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1922 Graphene – Synthesis, Characterization, Properties andGrAapphepnelicSyanthieosnis s where A(γ) is unknown amplitude. This integral can be evaluated numerically, and the corresponding algorithm is very simple. The rectangular or Simpson formula may be used. The basic idea of the approximation is that the total fan of classical trajectories corresponding to the discrete set of angular parameter γ ∈ [0, 2π] must stretch for the values of s as large as possible thus securing total and uniform covering of the observation point x. Then, all the components of the Gaussian beam asymptotic solution (22), that is, S0(s), S1(s), a(s), z(s), p(s) must be computed for various values of the discrete set of γ ∈ [0,2π]. This includes the coordinates s, n of the observation point with respect to every trajectory determined by γ. The parameter w is used in the construction of the complex z(s), p(s) z(s) z1(s) z2(s) p(s) = p1(s) + iw p2(s) where the real functions z1(s), p1(s), z2(s), p2(s) satisfy (21), and the initial conditions z1 (0) 1 z2 (0) 0 The parameter w determines the width of localized Gaussian beams. The thinner Gaussian beams, the more accurate approximation for solution may be obtained ((16)), ((17)). For example, for the case U(x) = 0, we obtain the equation in variations in the form p1(0) = 0 , p2(0) = 1 . z ̈ + R − 2 z = 0 , z=coss+iwsins, w>0, and the corresponding solution leads to The amplitude A(γ) can be determined by the steepest descent method (see ((16)), ((17))). In the region close to x(0), where the structure of electron classical trajectories is regular away from caustics, the approximation (23) has to coincide with the ray asymptotic solution RR I m ( Γ ) = a I m ( z ̇ ) = a s > 0 , a = E / v F . z R[cos2 s +w2sin2 s] RR i S(t,γ0)+iπ/4 −i θ−γ0 −i θ+γ0 G ( x , x ( 0 ) ) = 1 Jα(t,γ0) e 2 e 2 ( 1 + O ( h ̄ ) ) , ( 2 4 ) iθ+γ0 iθ−γ0 e 2 e 2 k e h ̄ 2h ̄ 2π where γ0 determines the trajectory connecting x(0) and x. Taking into account that asymptotically small neighbourhood of trajectories close to the trajectory γ = γ0 contribute into the integral (23) as h ̄ → 0, and using inside this neighbourhood the following approximations (see (17)) S = S 0 + S 1 n + 1 p ̃ ( s ) n 2 + O ( n 3 ) , 2 z ̃ ( s ) n = z ̃(s)(γ − γ0) + O((γ − γ0)2),PDF Image | GRAPHENE SYNTHESIS CHARACTERIZATION PROPERTIES
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