Calculation of three dimensional motion of particle in electron linear accelerator

 

I. Introduction

 

As a key equipment in modern scientific research and medical application, the motion state of the particles in the electron linear accelerator has an important influence on the performance and therapeutic effect of the equipment. In an electron linac, electrons are accelerated to a high energy state and move in three dimensions in an electromagnetic field. In order to understand and predict the trajectory of electron more accurately, the three-dimensional motion of particle in electron linear accelerator is calculated and analyzed in this paper.

Second, the basic principle of electron linear accelerator

 

The electron linear accelerator accelerates the electron through the microwave electric field, so that the electron obtains enough energy. In the process of acceleration, the electrons are subjected to the action of electromagnetic fields and carry out complex motions in three-dimensional space. In order to calculate the three-dimensional motion of the electron, we need to understand the basic theory of electromagnetic field and the force of the electron in the electromagnetic field.

 

3. Basic theory of electromagnetic field

 

In electron linac, electrons are mainly affected by electric and magnetic fields. The electric field is generated by the electrodes in the accelerator tube to accelerate the electrons. The magnetic field is generated by a magnetron or deflection coil to control the trajectory of the electrons. According to the Lorentz force formula, the force on an electron in an electromagnetic field can be expressed as:

(\vec{F} = q(\vec{E} + \vec{v} \times \vec{B}))

 

Where, (\vec{F}) is the force on the electron, (q) is the amount of charge on the electron, (\vec{E}) is the electric field strength, (\vec{v}) is the velocity of the electron, and (\vec{B}) is the magnetic induction strength.

 

4. Three-dimensional motion equation of electrons

 

In order to calculate the three-dimensional motion of the electron, we need to establish the motion equation of the electron. In the Cartesian coordinate system, the three-dimensional motion equation of the electron can be expressed as:

 

(\frac{d\vec{r}}{dt} = \vec{v})

 

(\frac{d\vec{v}}{dt} = \frac{q}{m}(\vec{E} + \vec{v} \times \vec{B}))

 

Where (\vec{r}) is the position vector of the electron, (t) is the time, and (m) is the mass of the electron. These two equations describe the change in the position and velocity of the electron with time, respectively.

 

V. Numerical calculation methods

 

Because the motion equation of electron in electromagnetic field is a differential equation, we can not directly solve the exact trajectory of electron motion. Therefore, we need to use numerical calculation method to approximate the solution of electron motion. The commonly used numerical calculation methods include Euler method, Runge-Kutta method and so on.

 

Using Euler's method as an example, we can make the time step (\Delta t) small enough, and then use the iterative way to gradually calculate the displacement and velocity change of the electron in each time step. The specific steps are as follows:

 

Set initial conditions: including the initial position of the electron (\vec{r}_0), initial velocity (\vec{v}_0), electric field strength (\vec{E}) and magnetic induction strength (\vec{B}).

Calculate the time step (\Delta t).

The displacement and velocity change of the electron in each time step is calculated using the Olefa iteration:

Calculate the electron acceleration in the current time step (\ vec = {a} \ frac {q} {m} (\ vec {E} + \ vec {n} \ times \ vec {B})).

Update electronic speed (\ vec {n} _ {n + 1} = \ vec} {v _n + \ vec {a} \ cdot \ Delta t).

Update the position of the electronic (\ vec = {r} _ {n + 1} \ vec \ {r} _n + vec} {v _n \ cdot \ Delta t).

Repeat Step 3 until the set calculation time is reached or other stop conditions are met.

6. Calculation results and analysis

 

By means of numerical calculation, we can obtain the three-dimensional motion trajectories of electrons in electron linac. The following is an analysis of the results:

 

Electron trajectories are affected by both electric and magnetic fields. The electric field mainly affects the acceleration process of electrons and makes them obtain energy. The magnetic field mainly affects the direction of electron movement, which makes the electron carry out complex trajectory movement in three-dimensional space.

The trajectories of electrons are affected by the initial conditions. Different initial positions and velocities lead to different trajectories of electrons. Therefore, in practical applications, we need to set the appropriate initial conditions according to the specific needs.

The trajectory of the electron is affected by the time step (\Delta t). The smaller the time step, the higher the accuracy of the calculation result, but the calculation amount will increase accordingly. Therefore, in practical applications, we need to choose the appropriate time step according to the needs of calculation accuracy and calculation amount.

By analyzing the trajectory of electrons, we can understand the acceleration process and energy change of electrons in the electron linear accelerator. This is of great significance for optimizing the performance of the device and improving the therapeutic effect.

Vii. Conclusion

 

In this paper, the calculation method of particle three-dimensional motion in electron linear accelerator is introduced. By establishing the electron motion equation in the electromagnetic field and solving the electron motion approximately by numerical calculation method, we can obtain the three-dimensional motion trajectory of the electron in the electron linear accelerator. By analyzing the trajectory of the electrons, we can understand the acceleration process and energy changes of the electrons, which can optimize the performance of the equipment and improve the treatment effect