_{1}

^{*}

The appearance of time advance (due to distortion by the non-resonant background) instead of the expected time delay in the region of a compound-nucleus resonance in the center-of-mass (C-) system is known. Here at the same conditions we study cross sections and durations of the neutron-nucleus scattering in the laboratory (L-) system. Here there are a review of papers where it is shown that such time advance is a virtual paradox and in the L-system the time-advance phenomenon does not occur and only the trivial time delay is observed. At the same time the transformations from C-system into the L-system appear to be different from the standard kinematical transformations because in the C-system the motion of a compound nucleus is absent but it is present in the L-system. Here we analyze the initial wave-packet motion (after the collision origin) and the cross section in the laboratory (L-) system. Also here (as physical revelations of profound general methodic and in very good consistent accordance with the experiment) several results of the calculated cross sections for the neutron-nucleus in comparison with the experimental data in the L-system at the range of one or two overlapped compound resonances are presented. It is shown in the space-time approach that the standard cinematic transformations of cross sections from the C-system to the L-system are not valid because it is necessary to consider the center-of-mass motion in the L-system. Finally on a correct self-consistent base of the space-time description of the nuclear processes in the laboratory system with 3 particles in the final channel, the validity of the former approach is shown, which is obtained for the space-time description of the nuclear processes with 2-particle channels earlier.

The phenomenon of time advance instead of expected time delay in the C-system was found in [

Then in [_{C}(E, θ) for the elastic scattering of nucleons by spherical nuclei near an isolated resonance in the C-system can be written as

where

here

Rewriting (1) in the form

where

we obtain the following expression for the scattering duration

in case of the quasi-monochromatic particles which have very small energy spreads^{C} is

with

From (3) one can see that, if 0 < Rea < G, the quantity Dτ(E,θ) appears to be negative in the energy interval ~ Rea around the center at the energy^{+}, the interference of the resonance and the background scattering can bring as much as desired large of the advance instead of the delay! Such situation is mathematically described by the zero^{C}(E, θ) (or the correspondent T-matrix) in the lower unphysical half-plane of the complex values for energy E. We should notice that a very large advance can bring to the problem of causality violation (see, for instance the note in [

For two overlapped resonances, the scattering amplitude for an elastic reaction can be written in center-of- mass system also in form (1):

where

and already

we obtain the following expression for the total scattering duration τ^{C}(E, θ)

for the quasi-monochromatic particles which have very small energy spreads

In

resonances [

In

_{0} with a very small time duration τ_{dir}, while ^{*} from point C_{0} to point C_{1}, where it decays by the final products y + Y after traveling a distance between C_{0} and C_{1} which is equal to ~V_{C}Dτ_{res} before its decay. Here V_{C} is the compound-nucleus velocity, equal to the

center-of-mass velocity, and ^{*} motion

before its decay [_{Z} = E^{*},

For the macroscopically defined cross sections, in the case of very large macroscopic distances r_{1} (near the detector of the final particle y) with very small angular and energy resolution (_{1} and_{1} and

where

_{l} is the l-wave scattering background phase shift. Formulas (8)-(11) were obtained for a quasi-monochromatic incident beam

For the simplicity we neglect here the spin-orbital coupling and we suppose also that the absolute values of all differences r_{n}/v_{n} ? r_{p}/v_{p} (n ¹ p = 1,2) are much less than the time resolutions. Here J_{C→L} is the standard Jacobian of pure cinematic transformations from the C-system to the L-system.

We underline that Formulas (8)-(13) for the cross section σ, obtained in [

differ from the standard kinematical transformation of

system, considering only the kinematical transformations of the energies and angles from the C-system (with φ = 0) to the L-system. Such difference arises because the formal expression for σ^{C}(E,θ) as taken without consideration of the microscopic difference between the processes in

We underline that Formulas (8)-(13) for the cross section σ, obtained here, are defined by the usual macroscopic way and also consider the real microscopic motion of the compound nucleus which strongly differ them from

the standard cinematic transformation

In the first my works (for instance, in [^{C}, θ^{C}}«{E^{L}, θ^{L}}, but also taking into account of the motion of the decaying compound nucleus along the distance V_{C}Dτ_{res}, as it was shown in

the both of cross section ^{C}(E,θ) without the microscopic motion

of the decaying compound nucleus from point C_{0} till point C_{1}. It is possible to evaluate the general duration of collision in L-system, taking the superposition of the wave packets of the direct scattering and of the scattering, going on with the formation of the intermediate compound nucleus (in the correspondence with diagrams 1a and 1b, respectively) which was obtained in [

where^{*} motion velocity on the k_{1,2}

direction, t_{i} is the initial time moment, defined by the amplitude phase of the initial weight factor g_{i}, chosen for

the simplicity in the Lorentzian form _{l} (l = 1, 2), correspondent to par

ticles y and Y, respectively. Тhen, utilizing the general approach from [

(with

Thus, we obtain the trivial mean time delay in the approximation ^{C},θ^{C}}«{E^{L},θ^{L}} in L-system does not have any practical physical sense. And the reason of it is connected with that we neglect the real motion of the compound nucleus.

For the case of two overlapped resonances [

Here^{*} on the vec

tors_{i} is initial moment of time.

To calculate the time of delay in the L-system we have to use this formula:

where

So, if we will take into account the movement of the compound-nucleus the advanced time vanishes also here.

We have calculated the excitation functions σ(E) for the low-energy elastic scattering of neutrons by nuclei ^{52}Cr and ^{56}Fe and in the region of distorted isolated resonances E_{res} = 50.5444 keV and G = 1.81 keV, Е_{res} = 27.9179 keV and 0.71 keV, respectively. The values of the parameters for the amplitudes of the direct and resonance scattering separately in C-system for l = 0 (and, naturally, without the Coulomb phases) in formulas (8)-(13) were selected with the help of the standard procedure. The fitting parameter c was chosen to be equal to 0.68p or 0.948π or 0.956π or p, respectively.

The calculation results were obtained with the help of Formulas (8)-(13) in the comparison with the experimental data, given from [

If we want to take into consideration the moving of the compound nucleus, we have to use another formula for cross section:

where

We can calculate phase Ф the same way, as in the case with the one resonance.

Other values can be found this way:

In

After approximation we had such values of the parameters

Time analysis of experimental data on nuclear processes is presented here to make the following conclusions and perspectives:

1) The simple application of time analysis of quasi-monochromatic scattering of neutrons by nuclei in the region of isolated resonances, distorted by the non-resonance background, brings in C-system to the delay-ad- vance paradoxical phenomenon near a resonance in any two-particle channel. Such phenomenon of the time- transfer delay in the time advance is usually connected with a minimum in the cross section, or zero in analytic plane of scattering amplitude (apart from the resonance pole) near the positive semi-axis of kinetic energies in lower non-physical semi-plane of the Riemann surface. Here this paradox is eliminated by the thorough space- time analysis in L-system with moving C-system.

2) Moreover, it is also revealed that the standard formulas of transformations from L-system into C-system are in-suitable in the presence of two (and more) collision mechanisms―quick (direct or potential) process when the center-of-mass is practically not displaced in the collision and the delayed process when the long-li- ving compound nucleus is moving in L-system. And revealed by our group the additional change of the amplitude phase in C ® L transformations now agrees with the elimination of the paradox of passing the usual time delay in the time advance. The obtained analytic transformations of the cross section from C-system into L-sys- tem are illustrated by the calculations of excitation functions for examples of the elastic scattering of neutrons by nuclei ^{52}Cr, ^{56}Fe and ^{58}Ni near the distorted resonances in L-system.

3) The presented results of time analysis for quasi-monochromatic nucleon-nucleus scattering near the isolated resonances, distorted by the non-resonance background, can be easily generalized to the scattering nucleons by nuclei near two - three overlapped resonances.

4) Of course, new Formulas (8)-(13) and (18)-(22) can be also used for the improvement of the existing general methods of analyzing resonance nuclear data for the two-particle channels in nucleon-nucleus collisions in L-system and, moreover, can be generalized for more complex collisions.

5) Applying time analysis to elastic nucleon-nucleus with 2 - 3 overlapping compound-resonances, it is possible also to obtain the paradoxical phenomenon of transition decay in advance in C-system. But the behavior of amplitudes and durations can be certainly more complex than for an isolated resonance. Therefore the study of such cases can be more complicated than for an isolated resonance, and it has to be rather interesting and perspective.

7) It is rather interesting to apply the results of the space-time description of direct and sequential (via compound-nucleus) processes in the L-system of nuclear reactions with three particles in the final channel for concrete investigations, elaborations and calculations of many concrete nuclear collisions.

V. S. Olkhovsky, (2015) Space-Time Description of Cross Sections and Durations of Neutron-Nucleus Scattering near 1 - 2 Resonances in the C- and L-Systems. Open Access Library Journal,02,1-10. doi: 10.4236/oalib.1101586