ЯДЕРНАЯ ФИЗИКА, 2009, том 72, № 5, с. 822-828
= ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
CHANNELING, SUPERFOCUSING, AND NUCLEAR REACTIONS
© 2009 Yu. N. Demkov*
Institute of Physics, The University of St. Petersburg, Russia Received October 14, 2008
A highly collimated beam of protons («1 MeV) entering the channel of a monocrystal film forms at a certain depth an extremely sharp (<0.005 nm) and relatively long (some monolayers of the crystal) focusing area where the increase of the flux density can reach hundreds times. Impinging atoms in this focusing area can undergo nuclear or atomic reactions with proper foreign dopants which disappear if the crystal is tilted from this position by only 10~3 rad. This effect can be called channeling superfocusing, in contrast to the ordinary flux peaking where the increase of flux density reaches only few times. Results are predicted by the exactly solvable quantum mechanical model calculations and confirmed by channeling Monte Carlo simulations accounting for several properties of the real lattice. Unfortunately, the mosaic structure of the film and statistical spread of the film and statistical spread of the axes of channels prevent the observation. Special technologies and materials choice are needed to minimize this effect.
PACS:61.85.+p, 34.20.-b, 34.70.+e, 34.80.-i
1. INTRODUCTION
A bit provocative title is chosen to attract the attention to these problems [1]. Considered together they may lead to some promising directions of research. The present development in microphysics and in microtechnology allows us to construct experimental devices on atomic scale. Channels in a monocrystal are one of such "devices". Combining macroscopic length with microscopic width they allow one to build a bridge between both regions.
Let us consider a thin monocrystal film with hundreds of monolayers and with channels perpendicular to the surface. All of them transmit charged particles (e.g., energetic protons MeV) with small energy loss and small momentum transfer across the whole film, the effective potential averaged over the direction of the channel can be calculated easily and then the deflection of the fast particle within the channel can be found. In many cases the potential of the central part of the averaged channel is cylindrically symmetric and harmonic to a good approximation. Then it produces isochronic oscillations in the plane normal to the direction of the channel. It is possible to adjust the strength of the central 2-dimensional channel paraxial potential, the velocity of the incident particles and their masses, so that they will be focused together into a sharp and relatively long focus (some tens of monolayers) on the rear side of the film and on the axis of the channel. The radius of this focus can be in principle very small, less than 10-2 nm. This looks
* E-mail: abramov472007@yandex.ru
fantastically small and is even less than the thermal vibrational amplitude of a single atom in the lattice. Such a possibility occurs because the geometrical position of the channel relative to the lattice can be defined much better than the position of a single atom in a lattice which can be estimated by the amplitude of its thermal vibrations. This is also connected with the long-range order within the lattice and with essential coupling of this order with the channel. So we have a needle-like focusing area where the flux density of particles increases hundreds and even thousand times relative to the initial one outside of the lattice! Such an unprecedented sharpness of the focusing peak allows us to call this effect the super-focusing. Previously there were experiments investigating the distribution of momenta of particles across the channel resulting in rainbow-like distributions [2], but spatial focusing was not considered, partly because it was not clear how to observe it. The above-mentioned super-focusing needs a very good collimation of the incident particles, the beam divergence has to be better than 2 x 10-4 rad. In this case many of the trajectories do not leave the paraxial harmonic region before focusing after a quarter of the transverse oscillation period T/4 occurs. Now, if this focusing occurs near the rear side of the film, we can impinge the ions on to a different sort of foreign atoms in the channels of the film near the rear surface and register their collisions by induced nuclear reactions [3] when the position of the foreign nuclei coincide with the focus. Tilting the film by such a small angle as 10-4 rad will reduce the collision probability and the nuclear reaction yield will
sharply decrease, one should keep in mind that only the paraxial part of the incident beam is focused and this is presumed to be of the order of ten percent of the whole beam (however, the other major part of the beam participates only weakly in nuclear reactions). This allows one to find the positions of impinging atoms relative to the channel and to analyze the sharpness of the focus. Further on, this arrangement can be seen as a two-dimensional periodical array of nuclear microscopes for the target nuclei with the periodicity of the focusing lattice. Eventually one can even improve the film detection efficiency further by adding to it epitaxially some number of layers with foreign atoms, with the interstitial position of these dopants just across the centers of the channels. The focused beam will coincide with these rows of atoms and the rate of nuclear reactions will increase further. Another possibility occurs if we illuminate the film from both sides by two different sorts of beams of light nuclei chosen to have a large cross section for a low-energy nuclear reaction (e.g., a deuterium—deuterium reaction [4]), so that both beams are exactly opposite to each other. The particles will then be focused in the center of the channels and near the middle of the film where the lattice is more regular than at the surface. Because both focused regions are elongated along the channel axis the overlapping could be maximized considerably. Periodical repetition of the focusing of both beams by T/2 oscillations will make the mutual reaction volume several times larger. Such a merging of the focuses within the channel in case of opposite beams makes the size of the focus and the size of the target almost equal, which is the most favorable case. An additional possibility will be the bunching of the beams in longitudinal direction by applying a saw-like electric field to the incident ions resulting in the formation of bunches which will meet just in the focusing area. The decrease of temperature of the ion source which leads to additional possibilities for the beam collimation and also cooling of the foil (improving it's structure, diminishing its thermal oscillations etc.) shall also be important for increasing the probability of nuclear reactions. However, in each case a detailed calculation and choice of proper pairs of nuclei is needed. The focusing and beam collision can even occur outside the film to prevent its destruction. Probably there exist other possible improvements of dynamic micro-regulations for beams of particles. Such dynamical manipulations with the system of particles using Liouville's theorem and transforming high concentration in the momentum space (collinearity) into the concentration in the coordinate space (focusing), which is demonstrated here, shows evidently how these theoretical facts can help to understand the possibilities in interaction of
beams and solids including, as a distant perspective, the macroscopical release of the nuclear energy.
These preliminary considerations can be supported by the solution of an exactly solvable quantum-mechanical model approximating the channel-averaged potential (independent of the channel direction z) by a 2-dimensional oscillator potential proportional to (x2 + y2)/2 for the transverse direction. For the energy of the beam considered and to separate the coordinates we can treat the z motion along the channel classically and replace z by the time t setting the velocity equal to unity. Then the Schrodinger equation for the x part of the potential will be:
1 d2
1
j—r 2dx2 2
^(x, t) = i
.d^(x, t)
dt
with a given starting wave function «(x, 0) = = (A/n)4exp(-x2/(2RQ)). Here, Rs is the initial radius of the beam area within the channel where our harmonic approximation of the potential holds. The value of Rs could be one half to one third of the channel radius under consideration. This nonstationary Schro dinger equation can be solved using a Green function approach for the harmonic oscillator [5]
G(x', x,t) = n-1/2(2n sin t)-1/2 x x exp[(i/2)(x2 + x'2) cot t - i(xx'/ sin t - n/4)]. From the general formula:
«CM) = / G<x' ...m^,
an important property of this oscillator Green function can be exploited: it depends exponentially on x and x' and is purely quadratic in the exponent. Therefore the integral can be calculated explicitly for any initial exponential wave function depending quadratically and linearly on x'. Performing these calculations we get:
= N 2(t) exp
№(x,V,t)\2 =
(x + px sin t)2 + y2
(Rs c°s t)2 + (Rmin sin t)2J
Here we multiplied «(x,t) by the corresponding «(y,t) but without the px term. This px term describes a possible tilting of the beam axis relative to the x axis of the channel leading to the appearance of the additional factor exp(ipxx) in the starting «(x, 0).
The crucial parameter defining the rate of focusing is i = Rs/Ro which is the relation between Rs, the oscillator-like size of the average potential, and R0, the size of the ground state transverse wave function. The product RsRm;n = Rq, where Rq = 1 in our units
Fig. 1. Surface of the area where the density |^(x, y,t)\ a drop to e-1/2, the particle enters from the top. On the left-hand side pt is zero (no tilting of the channel). The surface is "breathing" as a function of time (depth). The equilibrium solution of the Schrodinger equ
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