ABSTRACT
An interpretation of measured data in terms of simple physical terms, as well as direct and adequate comparisons with corresponding theoretical predictions assumes that a number of important experimental conditions are fulfilled. For example, if the target is comprized of N identical particles, then it is natural to suppose that the final experimental result will be equal to the corresponding N fold data for collisions on one particle of the target. Roughly speaking, this condition in the mentioned Rutherford experiment [1] on collisions of α particles on Au has been achieved by preparing the target in the form of very thin gold slices (≈ 1 micron ≡ 1µ). A detailed inspection of a typical scattering experiment reveals that the following eight most important conditions are necessary to secure a relatively solid basis which would justify subsequent meaningful comparison with theoretical predictions: • (i) The source of the projectiles must secure that the incident beam is not
of an overly weak intensity. This is because any statistically significant measurement must detect a large number of collisional events. The incident beam must not be too strong either, or otherwise interactions among individual particles in the beam would not be small. The existence of a non-negligible effect of the latter interactions would preclude the unequivocal definition of the initial state of the projectile-target system. • (ii) The dimension of the hole on the collimating diaphragms must not
be too small in order: (a) to eliminate as much as possible the undesirable diffraction effects of particles of the incident beam on the edges of the hole and (b) to describe the middle part of the impact beam via plane waves. Clearly, due to diffraction, there would be a number of particles from the incident beam that would scatter without having any collision whatsoever on the target. This would lead to unwanted enhancement of scattering angles because of diffraction. Obviously, one of the basic purposes of collimators is to prevent scattering of the incident beam prior to its arrival to the target. Likewise, collimating holes must not be too wide in order to avoid formation of a wide incident wave front. A consequence of such a wide wave front is that some particles from the projectile beam that are too far from the target would hit the detector without undergoing any collision. • (iii) The target B must contain a large number NB of scattering centers
in order to intensify the overall collisional effects in the sense of accumulating a sufficiently large number of elementary processes. In the theory, this condition permits computation of average probabilities for scattering through the
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replacement of sums by integrals over contributions from elementary probabilities for collisions on individual scattering centers. • (iv) Detectors must be sufficiently far from the target, so that it can be
said with certainty that they are located in the asymptotic region. This would guarantee that the differential cross section dQ/dΩi is correctly related to the absolute value of the scattering amplitude f(Ωi) which is the coefficient of the outgoing spherical wave for the stationary scattering states from (1.11). Simultaneously, the distance r between the detector and the target must be much larger that the radius d0 of the collimator (r À d0 ). This would enable us to distinguish scattered particles from those reaching the detector without collisions with the target. As an example, taking e.g. that d0 ≈ 10−1cm, it would be sufficient to choose r ≈ 102cm to fulfill the condition (iv) and, of course, this is trivially feasible in the usual experimental circumstances. It is a typical empirical fact that in scattering experiments with short-range potentials, measured cross sections become independent of the distance r′ ≈ r between the interaction domain and the detector for sufficiently large distances i.e. if the measuring instrument is situated in the asymptotic spatial region of scattering. In non-stationary theory, this fact is interpreted by saying that the corresponding observables must possess their limiting values when t → ∓∞. Thus, one of the tasks of the theory is to find the appropriate asymptotic constants of motion. In the experiment, the initial information related to such constants is obtained through the analysis of the incident beam by removing the target from its place. Namely, at an arbitrarily chosen initial time t, we would first measure the probability distribution of these constants of motion (impulse, spin, isospin, etc) that characterize the incident beam without the presence of the target. Then we assert that such a determined distribution corresponds to the incident beam in the remote past (t → −∞) i.e. a long time before collision takes place in the actual experiment when the target is placed at its location1. This is a realistic assertion, since a typical collision time T0 is short (of the order of e.g. ≈ 10−15s or less). Such a small value of T0 justifies the said interpretation even if the projectile is released from its source immediately before the collision. Similarly, and in a symmetric reasoning, due to the shortness of the time duration T0 of a typical collision, every finite time t À T0 chosen as the beginning of measuring the examined observable, which is associated with the stationary scattering state, could justifiably be considered as the distant future (t → +∞). Thus, if measurements of the final information (after the collision) on the distribution of the investigated constants of motion yield approximately the same result as the corresponding data from the initial configuration, we can be reasonably sure that indeed we are talking about the asymptotic constants of motion. We then say that such constants of motion are practically the same for the remote past and the distant future (of course, having in mind the relative meaning of the time
T0). In the corresponding theoretical description, we must strictly apply the limits t → ∓∞, since there does not exist an alternative way which would guarantee that we obtained the asymptotic stationary state vectors with the outgoing/incoming spherical waves, respectively. From the given arguments, the importance of establishing symmetry between the past and the future in the scattering problem can be clearly seen. This is because such a symmetry can help in establishing a meaningful correspondence between experimentally measured and theoretically predicted quantities. It should be emphasized that what is usually considered to be the main
result of a scattering experiment is not the answer to the otherwise customary question: if we know the state Ψi(t0) of a given physical system at the initial time t0, what is the probability to find this system in a different state Ψf (t) at a later time t > t0? The appropriate modification of this question relevant to the scattering experiment is: for a given initial state Ψi(t0) established in the considered system when the target is removed at an infinite separation from the projectile source, what is the probability that the system is found in the state Ψf (t) when t→ +∞ after returning the target to its original place? • (v) The distance among the particles of the target must be much larger
than the range of the perturbation interactions. This range is the characteristic radius after which the potential V can be ignored as e.g. in nuclear collisions. For example, if the target is in its solid state, then the typical distance r0 among the constituent particles of the target is of the order of the Bohr radius a0 i.e. r0 ≈ a0 ≈ 10−8cm. Certainly, this distance is much greater from the range R of nuclear interactions, since R is of the order of one fermi (R ≈ 10−13cm ≡ 1f). The reason for imposing the condition (v) is in attempting to eliminate a situation where the incident particle on a given scattering center senses the fields of neighboring scattering centers. Of course, here it is understood that the scattering centers in the target are distributed in a random fashion. Moreover, such centers are assumed to be distributed in such a way that, on the average, they appear uniformly and without mutual overlap. This would eliminate phase correlations among waves scattered at different scattering centers. The theory of scattering examines single events at a time, such as a collision of one particle from the incident beam with one typical scattering center from the target. A failure in a measurement to secure that the coherent phase effects from different scattering centers are approximately eliminated would invalidate any comparison between the experimental and theoretical data. • (vi) Each individual particle from the incident beam must be scattered
only on one particle of the target (binary collisions). In other words, double, triple or multiple collisions of the projectile with the target must be eliminated along the way of the incident beam through a target of a given thickness. For a given target in the solid state, this condition can be approximately achieved by preparing the target in the form of thin slices (foils). If such a solid state target is imagined as a set of parallel layers separated by the distance r0 ≈ 10−8cm, then the probability W of scattering of the projectile
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on each layer will be approximately equal to Q/(pir20). Here, Q is the total cross section and pir20 is the classical geometric cross section. To get a rough estimate for W, we can take the value r0 ≈ 10−8cm as the standard distance between the individual particles of the target and using Q < 10−25cm2 as a typical cross section measured in experiments on nuclear collisions, it follows W ≈ Q/(pir20) < 10−9. Thus, in a sufficiently thin target e.g. of thickness of the order ≈ 10−2cm, the probability of multiple scattering is about 10−3 ¿ 1. Under such circumstances, the condition (vi) can be considered as fulfilled.
