February 26, 2016
Basic, nuclear, Scattering, theory
Bessel function, Cross section, Legendre polynomial, Partial wave, phase shift, radio, Runge-Kutta, S-matrix, scattering, scattering amplitude
In a scattering problem, the main objective is solving the Schrödinger equation
where H is the total Hamiltonian of the scattering system in the center of momentum, K is the kinetic energy and V is the potential energy. We seek for a solution ,
The solution can be decomposed
The solution of can be solve by Runge-Kutta method on the pdf
where and .
For , the solution of is
where and is the Riccati-Bessel function. The free wave function is
where is the Legendre polynomial.
Note that, if we have Coulomb potential, we need to use the Coulomb wave instead of free wave, because the range of coulomb force is infinity.
For , the solution of can be found by Runge-Kutta method, where R is a sufficiency large that the potential is effectively equal to 0. The solution of is shifted
where is the scattering matrix element, it is obtained by solving the boundary condition at . The scattered wave function is
put the scattered wave function and the free wave function back to the seeking solution, we have the
and the differential cross section
In this very brief introduction, we can see
- How the scattering matrix is obtained
- How the scattering amplitude relates to the scattering matrix
But what is scattering matrix? Although the page did not explained very well, especially how to use it.
December 23, 2010
Basic, experimental, Porperties, Scattering, theory
amplitude, Coulomb, Cross section, density, differential, electron, energy, example, fourier transform, intensity, momentum, potential, proton, scattering, scattering amplitude, spectrum, total, yield
In nuclear physics, cross section is a raw data from experiment. Or more precisely differential cross section, which is some angle of the cross section, coz we cannot measure every scatter angle and the differential cross section gives us more detail on how the scattering going on.
The differential cross section (d.s.c.) is the square of the scattering amplitude of the scatter spherical wave, which is the Fourier transform of the density.
Where the angle θ come from the momentum change. So, sometime we will see the graph is plotted against momentum change instead of angle.
By measuring the yield of different angle. Yield is the intensity of scattered particle. We can plot a graph of the Form factor, and then find out the density of the nuclear or particle.
However, the density is not in usual meaning, it depends on what kind of particle we are using as detector. For example, if we use electron, which is carry elected charge, than it can feel the coulomb potential by the proton and it reflected on the “density”, so we can think it is kind of charge density.
Another cross section is the total cross section, which is sum over the d.s.c. in all angle. Thus, the plot always is against energy. This plot give us the spectrum of the particle, like excitation energy, different energy levels.