Scattering phase shift

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for a central potential, the angular momentum is a conserved quantity. Thus, we can expand the wave function by the angular momentum wave function:

\sum a_l Y_{l , m=0} R_l(k, r)

the m=0 is because the spherical symmetry. the R is the radial part of the wave function. and a is a constant. k is the linear momentum and r is the radial distance.

for free particle, potential equal to zero,

R_l(k,r) \rightarrow J_{Bessel} (l, kr )

which is reasonable when r is infinite and the nuclear potential is very short distance. when r goes to infinity,

J_{Bessel} (l,kr) \rightarrow \frac {1}{kr} sin( k r - \frac{1}{2} l \pi )

for elastic scattering, the probability of the current density is conserved in each angular wave function, thus,

the effect of the nuclear potential can only change the phase inside the sin function:

\frac{1}{kr} sin( k r - \frac {1}{2} l \pi +\delta_l )

with further treatment, the total cross section is proportional to sin^2(\delta_l).

thus, by knowing the scattering phase shift, we can know the properties of the nuclear potential.

for more detail : check this website

Differential Cross Section

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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.

d.s.c = |f(\theta)|^2 = Fourier ( \rho (r), \Delta p , r )

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.