#Klein-Nishina Formula

[truphysics]

Klein-Nishina Formula

Introduction The Klein-Nishina formula is an essential result in quantum electrodynamics that describes the angular distribution of photons scattered off a single, free electron. Named after Oskar Klein and Yoshio Nishina, who derived the formula in 1928, it takes into account the relativistic and quantum mechanical effects of the scattering process, particularly in the case of high-energy (or…

[katakbara]

Photon Travel

Hi there, It is been a long time no write. Do something is always better than nothing, doesn’t it? It feels like heavy to get the first step to move. However, lying on the bed and just think it in our mind is the really dangerous situation. Feel fresh, feel great, and make a step.

Anyway, I would like to discuss a simple monte carlo simulation on matlab. The case is a photon transport but without any other produced particle from the interaction (it will be great if we can do it in the next post) and also the photon interaction is limited to compton scatter which is the highest probability for high energy photon. 

A monte carlo became a good method for the particle transport due to the randomly distributed number which is produced. The main idea in photon transport is moving one particle photon to the interaction site and simulate the interaction. A photon that travels in the matter follows a decay function and a ray tracing algorithm. The travel formula can be written:

In this case, we assume that the interaction material is homogeneous and have a linear attenuation equal to 1. Then, we simulate the compton scatter interaction based on the following equation (Klein-Nishina)

That’s all the brief theory. From now, we continue to the matlab based.

The first code is to determine the simulation parameters. E = 6 MeV photon energy is chosen to this simulation. Eph will be the energy of photon and Ee is electron energy. The location and distance also have to initiate first. r, theta, and phi because photon spread out in the radial direction but in matlab, we do the cartesian coordinate. x, y, z, become the initial position and dx, dy, dz are the change of position.

And then we do the random number for the r position according to the first equation, also we have to initiate the second formula (compton scattering). cosTheta = 1 is described the maximum cross section/angular in this case and cosTheta = 1-2*rand() is random number method for determining the photon direction. Zwat is an atomic number of water, ro and moc2 are a classical radius of electron and mass-energy of electron, respectively.

The last section is moving the photon to the next interaction site and calculates the energy of scattered photon and electron. The electron energy is not used in this simulation due to the limit that already told. In this code, there is a rejection formula to determine theta (If a random number times maximum cross section less than or equal to corresponding cross section, the theta is accepted). Viola, the 3D distributed for 1 photon is done. The black is the photon path, the dotted blue is the initial and the cycle red one is the site of interaction. For a better accuracy, monte carlo simulation should be done on many particles and histories.

It will be great if you correct my formula or english spelling because I still have to study much more and more about these things.