1. Introduction
Spallation mass-yield curves in nuclear interactions with thin targets were systematically studied in many nuclear chemistry laboratories for decades around the world. These observed spallation mass-yield curves strictly obey wellknown concepts of “limiting fragmentation” and “factorisation” (see section 2.2) and are thus well understood within current theoretical models. This applies for nuclear reaction studies induced by ions from Ekinetic < 1 GeV and is extending up to 80 GeV 40Ar irradiations. Limited studies extend up to proton induced reactions with Ekinetic = 300 GeV (see [1] for details).
Several articles have recently appeared describing “unresolved problems” in the study of nuclear interactions in thick targets induced by relativistic ions and their secondary reaction products [1,2]. Product yield distributions in thick copper targets from irradiations with 72 GeV 40Ar (at the LBNL, Berkeley), 44 GeV 12C (at the JINR, Dubna), and 48 GeV 4He (at CERN, Geneva) [3] cannot be understood with well-established theoretical concepts, thus constituting “unresolved problems”. Moreover, exceedingly large neutron emission during the irradiation of thick copper, lead and uranium targets with high energy heavy ion beams having Ekinetic > 30 GeV have been observed in several laboratories; where an exceedingly large neutron multiplicity is also considered to be an “unresolved problem”.
Several authors [4,5] confirm the existence of experimentally observed unresolved problems, however, they reject in very clear and strong terms any attempt to interpret these unresolved problems, even with unconventional approaches. Hartmann and Brandt have recently published one such unconventional approach [6].
All attempts to characterise unresolved problems in thick-target nuclear reactions since about 1954 [7] have borne no fruit; the problem being that there are no defined combinations of ion energy, projectile mass, and target mass where these unresolved problems systematically occur.
In this paper the following approach will be introduced:
One calculates on a purely hypothetical basis the centreof-mass energy ECM per nucleon in the entrance channel of the nuclear interaction. This entrance channel is defined by the kinetic energy EP of the primary ion (projecttile) with mass AP and the target mass AT. The value of ECM /u in units of MeV is calculated as:
(1)
In thick targets experimentally observed phenomena are produced both by primary ions (primaries) up to the end of their range and in addition by secondary fragments (secondaries) making nuclear interactions in the thick target. The relative importance of nuclear reactions in thick targets due to secondaries compared to primaries increases with the thickness of the target [8]. One may correlate the value ECM/u—which might be taken as the hypothetical average excitation energy of each nucleon in the entrance channel of the reaction—with experimentally observed phenomena.
Some correlations are presented in Section 2 for increasing ECM/u. Obviously any observed correlation between ECM/u in the entrance channel and interactions of secondary fragments in thick targets will not explain the reason for unresolved problems. However, one does find a systematic dependence which allows a priori classification and selection of experiments where unresolved results are to be expected. In Section 3 we will present some considerations which may be helpful to understanding the observed order as presented in Section 2. Section 4 contains our conclusions on the subject and new experiments are suggested which may help to shed light onto this rather old and complex set of unresolved problems. In the Appendix few known and published experiments on thick targets irradiated with very highenergy ions having Ekinetic > 100 GeV will be reported.
2. Correlations between ECM/u and Unresolved Problems
Unresolved problems as discussed in detail in [1,2] are observed only in high energy nuclear interactions with thick targets. Three types of experiments which reveal unresolved problems are described in more detail below.
2.1. Production of 24Na in Two Copper Discs in Contact
The quantification of the isotope 24Na (T1/2 = 15 h) produced in a thick copper target consisting of two Cu-disks of 8 cm diameter and 1 cm thickness each in irradiations with relativistic ions requires just conventional gammaray spectrometry. Irradiations of two copper disks at various accelerators lasted only a few hours. After the irradiation, radioactive decay of 24Na was measured in order to calculate with an accuracy of about ±1% the activity ratio:
(2)
where “upstream Cu” denotes the Cu-disk which is first hit by the beam and “downstream Cu” is the following Cu disk. There may be several downstream disks in a very thick target stack.
Correlations of ECM/u with experimental R0(24Na)- values are presented in Table 1 (column 3). In [1] it was shown that R0(24Na) > 1.0 constitutes an unresolved problem, i.e. one would normally expect more production of the very far spallation product 24Na in the first disk than in the second one. This case of an unresolved problem is systematically observed for ECM/u > 192 MeV as seen from the data presented in the third column of Table 1. The third column is divided into two sub-columns in which consistent and inconsistent (= unresolved) data are listed separately. R0(24Na) was not determined in reactions of 44 GeV 12C on Pb and U.
2.2. Maximum of Spallation Product Yields in Two Copper Discs in Contact
Similar to the activity ratio for the very distant spallation product 24Na, one can determine the yield ratio for any spallation product having nuclear charge Z and mass A as:
(3)
Most product cross-sections that were measured some time after the end-of-bombardment are actually cumulative. One expects from the standard-model of nuclear reactions, based on the concept of “limiting fragmentation” and “factorisation”, that R0(A) distributions have a maximum value close to the mass of the target nucleus, i.e. close to A = 63 in a Cu target. The R0(A) distribution should then decrease continuously with decreasing product mass A, i.e. with increasing mass difference ΔA from the target mass. Detailed supporting arguments for this statement are given in [1,2], and in particular in [6].
An example of thin-target reactions induced by relativistic particles is shown in Figure 1 where mass distributions measured in thin copper targets that were irradiated with various projectiles are shown. The distributions of reaction products are characterised by the principles of “limited fragmentation” and “factorisation”. The term “limiting fragmentation” means that the shapes of spallation product mass distributions do not change between reaction systems and the term “factorisation” means that
Table 1. Correlations in nuclear interactions between ECM/u and several observables.
Table 2. Correlations in nuclear interactions between ECM/u and the observed neutron multiplicities on GAMMA-2.
the height (cross-section) of the distributions is simply dependent on the projectile mass.
For thick targets two figures shall clarify the situation of resolved vs. unresolved results: In Figure 2 the R0(A) distribution is shown for the reaction of 7.3 GeV 2H on a thick Cu target [10]. The maximum value of spallation product ratios is found around mass A = 57 from where on the distribution goes slowly down with rising ΔA, which is perfectly consistent with standard theoretical model results. On the other hand, the R0(A) distribution measured from interactions of 72 GeV 40Ar on a thick Cu target [12] shown in Figure 3 has its maximum around mass A = 51 which is far below the target mass and which cannot be reproduced by current models. In the former experiment (Figure 2) the cross-section ratio for the very distant spallation product 24Na is below unity (0.90 ± 0.05), whereas it exceeds unity (1.51 ± 0.02) in the latter experiment (Figure 3). The reaction system of Figure 2 is well resolved and in agreement with calculations, whereas the reaction system of Figure 3 is an unresolved problem.
Figure 1. Spallation mass distributions from interactions of various relativistic projectiles with a thin copper target. For details, see text.
Figure 2. R0(A) distribution for the reaction of 7.3 GeV 2H on a thick two-disks Cu target. The maximum is around mass 57 which is consistent with model calculations and it constitutes NO unresolved problem.
Figure 4 gives slightly modified R(AZ) distributions from various projectiles hitting a 20 cm thick copper target, consisting of 20 disks of 1 cm thickness each. This Figure is taken from [1] and it is discussed in detail therein. Suffice it to say that the observed R(AZ) distributions for 44 GeV 12C and 18 GeV 12C constitute “unresolved problems” whereas the distributions for 7.3 GeV 2H and 3 GeV 2H indicate no problem, as Amax is close to the target mass. Identical conclusions derived from the distribution of 24Na throughout thick targets in the same reaction systems as shown in Figure 4 are drawn in [13].