, Switzerland. 2 Present address: Physics Department, Ohio State University, Columbus, OH 43212, USA. 3 Work supported in part by the US Department of Energy. 0 031-9163/82/0000-0000/$02.75 © 1982 North-HoUand
the primary analytic tool of experimentalists. Often one or more of the interpreted parameters (e.g. the "temperature" of the source) are allowed to vary freely [6-10, 14]. In this letter we present measurements of lowenergy ( 9 - 1 5 MeV/u) (t~, xn) and (3He, xn) reactions which, together with the results of higher-energy (ct, xn) studies [6,7,9], call the applicability of the thermal moving-source model to these reactions into question. In particular, we present cases in which the model successfully predicts the reaction cross section with either On or E n held constant, but fails when both are allowed to vary. We also show that the qualitative beam-energy dependence of the model does not agree with experimental results for the (t~, xnT) reaction. Both inclusive and exclusive neutron spectra and angular distributions were measured for 35 MeV (ct, xn) and 33 and 43 MeV (3He, xn) reactions on targets with A between 85 and 208 using the n - 7 coincidence system at the University of Colorado cyclotron [8]. The results of these measurements are presented in detail elsewhere [10,15] ; here we focus on the description of these reactions using the thermal moving-source model. The simplicity of the moving-source model derives from the assumption that the prompt particles are emitted isotropically in the rest frame of the projectile with a momentum distribution given [1,2] by exp(-p2/2a2), where o is proportional to the Fermi momentum PF of the ejectile A ' (mass m)while bound in the projectile A (massM) [16]:
81
Volume 114B, number 2,3
PHYSICS LETTERS
02 -_gIp F2 A , (A - A ' ) / ( A - 1).
(1)
Following Young et al. (ref. [14], eq. (3)), this expression can be transformed to lab coordinates: d2o dEdg2[ prompt =NN/~(°tEF)-3/2 exp{--(1/°tEF) X {E + (m/M) Ep -
2 [(m/M)EpE] 1/2 cos 0}}, (2)
where E is the ejectile energy,Ep the projectile energy, E F the Fermi energy per particle p2/2m, N the overall normalization, and a = ~A/(A - 1). This formulation is similar to that employed by Sujkowski [17] except that Coulomb and Q-value effects are neglected. For neutron emission the ejectile Coulomb energy is zero; moreover, comparison of spectral shapes and angular distributions from (a, xn) and (3He, xn) reactions on targets with 37 < Z < 82 shows that the projectiletarget Coulomb interaction can be neglected [10,11 ]. The effect of including Q-values in the calculations will be discussed below. Goldhaber [16] has shown that the width parameter given by eq. (1) characterizes two qualitatively distinct processes: the "sudden" or "direct" breakup of the projectile when it collides with the target, or "sequential" processes in which the projectile is excited as an entity, and then decays thermally before colliding with the target. The measurement of momentum distributions alone cannot distinguish between these two processes in this model. We note that this ambiguity goes even deeper: If the projectile decays after being locally absorbed by the target, and if the ejectile is not perturbed significantly by the surrounding medium, the momentum distribution will not differ in form from (1). Thus either direct or sequential breakup followed by fusion (of the remaining projectile-like fragment) is indistinguishable in form from fusion followed by fast-particle emission from a small excited region if the surrounding medium is transparent to the ejectile. While the value of M in eq. (2) would change with the mass of the excited region, with a corresponding change in the effective PF, the qualitative features of the process (e.g. the beamenergy dependence of the momentum distribution for constant PF and M) would remain the same as in the breakup followed by fusion reaction. We now apply these considerations to the analysis of fast neutron production in (3He, xn) and (a, xn) 82
22 July 1982
reactions. Since o 2 is independent ofEp [eq. (1)], we assume that the beam-energy dependence of the reaction is specified completely by the occurrences of Ep in eq. (2); in particular, we assume that only one prompt neutron is emitted. If any thermal "moving source" model of the reaction (either breakup followed by fusion or fusion followed by decay of a small moving source as described in ref. [7]) is adequate, then we can qualitatively expect that both the average energy of the emitted neutrons and the forward-peaking of their angular distributions in the laboratory frame will increase with the beam energy. Moreover, both energy spectra and angular distributions should be fit by eq. (2) with reasonable values for any adjustable parameters. The only relevant parameters are M, E F and overall and relative (to the low-energy equilibrium spectrum) normalization. In order to constrain the theory, we set M = M (projectile); however, the qualitative results (beam energy and target dependence of d2a/dEd~) obtained below hold for any value of M. With this assumption, E F is the projectile Fermi energy per nucleon. Fig. la shows 30 ° inclusive 124Sn(3He, xn) spectra at 33 and 43 MeV together with calculations using eq. (2) with E F = 2.6 MeV, one third the total binding energy for 3He. The variation of spectral shape with projectile energy is clearly reproduced quite well by the model (the evaporative component has k T e = 1.4 MeV at 33 MeV, 1.6 MeV at 43 MeV), which assumes that the +8.7 MeV Q-value for (3He, n) has no effect. Moreover, the spectral shape does not vary significantly with target, consistent with quasi-free breakup. However, the agreement between the model and the experimental angular distributions, shown in fig. lb, is extremely poor. Correctly reproducing the observed angular distributions with eq. (2) requires the use of a value o f E F roughly ten times larger than that required to reproduce the 30 ° spectra shown in fig. 1a. While these distributions are measured in coincidence with 7-rays, we assume that n - 7 correlation effects are small (e.g. the distributions of neutrons gated by 7rays of different multipolarity do not vary greatly [8, 101). Similar data and calculations for 35 MeV (a, xn) reactions are shown in fig. 2. The two sets of calculations shown use Fermi energies calculated from the total binding energy (E F = 7.0 MeV) and from the single-neutron separation energy (E F = S n = 21.5 MeV)
Volume 114B, number 2,3
PHYSICS LETTERS
22 July 1982
b) 'Z4Sn(3He ,xn)
o) 3 0 ° (SHe,xn) 33MeV
~t 124Sn
IX
15°Nd \
5 .ci
4 3 MeV
43MeV
÷ 124~
÷
102
E.-12
f
MeV
÷
lOS b I0'
f
104
,/
\ \
0
I
I
I
I
I
4
8
12
16
20
/
\
t', L I I0
I 24
t
~'
I 14
I 18
I 22
I 26
I0 ~
I 30
20
I
1 60
E(MeV)
I
I
I
~1~ (deg)
Fig. 1. (a) 30 ° inclusive neutron spectra for (3He, xn) reactions. Long dashed lines are equilibrium
( d o / d E n o: E n e x p ( _ E n [ k T e )
,
short dashed lines breakup components as given by eq. (2). The solid line shows the sum. (b) Angular distributions of neutrons ( 1 0 - 1 5 and 1 0 - 2 0 MeV bins) from 43 MeV 124Sn(aHe, xn). Solid lines are breakup calculations (eq. (2)).
sonably well by the breakup calculations with E F = 21.5 (fig. 2b), but neither calculation successfully reproduces the high-energy part of the 30 ° spectrum. Moreover, the spectra from the two targets differ signifi-
and (as above) assume that Q-values have no effect. Here the situation is just the reverse of the (3He, xn) case: the angular distributions from 35 MeV (ct, xn) reactions (on many targets [10]) are reproduced rea-
l
o) 3 0 * , 3 5 M e V (a, xn) F~'X
i05
{
'~°Nd ((2, xn)
b) 35 MeV
'24Sn
-- -- E F = 7.0 MeV -- -EF :21.5 MeV
I~°Nd
\
- - - - E F= 7 0 M e V --- - E F = 21.5 MeV
.'x
102
~ " f\'l,
..
\ -e
N
~'-- ..
E:n= 12 MeV
\
104
g
%.
,o
Io'
•
-o
\
I I
0
IO~
\
En= 17MeV
\
~\
\
k3'
\ "~~..
12 E n (MeV)
16
20
24
20
'
60
I00
140
~L.ob (deg)
Fig. 2. Same as fig. 1 but for (a, xn) reactions. Different breakup calculations are labelled by the appropriate value of E F. Neutron energy bins are 1 0 - 1 5 and 1 5 - 2 0 MeV.
83
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PHYSICS LETTERS
cantly above E n = 12 MeV. We have shown earlier [18] that this variation in do/dE is strongly correlated with target neutron excess;the assumption of quasi-free breakup cannot explain such an effect. Neutron spectra from (a, xn) reactions at higher energy (70 MeV, ref. [7], 90 MeV, ref. [9], and 120 MeV, ref. [6]) have been measured by the Osaka group. While only exclusive spectra have been presented, these together with the relative cross sections for various neutron multiplicities evident from the 7-ray spectra suggest that even at 120 MeV very few neutrons are emitted with E > 20 MeV (ref. [19]). Indeed the centroid of the "pre-equilibrium peak" remains near 14 MeV, its position in 35 MeV (a, xn). This behavior can be explained by assuming that the projectile is locally absorbed by the target, and that the resulting source cools significantly (thus increasing in mass) before emitting particles. Several particles are thus emitted at the "pre-equilibrium" stage of the reaction with velocities considerably less than that of the beam. While a moving-source parameterization of the reaction which uses (and corroborates refs. [6,7]) this assumption is possible, it is inconsistent with the qualitative assumptions of the model. The "sudden" breakup approximation should improve as Ep becomes much larger than E F. The data for (a, xn) show that (between 35 and 120 MeV)it does not. In view of this result, we suggest that the agreement between the 35 MeV (~, xn) angular distributions and the calculations in fig. 2b may be entirely fortuitous. In fig. 3 we present angular distributions from the present work together with those measured for (a, xn) at higher energy [6,9] and for 152 MeV 154Sm(160, xn) [14]. These distributions are all very similar; indeed their general features are consistent with those noted by Kalbach and Mann [20] in a study of low-energy charged-particle emitting reactions: the dashed lines in fig. 3 are calculations using their phenomenological description. The forward-to-backward angular asymmetry of these distributions can be characterized by the value of the coefficient of P1 (cos 0) in a Legendrepolynomial expansion of do/d~2. In a moving-source description, this coefficient is proportional to the beam (source) velocity [7]. (Note the contrast with the usual direct-reaction (e.g. DWBA) prediction that forwardpeaking decreases with increasing l-transfer.)Reconciling the thermal-source model to these angular distribution data requires postulating that the Fermi energy 84
22 July 1982
Kalbach and Mann [,6] E,~ : 12MeV - - -- E. : 17 MeV
+ 35MeV '~°Nd(a,xn) lp,,se~, ,} 43MeV t~aSn (3He,xn)J .o,b, ¢ 152MeV m4Sm('eO, xn)[,3] 70aeV "~SGd(. . . . )[9]
-J 4._+
+ ,20MeV "~'OyC.... [4
IO~
10 2
I0'
÷ 20
k
l
i
40
60
80
i
IOQ
i
120
1 i
r40
i
160
~Lob {deg )
Fig. 3. Neutron angular distributions from several reactions ( 1 0 - 1 5 and 1 5 - 2 0 MeV bins). Dashed lines are calculations using the Kalbach-Mann [1_6] phenomenological parameterization for neutron energy E n assuming 100% MSD process.
increases with the beam energy [inconsistent with eq. (1)] or that the source velocity decreases with beam energy as the exciton mass increases. The latter assumption is probably correct [6,7], but again entails the statement that the breakup approximation is poorer at higher beam energy. In summary, the available data for fast neutron production in (3He, xn) reactions [E(3He) = 11-15 MeV/u] indicate that, while the spectra can be successfully described by a simple breakup model, such a model does not provide a good description of the angular distributions. The available data for (a,xn) reactions [E(a) = 9 - 3 0 MeV/u] suggest that no thermal movingsource picture is adequate for the description of either the spectra or the angular distributions; in neither case is the correct (even qualitatively) incident-energy dependence observed. The differences between these two reactions may be traceable to the large difference in binding energies (a factor of 4) between 3He and a; however, we note that the emission of fast protons in a-induced reactions between 42 and 160 MeV (refs. [5,21,22]) appears to be at least qualitatively consistent with what would be expected in a moving-source approach (although Lieder et al. [23] report the ob-
Volume 114B, number 2,3
PHYSICS LETTERS
servation of structures in (ct, p) spectra at Ea = 4 5 - 7 5 MeV which might be difficult to explain using such an approach). In particular, the spectral shapes from (3He, xn) appear to be target-independent, in contrast to those observed in (a:, xn) [18]. At least some o f the difficulties in interpreting the (a, xn) data may therefore be due to features o f the reaction other than the t~-particle binding energy. In view o f the problems with the semi-classical description o f these reactions, we suggest that a treatment of (3He, xn) and (a, xn) which takes the nature of the nuclear potential explicitly into account, such as the breakup-fusion theory which Udagawa and Tamura have applied to the (14N, ct) reaction [12], or the DWBA techniques of Aarts et al. [13], would be very interesting. In particular, such an account may explain the apparent beam-energy independence o f the neutron angular distributions. We would like to thank Dr. R.J. de Meijer (KVI), Dr. M.L. Halbert (ORNL), and Dr. H. Sakai (Osaka) for helpful discussions.
22 July 1982
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
R. Serber, Phys. Rev. 72 (1947) 1008. H. Feshbach and K. Huang, Phys. Lett. 47B (1973) 300. C.K. Gelbke et al., Phys. Lett. 70B (1977) 415. K. Siwek-Wilczynska et al., Nucl. Phys. A330 (1979) 150. J.R. Wu et al., Phys. Rev. C20 (1979) 1284. H. Sakai et al., Phys. Rev. C20 (1979) 464. H. Ejiri et al., J. Phys. Soc. Japan 49 (1980) 2103. C.A. Fields et al., Nucl. Phys. A366 (1981) 38. M.J.A. de Voight et al., to be published, private communication. [10] C.A. Fields et al., Nucl. Phys. A377 (1982) 217. [11] H. Feshbach, Intern. Symp. on Highly excited states in nuclear reactions (Osaka, 1980). [12] T. Udagawa and T. Tamura, Phys. Rev. Lett. 45 (1980) 1311. [13] E.H.L. Aarts et al., to be published; R.J. de Meijer, private communication. [14] K.G. Young et al., Phys. Rev. C23 (1981) 2479. [15] C.A. Fields et al., to be published. [16] A.S. Goldhaber, Phys. Lett. 53B (1974) 306. [17] Z. Sujkowski, Proc. XIX Intern. Winter Meeting on Nuclear Physics (Bormio, Italy, 1981). [18] C.A. Fields et al., Phys. Lett. 106B (1981) 453. [ 19] H. Sakai, private communication. [20] C. Kalbach and F.M. Mann, Phys. Rev. C23 (1981) 112. [21] R.W. West, Phys. Rev. 141 (1966) 1033. [22] A. Chevarier et al., Phys. Rev. C8 (1973) 2155. [23] R.M. Lieder et al., Phys. Scr. 24 (1981) 123.
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PHYSICS LETTERS
22 July 1982
BREAKDOWN OF THE THERMAL MOVING-SOURCE DESCRIPTION OF FAST NEUTRON PRODUCTION IN (ez, xn)AND (3He, xn) REACTIONS C.A. FIELDS, F.W.N. DE BOER 1, R.A. RISTINEN and E. SUGARBAKER 2 Nuclear Physics Laboratory a, Department of Physics, University of Colorado, Boulder, CO 80309, USA Received 4 March 1982 Revised manuscript received 29 April 1982
The production of fast neutrons in (a, xn) and (aHe, xn) reactions is analyzed in terms of a thermal moving-sourcemodel. Surprising discrepancies are found between the behavior of the data and the predictions of the model.
Semi-classical moving-source models provide a simple method for calculating the energy and angular distributions of energetic particles emitted in nuclear reactions at sufficiently high energy. The simplest such model is the quasi-free projectile breakup model of Serber [1 ], which was originally proposed to explain the continuum neutron spectrum from the 190 MeV (d, n) reaction. This model has more recently been applied with considerable success to the analysis of composite-particle emission in various heavy-ion (notably 12C and 160) induced reactions (see, e.g. refs. [ 2 - 4 ] ) , and to the production of fast protons (as well as d, t, and 3He) in 80 and 160 MeV a-induced reactions [5]. This breakup picture can easily be extended to the case in which the projectile is absorbed by the target: here the moving source is the excited "hot spot" formed in the collision. The production ot~fast neutrons in a-induced reactions at various energies [6-10] has been successfully parameterized with the use of such a model. The central feature of such models is that fast-particle emission can be regarded as a thermal process. While various more realistic approaches to continuum scattering have been developed in recent years (e.g. refs. [ 11-13]), the thermal moving-source approach remains 1 Present address: Physics Department, University of Fribourg, c/o SIN, CH-5234 Villigen, Switzerland. 2 Present address: Physics Department, Ohio State University, Columbus, OH 43212, USA. 3 Work supported in part by the US Department of Energy. 0 031-9163/82/0000-0000/$02.75 © 1982 North-HoUand
the primary analytic tool of experimentalists. Often one or more of the interpreted parameters (e.g. the "temperature" of the source) are allowed to vary freely [6-10, 14]. In this letter we present measurements of lowenergy ( 9 - 1 5 MeV/u) (t~, xn) and (3He, xn) reactions which, together with the results of higher-energy (ct, xn) studies [6,7,9], call the applicability of the thermal moving-source model to these reactions into question. In particular, we present cases in which the model successfully predicts the reaction cross section with either On or E n held constant, but fails when both are allowed to vary. We also show that the qualitative beam-energy dependence of the model does not agree with experimental results for the (t~, xnT) reaction. Both inclusive and exclusive neutron spectra and angular distributions were measured for 35 MeV (ct, xn) and 33 and 43 MeV (3He, xn) reactions on targets with A between 85 and 208 using the n - 7 coincidence system at the University of Colorado cyclotron [8]. The results of these measurements are presented in detail elsewhere [10,15] ; here we focus on the description of these reactions using the thermal moving-source model. The simplicity of the moving-source model derives from the assumption that the prompt particles are emitted isotropically in the rest frame of the projectile with a momentum distribution given [1,2] by exp(-p2/2a2), where o is proportional to the Fermi momentum PF of the ejectile A ' (mass m)while bound in the projectile A (massM) [16]:
81
Volume 114B, number 2,3
PHYSICS LETTERS
02 -_gIp F2 A , (A - A ' ) / ( A - 1).
(1)
Following Young et al. (ref. [14], eq. (3)), this expression can be transformed to lab coordinates: d2o dEdg2[ prompt =NN/~(°tEF)-3/2 exp{--(1/°tEF) X {E + (m/M) Ep -
2 [(m/M)EpE] 1/2 cos 0}}, (2)
where E is the ejectile energy,Ep the projectile energy, E F the Fermi energy per particle p2/2m, N the overall normalization, and a = ~A/(A - 1). This formulation is similar to that employed by Sujkowski [17] except that Coulomb and Q-value effects are neglected. For neutron emission the ejectile Coulomb energy is zero; moreover, comparison of spectral shapes and angular distributions from (a, xn) and (3He, xn) reactions on targets with 37 < Z < 82 shows that the projectiletarget Coulomb interaction can be neglected [10,11 ]. The effect of including Q-values in the calculations will be discussed below. Goldhaber [16] has shown that the width parameter given by eq. (1) characterizes two qualitatively distinct processes: the "sudden" or "direct" breakup of the projectile when it collides with the target, or "sequential" processes in which the projectile is excited as an entity, and then decays thermally before colliding with the target. The measurement of momentum distributions alone cannot distinguish between these two processes in this model. We note that this ambiguity goes even deeper: If the projectile decays after being locally absorbed by the target, and if the ejectile is not perturbed significantly by the surrounding medium, the momentum distribution will not differ in form from (1). Thus either direct or sequential breakup followed by fusion (of the remaining projectile-like fragment) is indistinguishable in form from fusion followed by fast-particle emission from a small excited region if the surrounding medium is transparent to the ejectile. While the value of M in eq. (2) would change with the mass of the excited region, with a corresponding change in the effective PF, the qualitative features of the process (e.g. the beamenergy dependence of the momentum distribution for constant PF and M) would remain the same as in the breakup followed by fusion reaction. We now apply these considerations to the analysis of fast neutron production in (3He, xn) and (a, xn) 82
22 July 1982
reactions. Since o 2 is independent ofEp [eq. (1)], we assume that the beam-energy dependence of the reaction is specified completely by the occurrences of Ep in eq. (2); in particular, we assume that only one prompt neutron is emitted. If any thermal "moving source" model of the reaction (either breakup followed by fusion or fusion followed by decay of a small moving source as described in ref. [7]) is adequate, then we can qualitatively expect that both the average energy of the emitted neutrons and the forward-peaking of their angular distributions in the laboratory frame will increase with the beam energy. Moreover, both energy spectra and angular distributions should be fit by eq. (2) with reasonable values for any adjustable parameters. The only relevant parameters are M, E F and overall and relative (to the low-energy equilibrium spectrum) normalization. In order to constrain the theory, we set M = M (projectile); however, the qualitative results (beam energy and target dependence of d2a/dEd~) obtained below hold for any value of M. With this assumption, E F is the projectile Fermi energy per nucleon. Fig. la shows 30 ° inclusive 124Sn(3He, xn) spectra at 33 and 43 MeV together with calculations using eq. (2) with E F = 2.6 MeV, one third the total binding energy for 3He. The variation of spectral shape with projectile energy is clearly reproduced quite well by the model (the evaporative component has k T e = 1.4 MeV at 33 MeV, 1.6 MeV at 43 MeV), which assumes that the +8.7 MeV Q-value for (3He, n) has no effect. Moreover, the spectral shape does not vary significantly with target, consistent with quasi-free breakup. However, the agreement between the model and the experimental angular distributions, shown in fig. lb, is extremely poor. Correctly reproducing the observed angular distributions with eq. (2) requires the use of a value o f E F roughly ten times larger than that required to reproduce the 30 ° spectra shown in fig. 1a. While these distributions are measured in coincidence with 7-rays, we assume that n - 7 correlation effects are small (e.g. the distributions of neutrons gated by 7rays of different multipolarity do not vary greatly [8, 101). Similar data and calculations for 35 MeV (a, xn) reactions are shown in fig. 2. The two sets of calculations shown use Fermi energies calculated from the total binding energy (E F = 7.0 MeV) and from the single-neutron separation energy (E F = S n = 21.5 MeV)
Volume 114B, number 2,3
PHYSICS LETTERS
22 July 1982
b) 'Z4Sn(3He ,xn)
o) 3 0 ° (SHe,xn) 33MeV
~t 124Sn
IX
15°Nd \
5 .ci
4 3 MeV
43MeV
÷ 124~
÷
102
E.-12
f
MeV
÷
lOS b I0'
f
104
,/
\ \
0
I
I
I
I
I
4
8
12
16
20
/
\
t', L I I0
I 24
t
~'
I 14
I 18
I 22
I 26
I0 ~
I 30
20
I
1 60
E(MeV)
I
I
I
~1~ (deg)
Fig. 1. (a) 30 ° inclusive neutron spectra for (3He, xn) reactions. Long dashed lines are equilibrium
( d o / d E n o: E n e x p ( _ E n [ k T e )
,
short dashed lines breakup components as given by eq. (2). The solid line shows the sum. (b) Angular distributions of neutrons ( 1 0 - 1 5 and 1 0 - 2 0 MeV bins) from 43 MeV 124Sn(aHe, xn). Solid lines are breakup calculations (eq. (2)).
sonably well by the breakup calculations with E F = 21.5 (fig. 2b), but neither calculation successfully reproduces the high-energy part of the 30 ° spectrum. Moreover, the spectra from the two targets differ signifi-
and (as above) assume that Q-values have no effect. Here the situation is just the reverse of the (3He, xn) case: the angular distributions from 35 MeV (ct, xn) reactions (on many targets [10]) are reproduced rea-
l
o) 3 0 * , 3 5 M e V (a, xn) F~'X
i05
{
'~°Nd ((2, xn)
b) 35 MeV
'24Sn
-- -- E F = 7.0 MeV -- -EF :21.5 MeV
I~°Nd
\
- - - - E F= 7 0 M e V --- - E F = 21.5 MeV
.'x
102
~ " f\'l,
..
\ -e
N
~'-- ..
E:n= 12 MeV
\
104
g
%.
,o
Io'
•
-o
\
I I
0
IO~
\
En= 17MeV
\
~\
\
k3'
\ "~~..
12 E n (MeV)
16
20
24
20
'
60
I00
140
~L.ob (deg)
Fig. 2. Same as fig. 1 but for (a, xn) reactions. Different breakup calculations are labelled by the appropriate value of E F. Neutron energy bins are 1 0 - 1 5 and 1 5 - 2 0 MeV.
83
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cantly above E n = 12 MeV. We have shown earlier [18] that this variation in do/dE is strongly correlated with target neutron excess;the assumption of quasi-free breakup cannot explain such an effect. Neutron spectra from (a, xn) reactions at higher energy (70 MeV, ref. [7], 90 MeV, ref. [9], and 120 MeV, ref. [6]) have been measured by the Osaka group. While only exclusive spectra have been presented, these together with the relative cross sections for various neutron multiplicities evident from the 7-ray spectra suggest that even at 120 MeV very few neutrons are emitted with E > 20 MeV (ref. [19]). Indeed the centroid of the "pre-equilibrium peak" remains near 14 MeV, its position in 35 MeV (a, xn). This behavior can be explained by assuming that the projectile is locally absorbed by the target, and that the resulting source cools significantly (thus increasing in mass) before emitting particles. Several particles are thus emitted at the "pre-equilibrium" stage of the reaction with velocities considerably less than that of the beam. While a moving-source parameterization of the reaction which uses (and corroborates refs. [6,7]) this assumption is possible, it is inconsistent with the qualitative assumptions of the model. The "sudden" breakup approximation should improve as Ep becomes much larger than E F. The data for (a, xn) show that (between 35 and 120 MeV)it does not. In view of this result, we suggest that the agreement between the 35 MeV (~, xn) angular distributions and the calculations in fig. 2b may be entirely fortuitous. In fig. 3 we present angular distributions from the present work together with those measured for (a, xn) at higher energy [6,9] and for 152 MeV 154Sm(160, xn) [14]. These distributions are all very similar; indeed their general features are consistent with those noted by Kalbach and Mann [20] in a study of low-energy charged-particle emitting reactions: the dashed lines in fig. 3 are calculations using their phenomenological description. The forward-to-backward angular asymmetry of these distributions can be characterized by the value of the coefficient of P1 (cos 0) in a Legendrepolynomial expansion of do/d~2. In a moving-source description, this coefficient is proportional to the beam (source) velocity [7]. (Note the contrast with the usual direct-reaction (e.g. DWBA) prediction that forwardpeaking decreases with increasing l-transfer.)Reconciling the thermal-source model to these angular distribution data requires postulating that the Fermi energy 84
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Kalbach and Mann [,6] E,~ : 12MeV - - -- E. : 17 MeV
+ 35MeV '~°Nd(a,xn) lp,,se~, ,} 43MeV t~aSn (3He,xn)J .o,b, ¢ 152MeV m4Sm('eO, xn)[,3] 70aeV "~SGd(. . . . )[9]
-J 4._+
+ ,20MeV "~'OyC.... [4
IO~
10 2
I0'
÷ 20
k
l
i
40
60
80
i
IOQ
i
120
1 i
r40
i
160
~Lob {deg )
Fig. 3. Neutron angular distributions from several reactions ( 1 0 - 1 5 and 1 5 - 2 0 MeV bins). Dashed lines are calculations using the Kalbach-Mann [1_6] phenomenological parameterization for neutron energy E n assuming 100% MSD process.
increases with the beam energy [inconsistent with eq. (1)] or that the source velocity decreases with beam energy as the exciton mass increases. The latter assumption is probably correct [6,7], but again entails the statement that the breakup approximation is poorer at higher beam energy. In summary, the available data for fast neutron production in (3He, xn) reactions [E(3He) = 11-15 MeV/u] indicate that, while the spectra can be successfully described by a simple breakup model, such a model does not provide a good description of the angular distributions. The available data for (a,xn) reactions [E(a) = 9 - 3 0 MeV/u] suggest that no thermal movingsource picture is adequate for the description of either the spectra or the angular distributions; in neither case is the correct (even qualitatively) incident-energy dependence observed. The differences between these two reactions may be traceable to the large difference in binding energies (a factor of 4) between 3He and a; however, we note that the emission of fast protons in a-induced reactions between 42 and 160 MeV (refs. [5,21,22]) appears to be at least qualitatively consistent with what would be expected in a moving-source approach (although Lieder et al. [23] report the ob-
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servation of structures in (ct, p) spectra at Ea = 4 5 - 7 5 MeV which might be difficult to explain using such an approach). In particular, the spectral shapes from (3He, xn) appear to be target-independent, in contrast to those observed in (a:, xn) [18]. At least some o f the difficulties in interpreting the (a, xn) data may therefore be due to features o f the reaction other than the t~-particle binding energy. In view o f the problems with the semi-classical description o f these reactions, we suggest that a treatment of (3He, xn) and (a, xn) which takes the nature of the nuclear potential explicitly into account, such as the breakup-fusion theory which Udagawa and Tamura have applied to the (14N, ct) reaction [12], or the DWBA techniques of Aarts et al. [13], would be very interesting. In particular, such an account may explain the apparent beam-energy independence o f the neutron angular distributions. We would like to thank Dr. R.J. de Meijer (KVI), Dr. M.L. Halbert (ORNL), and Dr. H. Sakai (Osaka) for helpful discussions.
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References [1] [2] [3] [4] [5] [6] [7] [8] [9]
R. Serber, Phys. Rev. 72 (1947) 1008. H. Feshbach and K. Huang, Phys. Lett. 47B (1973) 300. C.K. Gelbke et al., Phys. Lett. 70B (1977) 415. K. Siwek-Wilczynska et al., Nucl. Phys. A330 (1979) 150. J.R. Wu et al., Phys. Rev. C20 (1979) 1284. H. Sakai et al., Phys. Rev. C20 (1979) 464. H. Ejiri et al., J. Phys. Soc. Japan 49 (1980) 2103. C.A. Fields et al., Nucl. Phys. A366 (1981) 38. M.J.A. de Voight et al., to be published, private communication. [10] C.A. Fields et al., Nucl. Phys. A377 (1982) 217. [11] H. Feshbach, Intern. Symp. on Highly excited states in nuclear reactions (Osaka, 1980). [12] T. Udagawa and T. Tamura, Phys. Rev. Lett. 45 (1980) 1311. [13] E.H.L. Aarts et al., to be published; R.J. de Meijer, private communication. [14] K.G. Young et al., Phys. Rev. C23 (1981) 2479. [15] C.A. Fields et al., to be published. [16] A.S. Goldhaber, Phys. Lett. 53B (1974) 306. [17] Z. Sujkowski, Proc. XIX Intern. Winter Meeting on Nuclear Physics (Bormio, Italy, 1981). [18] C.A. Fields et al., Phys. Lett. 106B (1981) 453. [ 19] H. Sakai, private communication. [20] C. Kalbach and F.M. Mann, Phys. Rev. C23 (1981) 112. [21] R.W. West, Phys. Rev. 141 (1966) 1033. [22] A. Chevarier et al., Phys. Rev. C8 (1973) 2155. [23] R.M. Lieder et al., Phys. Scr. 24 (1981) 123.
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