## Optical Model

In nuclear physics, the Optical Model means, we are treating the scattering problem is like optical wave problem. due to the incident beam can be treated as a wave-function. and this wave will be scattered by the target.

when the beam is far away from the target, the wave function of the incident beam should satisfy the Schrödinger equation in free space :

$\left( \frac {\hbar^2 } {2m} \nabla^2 + V(r) \right) \psi( \vec{r} ) = E \psi ( \vec{r} )$

and the plane wave solution is

$\psi ( \vec{r} ) \sim Exp ( \pm i \vec{k} \cdot \vec {r} )$

after the scattering, there will be some spherical wave come out. the spherical wave should also satisfy the free-space Schrödinger equation.

$\psi( \vec{r} ) \sim Y(\theta, \phi) \frac {Exp( \pm i \vec{k} \cdot \vec{r} ) }{r}$

Thus, the process of scattering can be think in this way:

$Exp( \pm i k z ) \rightarrow Exp( i k z ) + f ( \theta ) \frac { Exp ( i k r ) } {r}$

where f(θ) is a combination of spherical wave.

one consequence of using Optical Model is, we use complex potential to describe the nuclear potential terms in quantum mechanics.

when using a complex potential, the flux of the incident beam wave function can be non-zero. meanings that the particles in the beam are being absorbed or emitted. This corresponding to the inelastic scattering.

The reason for the “OPTICAL” is come form the permittivity and permeability of the EM field. for metallic matter, their permittivity or permeability may have a imaginary part. and this imaginary part corresponding to the absorption of the light. so, nuclear physics borrow the same idea.

the flux is defined as:

$J = \frac { \hbar }{ 2 i m} ( \psi^*(r) \nabla \psi(r) - \psi(r) \nabla \psi^* (r) )$

and the gradient of the flux, which is the absorption (sink) or emission ( source ) is:

$\nabla J = \frac {\hbar }{ 2 i m }( \psi^* \nabla^2 \psi - \psi \nabla^2 \psi^* )$

The Schrödinger equation gives the equation for the wave function:

$\nabla^2 \psi(r) = \frac { 2m} {\hbar^2} ( E - V(r)) \psi(r)$

when sub the Schrödinger equation in to the gradient of flux, we have:

$\nabla J = \frac {1} {i \hbar } ( V(r) - V^*(r) ) | \psi |^2 = \frac { 2} {\hbar } Im ( V) | \psi |^2$

we can see, if the source and the sink depend on the complex part of the potential. if the imaginary part is zero, the gradient of the flux is zero, and the wave function of the beam is conserved.

## On NMR signal

The NMR signal is obtained by the coil, which also generate the Rabi field or a radio frequency to flip the spin.

the origin of the NMR signal is the transverse magnetization. for spin-½ system. the transverse component of the magnetization is:

$M_T = ( M_x, M_y ) = A ( cos(\omega_0 t), sin(\omega_0 t))$

where $A$ is the amplitude and $\omega_0$ is the Larmor frquency. for consistency and cross reference in this blog, i keep the 0 with the $\omega$.

the magnetization is proportional a changing magnetic field. a changing magnetic field will induce an e.m.f on a coil. if the coil is perpendicular to an oscillating magnetic field a maximum e.m.f will be obtained. however, since the magnetization is rotating, the coil can be point at any direction to give the same e.m.f. . without lost of generality, the coil will define the x-axis of the system.

$B = B_{NMR} ( cos (\omega_0 t ), sin ( \omega_0 t) )$

and the Maxwell’s equation:

$\nabla \times E = \frac { d}{dt} B$

$\nabla \times E = B_{NMR} \omega_0 ( - sin (\omega_0 t), cos(\omega_0 t))$

we can see that the amplitude of the E field in the coil, which is the NMR signal strength, is depending on the Larmor frequency $\omega_0$. That explained why NMR always looking for strong magnetic field, now can go to 22 Tesla ( earth magnetic field is just $5 \times 10^{-5}$ Tesla ), a higher magnetic field strength, the higher Larmor frequency, and a stronger signal.

Moreover, the magnetic field produced by the sample is proportional to number of NMR center, the polarization and a factor on how the spin ensemble to combine to be a giant single field. and also, the change of the flux of the NMR coil is depends on how the area was integrated. These all factor are not just related to the NMR coil but also on the particular sample.

## type of accelarator I (Linac)

The machine used in nuclear physics is call Accelerator. Because it accelerate, speed up the particle.

There are basically 2 types, one is called Linear Accelerator (Linac) , another type is called Circular Accelerator.

in fact, there are many other type of accelerator, as long as they can accelerate particle, by definition, it can be called as accelerator. for example, The vacuum tube in old day TV is an electron accelerator! However, most other type of accelerators can only speed up particle at low speed, not comparable to speed of light. So, in modern nuclear physics, we don`t use them. Nevertheless, the mechanism of them may be reviewed and other type of accelerator may be invented in future, who know!

so, far, all Accelerator can only handle charged particle.

Linac

The particle being accelerated in Linac is moving in a straight line. that is why the name Linear. Linac constitutes of  many sectors, each sector is a mini accelerator, which speed up the particle by adding energy into it.

The simplest type of Linac is 2 parallel plate with a hole at the middle, and has the electric potential different V. when an electron passed through it, it will gain eV of energy and then speed up. so, if there is n sector, and each sector are identical, the final energy is neV. This LINEAR behavior also address the name.

Working Principle

Now, imagine you have many plates, when the particle passed plate 1, a voltage applied on plate 2 to speed up the particle. after the particle passed plate 2, you have to turn off the voltage and apply the voltage on plate 3, and the process go on. thus, the voltage on each plate is oscillating, which is the working frequency of Linac. Moreover, when the particle speed up, the time for it with in each section will be smaller and smaller, thus, the working frequency has to be increase and matching the particle speed. the other way around it, building each sector in different length, but this method will set a definite frequency of the Linac and the output energy.

The advantage of Linac is that :

1. it can produced continuous beam
2. less energy lost during acceleration. where all charged particle will radiate energy while accelerating.

The disadvantage is :

1. limited length, so the Maximum energy is limited
2. The working frequency has to be tuned so accurate. it is harder to operate a Linac.

## Objects of Interest

Nuclear Physics is a study on nuclear matter which is fundamental building block of the world.

electron, proton , neutron, deuteron, tritium, etc… those are objects in nuclear, we call them “particle”. the most simple particle in here is electron, proton and neutron.

The different between nuclear and atom is:

Nuclear core (sit in the center) + Electrons (moving around) = Atom

the mass of atom is almost contributed by nuclear. This is because the mass of proton is about 1830 times bigger than electron, and neutron’s mass is only heavier a bit then proton.

There are many properties contained in each particle. there are electric charge, mass, spin, kinetic energy, etc… and the objective of nuclear physics is understand all these properties and how these properties affect the inter-reaction among them. for example, how a proton and neutron form a nuclear core in deuteron? how they attract each other?

these properties, some may say, are ASSIGNED to the particles. Basically, we can only measure the effect or the result from each interaction. we think, there is a FORCE to make particles able to INTER-ACT with each others. simple to say, when an electron meets another electron, they affect each other by ELECTROMAGNETIC force. but when consider an electron meet a neutron, they don’t interact by electromagnetic(EM) force. in order to distinguish these. we assign an electric CHARGE to electron, and no charge for neutron.

so, basically, Nuclear Physics is study the PROPERTIES of particles and the INTERACTION among them.

There are 3 major forces/interactions, Weak force, EM force and Strong force. Until this moment, we only know the weak and EM force and not fully understand the strong. We neglect the gravity in here, because it is very weak and do no observable effect.

## Range

Strong 10,000 10-15m
EM 1000 long
Weak 1 10-18m