## Resonance Scattering

We will talk about the basics theory of resonance scattering or s-wave in this post.

The asymptotic behaviour of the scattered wave is (this post or this post), ignore th noramlization factor.

$\displaystyle u_l(r \rightarrow \infty) = \hat{h}^{-}(kr) - S_l \hat{h}^+(kr)$

For the s-wave,

$\displaystyle u_0(r \rightarrow \infty) = e^{-ikr} - S_0 e^{+ikr}$

The logarithmic derivative at the boundary $r=R$ is

$\displaystyle L^{II} = R \frac{u_0'(kR)}{u_0(kR)} = -ikR \frac{e^{-ikR} + S_0 e^{ikR}}{e^{-ikR} - S_0 e^{+ikR}} = -ikR \frac{1+ S_0 e^{2ikR}}{1 - S_0 e^{2ikR}}$

This must be matched with the logarithmic derivative of the inner wave,

$\displaystyle L^{I} = -ikR \frac{1+ S_0 e^{2ikR}}{1 - S_0 e^{2ikR}}$

Solve for $S_0$

$\displaystyle S_0 = e^{-2ikR} \frac{L^{I} + ik R}{L^I - ikR}$

For elastics scattering, $L^I$ is real, but in general, it can be a complex when a complex potential is used, that corresponding to an absorption, that other reaction channels are populated. Suppose

$L^I = a- i b$

$\displaystyle S_0 = e^{-2ikR} \frac{a + i(kR-b)}{a - i(kR + b) }$

Recall that the scattering amplitude

$\displaystyle f(\theta) = \frac{1}{2ik} \sum_{l} (2l+1) P_l(\cos(\theta)) (S_l -1)$

Assume only s-wave exist,

$\displaystyle f(\theta) = \frac{1}{2ik} (S_0 -1) = \frac{1}{2ik} \left( e^{-2ikR} \frac{a + i(kR-b)}{a - i(kR + b) } - 1\right)$

$\displaystyle f(\theta) =\frac{1}{2ik} \left( e^{-2ikR} \frac{a + i(kR-b)}{a - i(kR + b) } - e^{-2ikR} + e^{-2ikR} - 1\right) = f_R + f_S$

where

$\displaystyle f_S=\frac{1}{2ik} \left( e^{-2ikR} - 1\right)$

$\displaystyle f_R=\frac{e^{-2ikR}}{2ik} \left( \frac{a + i(kR-b)}{a - i(kR + b) } - 1\right) = \frac{e^{-2ikR}}{k} \left( \frac{kR}{a - i(kR + b) }\right)$

We can see that the phase shift of $f_S$ is $kR$. It is like a scattering of a hard sphere.

The term $f_R$ is respond to resonance when $a = 0$. We can imagine that $a$ is variate with the beam energy $E$. Express $a(E)$ in Taylor series, the 1st order around the resonance energy $E_R$.

$\displaystyle a(E) = \frac{da(E)}{dE} (E-E_R) = a'(E_R) (E-E_R)$

And define,

$\displaystyle \Gamma_e = - \frac{2kR}{a'(E_R)}$

$\displaystyle \Gamma_a = - \frac{2b}{a'(E_R)}$

$\displaystyle \Gamma = \Gamma_e + \Gamma_a$

where $\Gamma_e$ is scattering width, $\Gamma_a$ is absorption width, and $\Gamma$ is total width.

$\displaystyle f_R(E) \approx \frac{e^{-2ikR}}{k} \left( -\frac{\Gamma_e / 2}{(E-E_R) + i \Gamma/2 }\right)$

The cross section for s-wave

$\displaystyle \sigma = 4\pi |f_S+f_R|^2$

Near the resonance energy, $|f_r| >> |f_S|$,

$\displaystyle \sigma \approx 4\pi |f_R|^2 = \frac{\pi}{k^2} \frac{\Gamma_e^2}{(E-E_R)^2 + \Gamma^2/4 }$

At resonance energy $\displaystyle \sigma \approx \frac{4\pi}{k^2} \frac{\Gamma_e^2}{\Gamma^2 }$

For real potential, $b = 0$ and $\Gamma_e = \Gamma$.

## detail treatment on Larmor Precession and Rabi Resonance

a treatment on Larmor Precession and Rabi resonance

the pdf is a work on this topic. it goes through Larmor Precession and give example on spin-½ and spin-1 system.

then it introduce Density matrix and gives some example.

The Rabi resonance was treated by rotating frame method and using density matrix on discussion.

the last topic is on the relaxation.

the purpose of study it extensively, is the understanding on NMR.

the NMR signal is the transverse component of the magnetization.

## NMR (nuclear magnetic resonance)

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.

1. 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.
2. 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.
3. 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.
4. 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.