# Bounds of Integration for Random Oriented particle

• relskhan
In summary, Bounds of Integration for Random Oriented Particle refers to the process of determining the limits at which a random oriented particle can move within a given system. This is essential in understanding the behavior and movement of particles in various physical and chemical systems. By setting these bounds, researchers are able to accurately model and predict the movements and interactions of particles, which can have significant implications in fields such as materials science, biology, and engineering. The bounds of integration are typically determined through mathematical equations and simulations, taking into account factors such as size, shape, and orientation of the particles. Overall, understanding the bounds of integration for random oriented particles is crucial in gaining a deeper understanding of their behavior and potential applications.
relskhan
In the Stoner-Wohlfarth model, a uniaxial, non-interacting particle is cooled to very low temperature with no exposure to an external field. Therefore, the orientation of each particle is random, if you have a group of particles. In their paper, they integrate such that:
$$\langle \cos (\Theta )\rangle =\int_0^{\frac{\pi }{2}} \sin (\Theta ) \cos (\Theta ) \, d\Theta$$

I am having a hard time understanding why they only integrate from 0 to pi over two, instead of 0 to pi. Can anyone shine any enlightenment on this?

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A copy of the paper can be found at http://spin.nanophys.kth.se/spin/stoner-wohlfarth.pdf There's a lot of discussion around Fig. 4 where they talk about how the symmetries of the problem allow them to reproduce the solutions everywhere in parameter space from the region ##0 \leq \theta,\phi \leq \pi/2##.

relskhan
fzero said:
A copy of the paper can be found at http://spin.nanophys.kth.se/spin/stoner-wohlfarth.pdf There's a lot of discussion around Fig. 4 where they talk about how the symmetries of the problem allow them to reproduce the solutions everywhere in parameter space from the region ##0 \leq \theta,\phi \leq \pi/2##.

That helped me tremendously - thank you!

## 1. What is the significance of the bounds of integration for random oriented particles?

The bounds of integration for random oriented particles are important because they determine the range of possible orientations that a particle can have in a given system. This is crucial for understanding the behavior and properties of the particle, as well as for predicting its interactions with other particles in the system.

## 2. How are the bounds of integration for random oriented particles determined?

The bounds of integration are typically determined by the shape and size of the particle, as well as the physical conditions of the system, such as temperature and pressure. They can also be calculated using mathematical models and simulations.

## 3. Can the bounds of integration change over time?

Yes, the bounds of integration can change over time as the particle's orientation can be influenced by external forces, such as collisions with other particles or changes in the system's conditions. This is especially true for dynamic systems, where particles are constantly moving and interacting with each other.

## 4. How do the bounds of integration affect the behavior of a particle?

The bounds of integration play a crucial role in determining the behavior of a particle. For example, if the bounds are narrow, the particle will have limited freedom to move and rotate, which can affect its diffusion rate and interactions with other particles. On the other hand, wider bounds of integration can lead to more diverse orientations and potentially different behaviors.

## 5. Are the bounds of integration the same for all types of particles?

No, the bounds of integration can vary depending on the type and properties of the particle. For example, a spherical particle may have different bounds of integration compared to a rod-shaped particle. Additionally, the bounds can also be influenced by the environment and interactions with other particles in the system.

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