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Important Formulae and Their Derivation Appendix A Important Formulae and Their Derivation Abstract This chapter collects appendices to the previous chapters. A.1 Basics of Scattering Problems Here we summarize some basics for scattering problems such as the partial wave expansion of the scattering amplitude.1 A.1.1 Partial Wave Expansion (1) The asymptotic form of the radial wave function for the partial wave (the case of a short range force; k is the incident wave number, S is the scattering matrix, i.e., the S-matrix, and δ is the phase shift) 1 −i(kr− 1 π) i(kr− 1 π) R(r) ∼ e 2 − S()e 2 (A.1) kr 1 1 ∼ sin kr − π + δ . (A.2) kr 2 (2) Partial wave expansion of the elastic scattering amplitude2 (θ is the scattering angle, and α is the index of each state) 1We consider the scattering of particles without spin. See [1] for the scattering of particles with spin. 2When one takes the partial wave sum in the case of the scattering problem involving Coulomb force, it is traditional to separate out the Rutherford scattering amplitude. © Springer Japan 2017 223 N. Takigawa and K. Washiyama, Fundamentals of Nuclear Physics, DOI 10.1007/978-4-431-55378-6 224 Appendix A: Important Formulae and Their Derivation ∞ (el) 1 f (θ) = (2 + 1) Sα,α() − 1 P(cos θ) , (A.3) 2ik =0 2iδ Sα,α() = e . (A.4) (3) Partial wave expansion of the inelastic scattering amplitude (α and β are the state indices) (inel) 1 fβ=α (θ) = (2 + 1) Sβ,α()P(cos θ) . (A.5) 2ik (4) Cross sections of the elastic, inelastic and total inelastic scattering π (el) (el) 2 σ = σ = (2 + 1)|Sα,α() − 1| , (A.6) 2 kα π (inel) (inel) 2 σ = σ () = (2 + 1)|Sβ,α()| , (A.7) β=α β=α 2 kα π (inel) (inel) 2 σ = σ = (2 + 1) 1 −|Sα,α()| , (A.8) β=α 2 β=α kα (5) Unitarity condition 2 |Sβ,α()| = 1 . (A.9) β (6) Total cross section (ReS is the real part of the S-matrix) 2π σ (total) = σ (el) + σ (inel) = (2 + 1)(1 − ReS). (A.10) 2 kα (7) Optical theorem (Im f (el) is the imaginary part of f (el)) 4π σ (total) = Im f (el)(θ = 0). (A.11) kα A.1.2 Sommerfeld–Watson Transformation For high energy scattering, it is more convenient to use the integral representation of the scattering amplitude instead of the partial wave expansion. The integral repre- sentation sometimes becomes useful for scattering at low energies as well. Starting from the integral representation of the scattering amplitude and deriving character- istic closed form expression through some suitable approximation scheme such as the stationary phase approximation or the method of steepest descent, one can often Appendix A: Important Formulae and Their Derivation 225 Fig. A.1 The contour used for the Sommerfeld–Watson transformation obtain intuitive understanding for complicated patterns appearing in the differential cross section in terms of, e.g., the interference of various scattering waves or the caustics such as the rainbow scattering and the glory scattering [2–7]. The transformation from the partial wave sum to the integral representation can be achieved by using the knowledge of the contour integral in the complex plane. We first represent the scattering amplitude as 1 iπ 1 1 f (θ) =− d e + [S(k,)− 1] P(cos θ) (A.12) 2k C sin π 2 by using the integral along the contour C in the complex plane shown in Fig. A.1. The transformation from Eq. (A.3)toEq.(A.12) is known as the Sommerfeld–Watson transformation. One can easily confirm that Eq. (A.12) reduces to Eq. (A.3) by noting that the integrand has a pole of order one at each integer value of and that the corresponding residue is given by − 1 0 0 lim = (−1) (0 : integer). (A.13) →0 sin π π A.1.3 Poisson Sum Formula TheintegralinEq.(A.12) is a contour integral in the complex plane. Let us attempt to convert it into an integral along the real axis. To that end, we first rewrite Eq. (A.12)as i eiπ 1 f (θ) =− d + [S(k,)− 1] P(cos θ) . (A.14) iπ −iπ k C e − e 2 We then remark that eiπ eiπ(+iε) = lim (A.15) iπ − −iπ ε→ + iπ(+iε) − −iπ(+iε) e e 0 e e =−e2iπ 1 + e2iπ + e4iπ +··· (A.16) 226 Appendix A: Important Formulae and Their Derivation on the contour C+ above the real axis, and that iπ iπ(−iε) e = e π − π lim π(− ε) − π(− ε) (A.17) ei − e i ε→0+ ei i − e i i = 1 + e−2iπ + e−4iπ +··· (A.18) on the contour C− below the real axis, hence eiπ eiπ d h() = + d h() iπ −iπ iπ −iπ C e − e C+ C− e − e ∞ ∞ = d e2iπmh() , (A.19) − / m=−∞ 1 2 for a function h(). We finally obtain ∞ ∞ i m 2πimλ f (k,θ)=− (−1) dλe λ [S(k,λ− 1/2) − 1] Pλ− / (cos θ) k 1 2 m=−∞ 0 (A.20) by introducing the Langer replacement + 1/2 = λ, which is referred to also as the Langer correction or the Langer modification. Equation(A.20) is called the Poisson sum formula. In the semi-classical theory for scattering [8], m represents the number of times which the incident particle circulates the target particle. The m = 0term governs the scattering at high energies or if the attractive force between the projectile and target particles is weak. However, the m ≥ 1 terms also play important roles in the scattering at low energies if a strong attractive force is involved [6]. We obtain the following Poisson sum formula for the cross section of the total inelastic scattering by applying the similar procedure to Eq. (A.8), π ∞ ∞ (inel) 2 m 2iπmλ 2 σ = (−1) dλe λ 1 −|Sα,α| (A.21) k2 α m=−∞ 0 ∞ ∞ m 2iπmkb 2 = 2π (−1) dbbe 1 −|Sα,α(b)| (A.22) 0 m=−∞ ∞ 2 ≈ 2π dbb 1 −|Sα,α(b)| (for high energy). (A.23) 0 The relation between the angular momentum and the impact parameter λ = kb has been used in transforming Eqs. (A.21)to(A.22). Appendix A: Important Formulae and Their Derivation 227 A.2 Basics of Semi-classical Theory I: WKB Approximation Here, let us first study the WKB approximation3 among various semi-classical meth- ods. The WKB approximation is the method to approximately obtain the solution of the Schrödinger equation in the limit of taking the Planck constant to be zero. Physically, it is valid when the change of the potential energy within a wave length can be ignored compared with the magnitude of the local kinetic energy. Hence, it can be said to be a short wave length approximation. In this approximation, the wave function and the phase shift as well as the scattering matrix for scattering problems are given in terms of the action integrals in classical mechanics. A.2.1 Wave Function As an example, let us consider a problem where the potential is given by Vn(r) + VC (r), the former and the latter being the nuclear and the Coulomb potentials, re- spectively, and approximately obtain ψ(r) when the radial wave function R(r) of an eigenstate of energy or the scattering problem with the orbital angular momentum is expressed as R(r) ≡ ψ(r)/r.Theψ(r) obeys the following equation: 2 d2 − + V (r) − E ψ(r) = 0 , (A.24) 2μ dr 2 V (r) = Vn(r) + VC (r) + V(r), (A.25) 2 ( + 1/2)2 2 λ2 V(r) = = . (A.26) 2μ r 2 2μ r 2 We introduced the Langer replacement ( + 1) → ( + 1/2)2 ≡ λ2 in Eq. (A.26). In the WKB approximation, we express the wave function in the form of the product of the amplitude and the phase factor, and write ψ(r) = A(r)eiS(r)/ . (A.27) We leave out the index in Eq. (A.27) and throughout this section. Making the substitution Eq. (A.27)inEq.(A.24) and treating A(r), S(r) and V (r) as quantities independent of and collecting the terms with the same powers of , we obtain 3Wenzel–Kramers–Brillion approximation. It is also called the JWKB approximation by including thenameofJefferey. 228 Appendix A: Important Formulae and Their Derivation 1 dS 2 i dA dS d2 S 1 d2 A + V (r) − E A0 − 2 + A 1 − 2 = 0 . 2μ dr 2μ dr dr dr 2 2μ dr 2 (A.28) By considering the limit of → 0, and setting the coefficients of the zeroth and the first order power of to be zero separately, we obtain 1 dS 2 + V (r) − E = 0 , (A.29) 2μ dr d dS A2 = 0 . (A.30) dr dr Equation(A.29) corresponds to the Hamilton–Jacobi equation in classical mechanics, and suggests that S is given by the action integral in classical mechanics S =± 2μ(E − V (r))dr =± k(r)dr . (A.31) Equation(A.30) denotes the conservation of the current density. Together with Eq. (A.31), it shows that the amplitude A is inversely proportional to the square root of the wave number, i.e., 1 A ∝ √ . (A.32) k(r) Finally, the wave function is given by r r 1 1 ψ(r) = A+ √ exp i k(r )dr + A− √ exp −i k(r )dr , ( ) ( ) k r r0 k r r0 (A.33) where A+ and A− are constants.
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