CLASSICAL AND QUANTUM RAINBOW CHANNELING OF CHARGED PARTICLES IN VERY THIN SILICON CRYSTALS AND CARBON NANOTUBES
S. Petrović, M.Ćosić and N.Nešković
Laboratory of Physics, Vinča Institute of Nuclear Sciences, University of Belgrade, P. O. Box 522, 11001 Belgrade, Serbia, e-mail:
[email protected]
In this work the rainbow channeling effects are presented in the following cases: (i) the channeling of 2 MeV protons through the 55 nm thick straight and tilted (001) silicon crystals – the classical angular rainbows [1], (ii) the channeling of 68 MeV protons and 481 nm thick (001) silicon crystal – the classical spatial rainbows [2] and (iii) the channeling of 1 and 10 MeV positrons through 200 and 560 nm long (11, 9) single-wall carbon nanotubes (SWCNTs), respectively – the quantum angular rainbows [3]. The ion channeling is ion’s motion through the axial or planar crystal channels [4-6]. It occurs when the incident ion has the small angle with respect to major crystallographic directions or planes and is explained by the series of ion's correlated collisions with the atoms of the strings defining the axial or planar crystal channels. When the incident ion beam is tilted away from a major crystallographic direction, the angular distribution of channeled ions exhibits a ring-like shape. This effect, named the doughnut effect, has been studied by many researchers [7-9]. Recently, its origin has been revealed [1]. The rainbow, which is a consequence of sunlight scattering off water droplets, is one of the most prominent physical phenomena widely known since the ancient times. If the process of sunlight scattering is treated via the geometric optics, the rainbow effect is characterized by the singular intensity of light at the rainbow angle [10, 11]. It has been established that the rainbows also occur in the particle scattering. For example, they occur in atom or electron scattering from crystal surfaces, nucleus-nucleus scattering and atom or ion collisions with atoms or molecules. Nešković [12] showed that the rainbow effect occurred in the axial channeling of high energy ions through a very thin crystal. The approach was via the ionmolecule scattering theory assuming the momentum approximation. Petrović et al. [13] formulated the theory of crystal rainbows, which generalized the model introduced by Nešković [14]. This theory is based on analysis of the singularities of the mapping of the impact parameter plane to the transmission angle plane determine by the channeling process:
θ x = θ x ( x0 , y0 ) and θ y = θ y ( x0 , y0 ) ,
(1)
where x0 and y0 are the transverse components of initial ion position vector, i.e., the components of its impact parameter vector, θ x and θ y are the components of final ion channeling angle, i.e., the components of its transmission angle. It should be noted that the mapping (1) depends on the ion’s energy, crystal channel, its thickness, as well as, on the crystal tilt angle ϕ . Since the components of ion channeling angle are small (smaller than the critical angle for channeling) the ion differential transmission cross section is given by:
σ ( x0 , y0 ) =
1 , Jθ ( x0 , y0 )
(2)
where Jθ ( x0 , y0 ) = ∂ x0θ x ∂ y0θ y − ∂ y0θ x ∂ x0θ y is the Jacobian of the mapping (1). Thus, equation
Jθ ( x0 , y0 ) = 0
(3)
gives the rainbow lines in the impact parameter plane. The images of these lines determined by functions θ x ( x0 , y0 ) and θ y ( x0 , y0 ) are the rainbow lines in the transmission angle (TA) plane. Figure 1 show a sequence of the experimental angular distributions and the corresponding rainbow lines in TA plane for 2 MeV protons transmitted through a 55 nm thick (001) silicon crystal obtained by tilting the crystal away from the [001] direction for the tilt angles ϕ equal to 0.000, 0.060, 0.120 and 0.150. It is clear that the rainbows patterns are the ″skeletons″ of the angular distributions and that the doughnut effect corresponds to the dark side of the circular rainbow. The rainbow patterns presented in Figure 1 were calculated assuming the ZBL protonsilicon interaction potential [14]. In order to investigate dependence of the rainbow pattern on the interatomic interaction potential, we also calculated the rainbows assuming the Molière’s proton-silicon interaction potential [15]. Figure 2 shows the rainbow patterns assuming the ZBL – red lines and Molière’s -
a)
b)
c)
d)
Figure 1. The experimental angular distributions of 2 MeV protons transmitted through a 55 nm thick (001) silicon crystal for the tilt angle equal to a) 0.000, b) 0.060, c) 0.120 and d) 0.150. The red lines are the rainbow lines in the transmission angle plane. blue lines proton-silicon interaction potentials. Comparing Figures 1 and 2(b) one can conclude that the inner blue rainbow circular line (Molière’s potential) and the outer red rainbow lines (ZBL potential), are in good agreement with the experiment. This opens possibility for determination of an accurate proton-silicon interatomic potential which is ″in between″ the ZBL and Molière’s potentials by using the crystal rainbow morphological method [16]. In analogy with the expression (3) one can define spatial rainbow lines in the impact parameter plane via expression: J ρ ( x0 , y0 ) = 0, where J ρ ( x0 , y0 ) = ∂ x0 x∂ y0 y − ∂ y0 x∂ x0 y . In the case of the parabolic 2D ion-atom interaction potential (LHO), J ρ ( x0 , y0 ) = cos2 (ωt ) , where
a)
b)
Figure 2. The rainbow patterns in the a) impact and b) transmission angle planes for 2 MeV protons transmitted through a 55 nm thick (001) silicon crystal for ϕ = 0.12° calculated by the ZBL interaction potential – red lines – and the Molière interaction potential – blue lines.
ω is the angular frequency of the ion’s oscillatory motion. Therefore, the spatial rainbows are the rainbow points corresponding to t = T/4 + n T/2, n = 0, 1, 2 …, where T is period of the ion’s oscillatory motion. These points are called first, second, third etc. superfocusing points. This is illustrated in Figure 3 where the first superfocusing point is presented. In the case of (001) silicon crystal and 2 MeV protons, assuming the Molière’s proton-silicon interaction potential, the evolution of the spatial rainbow lines around the superfocusing point is presented in Figure 4 [2]. It should be noted that this evolution is independent on the ion’s
Figure 3. Motion of the channeled ion assuming parabolic (LHO) ion-atom interaction potential. The first superfocusing point is shown.
Figure 4. The evolution of the rainbow line in the transverse position plane in the vicinity of the superfocusing point for 2 MeV protons and the (001) silicon crystal.
energy (up to the scaling factor). We have shown that for 68 MeV protons and the (001) silicon crystal it is possible to focus the beam within the region of the radius well below the Bohr radius [17]. This provides the theoretical basis for development of a measurement technique with the picometer resolution – the rainbow subatomic microscopy. Quantum rainbow channeling has been studied for 1 and 10 MeV positrons transmitted thought 200 and 560 nm long (11, 9) SWCNTs, respectively [3]. a)
The quantum angular distributions were determined by solving (numerically) c) b)
Figure 5. a) Rainbow diagrams of 1 MeV positrons transmitted through a 200 nm long (11, 9) SWCN (full line) and of 10 MeV positrons transmitted through a 560 nm long (11, 9) SWCN (dashed line), b) the quantum (full line) the corresponding classical angular distribution (dashed line) for 1 MeV positrons transmitted through a 200 nm long (11, 9) SWCN, c) the quantum (full line) the corresponding classical angular distribution (dashed line) of 10 MeV positrons transmitted through a 560 nm long (11, 9) SWCN. the corresponding time-dependent Schrödinger equation, assuming that the positron beam can be represented via an ensemble of non-interacting Gaussian wave packets. In this approach divergence of the initial beam is taken into account implicitly. Figure 5(a) shows the classical rainbow diagram of 1 and 10 MeV positrons for 200 and 560 nm long (11, 9) SWCNs, respectively. It determines the positions of the classical rainbow angles 1a,1 and 1a,10, respectively. The quantum and the corresponding classical angular distributions for 1 MeV positrons are shown in Figure 5(b). The strongest pair of the quantum maxima being close to the pair of classical primary rainbow maxima, belongs to the quantum primary rainbow and the two weaker pairs of maxima to the first and second primary supernumerary rainbows. Figure 5(c) shows the quantum and the corresponding classical angular distributions for 10 MeV positrons. The three pairs of maxima lie closer to each other and to the classical primary rainbow maxima than the three pairs of maxima appearing for 1
MeV positrons. The strongest pair of maxima, being close to the pair of classical primary rainbow maxima belongs to the quantum primary rainbow and the two weaker pairs of maxima to the supernumerary primary rainbows. Therefore, one can anticipate that for much higher positron energies the supernumerary rainbows will be less pronounced, while the quantum primary rainbow will practically coincide with the corresponding classical rainbow. This represents an interesting example of quantum-classical smooth transition at the level of ensembles of the particles (the positron angular distributions).
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