# br Conclusions The results of

Conclusions
The results of the analysis lead us to conclude that the EDD acoustic spectra in the studied frequency and temperature ranges are better described by the thermodynamic theory of acoustic relaxation rather than by the non-local diffusion theory. It has been demonstrated that within the experimental error the EDD acoustic spectra in the studied frequency and temperature ranges consist of two simple regions of acoustic dispersion.
Additional experimental studies are needed to identify at which point the acoustic spectra should be described by the Isakovich–Chaban non-local diffusion theory instead of relaxation spectroscopy methods. In particular, EDD studies for n ≥ 7, as well as for other amphiphile liquids, are necessary, since it is known that for n ≥ 12, EDD are solid alkane-like compounds even at room temperatures.
These kinds of investigations may be useful for the theoretical description and laboratory studies of micelle formation processes without drawing hypothetical analogies from physical chemistry [20]. Besides, the results of calculating the relaxation and thermodynamic parameters may be used for developing combined technologies of increasing oil recovery by means of surfactant solutions and various physical fields and factors [23].

Introduction
Channeling is an effect of propagation of charged relativistic projectiles in a crystalline medium along crystal planes and axes [1]. Motion of projectiles in the CM-272 can be characterized by the average length of staying in a channel, the fraction of particles that are captured in a channel and the spectrum of radiation emitted by the particles.
The concept of a crystalline undulator (CU) as a source of undulator-like electromagnetic radiation in the high energy range up to the MeV region was formulated in Ref. [2] and further studied in [3,4]. In crystalline undulators the projectile particles follow periodically bent channels and emit undulator radiation in addition to channeling radiation characteristic for the case of channeling in linearly oriented crystals.
In recent years several experiments were performed [5–7] to detect the radiation from electron- or positron-based CU units. The most recent works in this field are now in progress at the Mainz Microtron (Germany) facility for the 195–855 MeV electrons, and at the SLAC facility (Stanford Linear Accelerator Center, USA) with 10–20 GeV electron beams. In these experiments planar channeling of electrons is studied in bent [8] and periodically bent crystals.
In order to simulate processes of planar and axial channeling a new module was developed for MBN Explorer [9].
MBN Explorer is a versatile software package for simulating molecular systems of various degrees of complexity. MBN Explorer utilizes a broad variety of interatomic potentials to describe different molecular systems, such as atomic clusters, fullerenes, nanotubes, polypeptides, proteins, DNA, composite systems, nanofractals and many more.
Computer simulations of planar channeling using the MBN Explorer software were performed in the previous works [10–14]. The effect of axial channeling in straight crystals was studied by different groups both experimentally [15] and numerically using the averaged potential method [16–18].

Physical model
The motion of an ultra-relativistic projectile of the charge q and the mass m in an external electrostatic field with potential energy U(r) can be described with relativistic equations of motion written in the following form:

The momentum p, written in terms of velocity, reads , where γ is the Lorentz factor
with ɛ being the projectile energy.
The differential equations (1) are to be integrated for t ≥ 0 using the initial values of the coordinates (x0, y0, z0) and the velocity components (v, v, v) of the particle. To ensure an accurate numerical integration the fourth-order Runge–Kutta scheme and a time step variation algorithm were implemented.