Jackson, Classical Electrodynamics, Section 14.8 Thomson Scattering of Radiation

Transcription

1 High Energy Cross Sections by Monte Carlo Quadrature Thomson Scattering in Electrodynamics Jackson, Classical Electrodynamics, Section 14.8 Thomson Scattering of Radiation Jackson Figures and 14.18: Differential scattering cross section of unpolarized radiation by a point charged particle initially at rest in the laboratory. The solid curve is the classical Thomson result. The PHY Computational Physics 2 1 Wednesday, April 16

2 dashed curves are the quantum-mechanical results for a spinless particle, with the numbers giving the values of ω/mc 2. For ω/mc 2 = 0.25, 1.0 the dotted curves show the results for spin 1 2 point particles (electrons). The differential scattering cross section is defined by dσ dω Energy radiated/unit time/unit solid angle = Incident energy flux/unit area/second Number of photons detected/unit time/unit solid angle = Number of photons incident/unit area/unit time The differential cross section for a monochromatic plane wave with wavevector k 0 = ke z and polarization ɛ 0 incident on a free particle of charge e and mass m is ( ) dσ e 2 2 dω = ɛ ɛ mc where k = kn, ɛ are the wavevector and polarization of the outgoing wave. The polarization vectors ɛ 1 = cos θ(e x cos φ + e y sin φ) e z sin θ, ɛ 2 = e x sin φ + e y cos φ represent polarization in and perpendicular to the scattering plane defined by the initial and final wavevectors k 0, n. For plane polarized incident radiation and summing over final polarizations { ɛ ɛ 0 2 cos 2 θ cos 2 φ + sin 2 φ for ɛ 0 = e x = ɛ 1,2 cos 2 θ sin 2 φ + cos 2 φ for ɛ 0 = e y PHY Computational Physics 2 2 Wednesday, April 16

3 which give the unpolarized differential and total cross sections dσ dω = 1 2 ( e 2 mc 2 ) 2 (1 + cos 2 θ), σ T = 8π 3 ( e 2 mc 2 ) 2. The Compton Scattering Cross Sections The Tompson cross sections are valid only at low frequencies ω/c when the momentum of the incident photon is much smaller than mc. In this limit the charged particle remains at rest and the energy of the photon does not change. When k mc, the charged particle will recoil and the photon will change in magnitude and direction, giving rise to Compton scattering. The Compton formula for the change in frequency of the photon is ω ω = 1 + ω mc2(1 cos θ) where θ is scattering angle in the rest frame of the particle before the collision. To derive this result, let k µ, k µ be the 4-momenta of the photon before and after the scattering, and p µ, p mu the initial and final 4-momenta of the target particle. In the rest frame of the target particle (laboratory frame) p µ = (mc, 0, 0, 0), k µ = (1, 0, 0, 1) ω c, k µ = (1, sin θ cos φ, sin θ sin φ, cos θ) ω c PHY Computational Physics 2 3 Wednesday, April 16

4 Using conservation of 4-momentum p p = m 2 c 2 = (p + k k ) (p + k k) = p 2 + 2p (k k ) + (k k ) 2 = m 2 c 2 + 2m (ω ω ) 2 2 ωω (1 cos θ) The cross section is ( ) dσ e 2 2 ( ) ω 2 dω = ɛ ɛ mc ω Compton derived these formulas simply by using relativistic kinematics and conservation of energy and momentum. They are valid for scattering of a photon from a spinless charged particle. The Klein-Nishina Formula in Quantum Electrodynamics The quantum field theory which describes the interaction of charged electrons and positrons with photons is Quantum Electrodynamics or QED. Relativistic quantum field theory predicts that virtual electron-positron pairs will modify scattering cross section. In QED, these corrections are computed as a perturbation series in the the Fine-structure constant α = e 2 / c 1/ The cross section for Compton scattering in to lowest order in α is determined by the two Feynman diagrams from Section 5.5 of Peskin and Schroeder PHY Computational Physics 2 4 Wednesday, April 16

5 The probability amplitude for the process is given in Eq. (5.74) [ im = ie 2 ɛ µ(k )ɛ ν (k)ū(p γµ kγ ν + 2γ µ p ν ) + γν kγ µ + 2γ ν p µ ] u(p), 2p k 2p k where u(p) is a 4-component positive energy spinor wavefunction solution of the Dirac equation and γ µ are Dirac gamma matrices. This amplitude defines the scattering of a photon from an electron for arbitrary photon polarizations and electron spin. To compare with the Compton cross section for spinless particles, the amplitude must be squared to obtain the scattering probability, which is then averaged over the two initial spin states of the electron and summed over the two final spin states. This is a straightforward calculation done in detail in Peskin and Schroeder. The final result in the rest frame of the initial electron is (Jackson footnote on page 697) ( ) dσ e 2 2 ( ) ω 2 [ ɛ dω = ɛ mc (ω ω ) 2 ] {1 + (ɛ ɛ) (ɛ ω 4ωω 0 ɛ 0)}. The second dot product can be simplified (ɛ ɛ) (ɛ 0 ɛ 0) = (ɛ ɛ 0 )(ɛ ɛ 0) (ɛ ɛ 0)(ɛ ɛ 0 ) = ɛ ɛ 0 2 ɛ ɛ 0 2 PHY Computational Physics 2 5 Wednesday, April 16

6 It vanishes for pure linear or circular polarization but can contribute for elliptical polarization. This is the Klein-Nishina formula for Compton scattering. Numerical Evaluation of the Klein-Nishina Formula To compare this theoretical prediction with experimental measurements in the laboratory frame we need to express the result in Barns. The Classical electron radius is defined to be r e = e2 mc 2 = cm, r 2 e = b 79.4 mb. Using dω = sin θ dθ dφ the following dimensionless cross sections can be defined: +1 2π 1 σ re 2 T = d(cos θ) dφ 1 dσ 1 0 re 2 dω 1 dσ 2π re 2 dθ = sin θ dφ 1 dσ 0 re 2 dω ( ) 1 dσ ω 2 [ ɛ dω = ɛ (ω ω ) 2 ( 1 + ɛ ɛ ω 4ωω 0 2 ɛ ɛ 0 2) ] r 2 e PHY Computational Physics 2 6 Wednesday, April 16

7 Polarization Vectors Polarization vectors are defined in Jackson: Figure 7.2 on the left shows the complex electric field vector for linear and Figure 7.3 on the right shows the field for circular polarization. The most general polarization is a linear superposition of two independent linear or helicity eigenstates E(x, t) = (ɛ 1 E 1 + ɛ 2 E 2 )e ik x iωt, E(x, t) = (ɛ + E + + ɛ E )e ik x iωt, ɛ ± = 1 2 (ɛ 1 ± iɛ 2 ), where ɛ + represents photons with positive helicity (left-circular polarization in optics) and ɛ represents negative helicity (right-circular polarization). PHY Computational Physics 2 7 Wednesday, April 16

8 Specifying Initial and Final State Polarization The Klein-Nishina formulas given above have been averaged over initial electron spin states and summed over final electron spin states. The photon polarizations ɛ 0, ɛ can be specified arbitrarily. The incident photon has wavevector ˆk 0 = e z in the positive direction and is scattered in the direction ˆk = n = (sin θ cos φ, sin θ sin φ, cos θ). The simplest and physically most meaningful way to specify high energy photon polarizations is to use the helicity basis. The incident polarization is specified by two complex numbers (E 0+, E 0 ) and the scattered state polarization by (E +, E ). Linear polarizations are more natural at lower frequencies. To evaluate the scalar products of the polarization vectors note that the unit vectors perpendicular and parallel to the scattering plane are ɛ 2 = ˆn k 0 ˆn k 0 = ( sin φ, cos φ, 0), ɛ 1 = ɛ 2 n ɛ 2 n The polarization vectors are ɛ 0 = (E 10, E 20, 0) and = (cos θ cos φ, cos θ sin φ, sin θ). ɛ = E 1 ɛ 1 + E 2 ɛ 2 = (E 1 cos θ cos φ E 2 sin φ, E 1 cos θ sin φ + E 2 cos φ, E 1 sin θ), which can be expressed in terms photon helicity components E 1 = 1 2 (E + + E ), E 2 = i 2 (E + E ). PHY Computational Physics 2 8 Wednesday, April 16

9 C++ Code to Evaluate the Cross Sections #include <cmath> #include <complex> #include <cstdlib> #include <iostream> using namespace std; const double pi = 4 * atan(1.0); #include "../tools/quadrature.hpp" compton.cpp double omega; complex<double> ep_10; complex<double> ep_20; complex<double> ep_1; complex<double> ep_2; // incident photon angular frequency // x-component of incident photon polarization // y-component // in-plane scattered photon polarization // perpendicular component double f(const Vector& x, const double wgt) { PHY Computational Physics 2 9 Wednesday, April 16

10 double theta = x[0]; double phi = x[1]; double omega_prime = omega / (1 + omega * (1 - cos(theta))); // compute polarization dot products complex<double> ep_star_dot_ep0 = conj(ep_1 * cos(theta) * cos(phi) - ep_2 * sin(phi)) * ep_10 + conj(ep_1 * cos(theta) * sin(phi) + ep_2 * cos(phi)) * ep_20; complex<double> ep_dot_ep0 = (ep_1 * cos(theta) * cos(phi) - ep_2 * sin(phi)) * ep_10 + (ep_1 * cos(theta) * sin(phi) + ep_2 * cos(phi)) * ep_20; } return pow(omega_prime / omega, 2.0) * (norm(ep_star_dot_ep0) + pow(omega - omega_prime, 2.0) / (4 * omega * omega_prime) * (1 + norm(ep_star_dot_ep0) - norm(ep_dot_ep0))); int main() { cout << " Relativistic Compton Scattering Cross Sections using VEGAS\n"; PHY Computational Physics 2 10 Wednesday, April 16

11 double tgral, sd, chi2a; Vector regn(4); regn[0] = 0.0; regn[2] = pi; // 0 < theta < pi regn[1] = 0.0; regn[3] = 2 * pi; // 0 < phi < 2 pi // set global parameters omega = ; // incident angular frequency cout << " omega = " << omega << endl; // define and normalize polarization vectors ep_10 = 1.0; ep_20 = 0.0; ep_1 = 1.0; ep_2 = 0.0; double n0 = sqrt(norm(ep_10) + norm(ep_20)), n = sqrt(norm(ep_1) + norm(ep_2)); ep_10 /= n0; ep_20 /= n0; ep_1 /= n; ep_2 /= n; int init = 0; // initialize grid PHY Computational Physics 2 11 Wednesday, April 16

12 int ncall = 50000; int itmx = 10; int nprn = 0; // number of function evaluations // number of iterations // print some info after each iteration } vegas(regn, f, init, ncall, itmx, nprn, tgral, sd, chi2a); cout << " sigma_tot = " << tgral << " +/- " << sd << endl; PHY Computational Physics 2 12 Wednesday, April 16