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.
January 27, 2011
Basic, no math
frequency, Larmor, Magnetic, NMR, polarization, precess, radio, relaxation, resonance, spin, time
NMR is a technique to detect the state of nuclear spin. a similar technique on electron spin is call ESR ( electron spin resonance)
The principle of NMR is simple.
- apply a B-field, and the spin will align with it due to interaction with surrounding and precessing along the B-field with Larmor frequency, and go to Boltzmann equilibrium. the time for the spin align with the field is call T1, longitudinal relaxation time.
- Then, we send a pule perpendicular to the B-field, it usually a radio frequency pulse. the frequency is determined by the resonance frequency, which is same as the Larmor frequency. the function of this pulse is from the B-field of it and this perpendicular B-field with perturb the spin and flip it 90 degrees.
- when the spin are rotate at 90 degrees with the static B-field, it will generate a strong enough signal around the coil. ( which is the same coil to generate the pule ) and this signal is called NMR signal.
- since the spins will be affected by its environment, and experience a slightly different precession frequency. when the time goes, they will not aligned well, some precess faster, some slower. thus, the transverse magnetization will lost and look as if it decay. the time for this is called T2, transverse relaxation time.
by analyzing the T1 and T2 and also Larmor frequency, we can known the spin, the magnetization, the structure of the sample, the chemical element, the chemical formula, and alot many others thing by different kinds of techniques.
For nuclear physics, the use of NMR is for understand the nuclear spin. for example, the polarization of the spin.